Package 'difR'

Title:	Collection of Methods to Detect Dichotomous Differential Item Functioning (DIF)
Description:	Provides a collection of standard methods to detect differential item functioning among dichotomously scored items. Methods for uniform and non-uniform DIF, based on test-score or IRT methods, for comparing two or more than two groups of respondents, are available (Magis, Beland, Tuerlinckx and De Boeck,A General Framework and an R Package for the Detection of Dichotomous Differential Item Functioning, Behavior Research Methods, 42, 2010, 847-862 <doi:10.3758/BRM.42.3.847>).
Authors:	David Magis (U Liege), Sebastien Beland (U Montreal) and Gilles Raiche (UQAM)
Maintainer:	David Magis <[email protected]>
License:	GPL (>= 2)
Version:	5.1
Built:	2024-12-11 07:18:09 UTC
Source:	CRAN

Help Index

Collection of methods to detect dichotomous differential item functioning (DIF) in psychometrics
Breslow-Day DIF statistic
Contrast matrix for computing generalized Lord's chi-squared DIF statistic
Comparison of DIF detection methods
Breslow-Day DIF method
Generalized logistic regression DIF method
Generalized Lord's chi-squared DIF method
Generalized Mantel-Haenszel DIF method
Logistic regression DIF method
General logistic regression DIF method
Lord's chi-squared DIF method
Likelihood-Ratio Test DIF method
Mantel-Haenszel DIF method
Raju's area DIF method
SIBTEST and Crossing-SIBTEST DIF method
Standardization DIF method
Transformed Item Difficulties (TID) DIF method
Comparison of DIF detection methods among multiple groups
Generalized logistic regression DIF statistic
Generalized Lord's chi-squared DIF statistic
Generalized Mantel-Haenszel DIF statistic
Item parameter estimation for DIF detection using Rasch (1PL) model
Item parameter estimation for DIF detection using 2PL model
Item parameter estimation for DIF detection using 3PL model
Item parameter estimation for DIF detection using constrained 3PL model
Item parameter estimation for DIF detection
Rescaling item parameters by equal means anchoring
Logistic regression DIF statistic
Lord's chi-square DIF statistic
Likelihood-Ratio Test DIF statistic
Mantel-Haenszel DIF statistic
Raju's area DIF statistic
Selection of one of the DIF detection methods
Selection of one of the DIF detection methods among multiple groups
SIBTEST DIF statistic
Standardization DIF statistic
Testing for DIF among subgroups with generalized logistic regression
Verbal Aggression Data Set

Collection of methods to detect dichotomous differential item functioning (DIF) in psychometrics

Description

The difR package contains several traditional methods to detect DIF in dichotomously scored items. Both uniform and non-uniform DIF effects can be detected, with methods relying upon item response models or not. Some methods deal with more than one focal group.

Methods currently available are:

Transformed Item Difficulties (TID) method (Angoff and Ford, 1973)
Mantel-Haenszel (Holland and Thayer, 1988)
Standardization (Dorans and Kullick, 1986)
Breslow-Day (Aguerri et al., 2009; Penfield, 2003)
Logistic regression (Swaminathan and Rogers, 1990)
SIBTEST (Shealy and Stout) and Crossing-SIBTEST (Chalmers, 2018; Li and Stout, 1996)
Lord's chi-square test (Lord, 1980)
Raju's area (Raju, 1990)
Likelihood-ratio test (Thissen, Steinberg and Wainer, 1988)
Generalized Mantel-Haenszel (Penfield, 2001)
Generalized logistic regression (Magis, Raiche, Beland and Gerard, 2011)
Generalized Lord's chi-square test (Kim, Cohen and Park, 1995).

The difR package is further described in Magis, Beland, Tuerlinckx and De Boeck (2010).

Details

Package:	difR
Type:	Package
Version:	5.1
Date:	2020-05-10
Depends:	R (>= 3.0.0)
Imports:	mirt, ltm, lme4, deltaPlotR
License:	GPL

Author(s)

Sebastien Beland
Collectif pour le Developpement et les Applications en Mesure et Evaluation (Cdame)
Universite du Quebec a Montreal
[email protected], http://www.cdame.uqam.ca/
David Magis
Department of Psychology, University of Liege
Research Group of Quantitative Psychology and Individual Differences, KU Leuven
[email protected], http://ppw.kuleuven.be/okp/home/
Gilles Raiche
Collectif pour le Developpement et les Applications en Mesure et Evaluation (Cdame)
Universite du Quebec a Montreal
[email protected], http://www.cdame.uqam.ca/

References

Aguerri, M.E., Galibert, M.S., Attorresi, H.F. and Maranon, P.P. (2009). Erroneous detection of nonuniform DIF using the Breslow-Day test in a short test. Quality and Quantity, 43, 35-44. doi:10.1007/s11135-007-9130-2

Angoff, W. H., and Ford, S. F. (1973). Item-race interaction on a test of scholastic aptitude. Journal of Educational Measurement, 2, 95-106. doi:10.1111/j.1745-3984.1973.tb00787.x

Chalmers, R. P. (2018). Improving the Crossing-SIBTEST statistic for detecting non-uniform DIF. Psychometrika, 83(2), 376–386. doi:10.1007/s11336-017-9583-8

Dorans, N. J. and Kulick, E. (1986). Demonstrating the utility of the standardization approach to assessing unexpected differential item performance on the Scholastic Aptitude Test. Journal of Educational Measurement, 23, 355-368. doi:10.1111/j.1745-3984.1986.tb00255.x

Holland, P. W. and Thayer, D. T. (1988). Differential item performance and the Mantel-Haenszel procedure. In H. Wainer and H. I. Braun (Dirs.), Test validity. Hillsdale, New Jersey: Lawrence Erlbaum Associates.

Kim, S.-H., Cohen, A.S. and Park, T.-H. (1995). Detection of differential item functioning in multiple groups. Journal of Educational Measurement, 32, 261-276. doi:10.1111/j.1745-3984.1995.tb00466.x

Li, H.-H., and Stout, W. (1996). A new procedure for detection of crossing DIF. Psychometrika, 61, 647–677. doi:10.1007/BF02294041

Lord, F. (1980). Applications of item response theory to practical testing problems. Hillsdale, NJ: Lawrence Erlbaum Associates.

Magis, D., Beland, S., Tuerlinckx, F. and De Boeck, P. (2010). A general framework and an R package for the detection of dichotomous differential item functioning. Behavior Research Methods, 42, 847-862. doi:10.3758/BRM.42.3.847

Magis, D., Raiche, G., Beland, S. and Gerard, P. (2011). A logistic regression procedure to detect differential item functioning among multiple groups. International Journal of Testing, 11, 365–386. doi:10.1080/15305058.2011.602810

Penfield, R. D. (2001). Assessing differential item functioning among multiple groups: a comparison of three Mantel-Haenszel procedures. Applied Measurement in Education, 14, 235-259. doi:10.1207/S15324818AME1403_3

Penfield, R.D. (2003). Application of the Breslow-Day test of trend in odds ratio heterogeneity to the detection of nonuniform DIF. Alberta Journal of Educational Research, 49, 231-243.

Raju, N. S. (1990). Determining the significance of estimated signed and unsigned areas between two item response functions. Applied Psychological Measurement, 14, 197-207. doi:10.1177/014662169001400208

Shealy, R. and Stout, W. (1993). A model-based standardization approach that separates true bias/DIF from group ability differences and detect test bias/DTF as well as item bias/DIF. Psychometrika, 58, 159-194. doi:10.1007/BF02294572

Swaminathan, H. and Rogers, H. J. (1990). Detecting differential item functioning using logistic regression procedures. Journal of Educational Measurement, 27, 361-370. doi:10.1111/j.1745-3984.1990.tb00754.x

Thissen, D., Steinberg, L. and Wainer, H. (1988). Use of item response theory in the study of group difference in trace lines. In H. Wainer and H. Braun (Eds.), Test validity. Hillsdale, NJ: Lawrence Erlbaum Associates.

Breslow-Day DIF statistic

Description

Computes Breslow-Day statistics for DIF detection.

Usage

breslowDay(data, member, match = "score", anchor = 1:ncol(data), 
     BDstat = "BD")
 breslowDay(data, member, match = "score", anchor = 1:ncol(data), 
     BDstat = "BD")

Arguments

`data`	numeric: the data matrix (one row per subject, one column per item).
`member`	numeric: the vector of group membership with zero and one entries only. See Details.
`match`	specifies the type of matching criterion. Can be either `"score"` (default) to compute the test score, or any continuous or discrete variable with the same length as the number of rows of `data`. See Details.
`anchor`	a vector of integer values specifying which items (all by default) are currently considered as anchor (DIF free) items. See Details.
`BDstat`	character specifying the DIF statistic to be used. Possible values are `"BD"` (default) and `"trend"`. See Details.

Details

breslowDay computes one of the Breslow-Day statistics (1980) in the specific framework of differential item functioning. It forms the basic command of difBD and is specifically designed for this call.

The data are supplied by the data argument, with one row per subject and one column per item. Missing values are allowed but must be coded as NA values. They are discarded from sum-score computation.

The vector of group membership, specified by the member argument, must hold only zeros and ones, a value of zero corresponding to the reference group and a value of one to the focal group.

The matching criterion can be either the test score or any other continuous or discrete variable to be passed in the breslowDay function. This is specified by the match argument. By default, it takes the value "score" and the test score (i.e. raw score) is computed. The second option is to assign to match a vector of continuous or discrete numeric values, which acts as the matching criterion. Note that for consistency this vector should not belong to the data matrix.

Option anchor sets the items which are considered as anchor items for computing Breslow-Day DIF statistics. Items other than the anchor items and the tested item are discarded. anchor must hold integer values specifying the column numbers of the corresponding anchor items. It is primarily designed to perform item purification.

Two test statistics are available: the usual Breslow-Day statistic for testing homogeneous association (Aguerri, Galibert, Attorresi and Maranon, 2009) and the trend test statistic for assessing some monotonic trend in the odss ratios (Penfield, 2003). The DIF statistic is supplied by the BDstat argument, with values "BD" (default) for the usual statistic and "trend" for the trend test statistic.

Value

A list with three arguments:

`res`	A matrix with one row per item and three columns: the first one contains the Breslow-Day statistic values, the second column indicates the degrees of freedom, and the last column displays the asymptotic p-values.
`BDstat`	the value of the `BDstat` argument.
`match`	a character string, either `"score"` or `"matching variable"` depending on the `match` argument.

Author(s)

References

Breslow, N.E. and Day, N.E. (1980). Statistical methods in cancer research, vol. I: The analysis of case-control studies. Scientific Publication No 32. International Agency for Research on Cancer, Lyon, France.

Penfield, R.D. (2003). Application of the Breslow-Day test of trend in odds ratio heterogeneity to the detection of nonuniform DIF. Alberta Journal of Educational Research, 49, 231-243.

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)

 # With all items as anchor items
 breslowDay(verbal[,1:24], verbal[,26])

 # With all items as anchor items and trend
 # test statistic
 breslowDay(verbal[,1:24], verbal[,26], BDstat = "trend")

 # Removing item 3 from the set of anchor items
 breslowDay(verbal[,1:24], verbal[,26], anchor = c(1:5, 7:24))

## End(Not run)
## Not run: 

 # Loading of the verbal data
 data(verbal)

 # With all items as anchor items
 breslowDay(verbal[,1:24], verbal[,26])

 # With all items as anchor items and trend
 # test statistic
 breslowDay(verbal[,1:24], verbal[,26], BDstat = "trend")

 # Removing item 3 from the set of anchor items
 breslowDay(verbal[,1:24], verbal[,26], anchor = c(1:5, 7:24))

## End(Not run)

Contrast matrix for computing generalized Lord's chi-squared DIF statistic

Description

This command sets the appropriate contrast matrix C for computing the generalized Lord's chi-squared statistics in the framework of DIF detection among multiple groups.

Usage

 contrastMatrix(nrFocal, model)
 contrastMatrix(nrFocal, model)

Arguments

`nrFocal`	numeric: the number of focal groups.
`model`	character: the logistic model to be fitted (either `"1PL"`, `"2PL"`, `"3PL"` or `"3PLc"`). See Details.

Details

The contrast matrix C is necessary to calculate the generalized Lord's chi-squared statistic. It is designed to perform accurate tests of equality of item parameters accross the groups of examinees (see Kim, Cohen and Park, 1995). This is a subroutine for the command genLordChi2 which returns the DIF statistics.

The number of focal groups has to be specified by the argument nrFocal. Moreover, four logistic IRT models can be considered: the 1PL, 2PL and 3PL models can be set by using their acronyms (e.g. "1PL" for 1PL model, and so on). It is also possible to consider the constrained 3PL model, where all pseudo-guessing values are equal across the groups of examinees and take some predefined values which do not need to be supplied here. This model is specified by the value "3PLc" for argument model.

Value

A contrast matrix designed to test equality of item parameter estimates from the specified model and with nrFocal focal groups. The output matrix has a number of rows equal to nrFocal times the number of tested parameters (one for 1PL model, two for 2PL and constrained 3PL models, three for 3PL model). The number of columns is equal to (nrFocal+1) times the number of tested parameters. See Kim, Cohen and Park (1995) for further details.

Author(s)

References

Examples

## Not run: 

 # Contrast matrices with 1PL model and several focal groups
 contrastMatrix(2, "1PL")
 contrastMatrix(3, "1PL")
 contrastMatrix(4, "1PL")

 # Contrast matrices with 2PL, constrained and unconstrained 3PL models and three 
 # focal groups
 contrastMatrix(3, "2PL")
 contrastMatrix(3, "3PLc")
 contrastMatrix(3, "3PL")

## End(Not run)
## Not run: 

 # Contrast matrices with 1PL model and several focal groups
 contrastMatrix(2, "1PL")
 contrastMatrix(3, "1PL")
 contrastMatrix(4, "1PL")

 # Contrast matrices with 2PL, constrained and unconstrained 3PL models and three 
 # focal groups
 contrastMatrix(3, "2PL")
 contrastMatrix(3, "3PLc")
 contrastMatrix(3, "3PL")

## End(Not run)

Comparison of DIF detection methods

Description

This function compares the specified DIF detection methods with respect to the detected items.

Usage

dichoDif(Data, group, focal.name, method, anchor = NULL, props = NULL, 
 	thrTID = 1.5, alpha = 0.05, MHstat = "MHChisq", correct = TRUE, 
 	exact = FALSE, stdWeight = "focal", thrSTD = 0.1, BDstat = "BD", 
 	member.type = "group", match = "score", type = "both", criterion = "LRT", 
 	model = "2PL", c = NULL, engine = "ltm", discr = 1, irtParam = NULL, 
 	same.scale = TRUE, signed = FALSE, purify = FALSE, purType = "IPP1",
 	nrIter = 10, extreme = "constraint", const.range = c(0.001, 0.999), 
 	nrAdd = 1, p.adjust.method = NULL, save.output = FALSE,
 	output = c("out", "default")) 
## S3 method for class 'dichoDif'
print(x, ...)
 dichoDif(Data, group, focal.name, method, anchor = NULL, props = NULL, 
 	thrTID = 1.5, alpha = 0.05, MHstat = "MHChisq", correct = TRUE, 
 	exact = FALSE, stdWeight = "focal", thrSTD = 0.1, BDstat = "BD", 
 	member.type = "group", match = "score", type = "both", criterion = "LRT", 
 	model = "2PL", c = NULL, engine = "ltm", discr = 1, irtParam = NULL, 
 	same.scale = TRUE, signed = FALSE, purify = FALSE, purType = "IPP1",
 	nrIter = 10, extreme = "constraint", const.range = c(0.001, 0.999), 
 	nrAdd = 1, p.adjust.method = NULL, save.output = FALSE,
 	output = c("out", "default")) 
## S3 method for class 'dichoDif'
print(x, ...)

Arguments

`Data`	numeric: either the data matrix only, or the data matrix plus the vector of group membership. See Details.
`group`	numeric or character: either the vector of group membership or the column indicator (within `Data`) of group membership. See Details.
`focal.name`	numeric or character indicating the level of `group` which corresponds to the focal group.
`method`	character: the name of the selected method. Possible values are `"TID"`, `"MH"`, `"Std"`, `"Logistic"`, `"BD"`, `"SIBTEST"`, `"Lord"`, `"Raju"` and `"LRT"`. See Details.
`anchor`	either `NULL` (default) or a vector of item names (or identifiers) to specify the anchor items. See Details.
`props`	either `NULL` (default) or a two-column matrix with proportions of success in the reference group and the focal group. See Details.
`thrTID`	numeric: the threshold for detecting DIF items with TID method (default is 1.5).
`alpha`	numeric: significance level (default is 0.05).
`MHstat`	character: specifies the DIF statistic to be used for DIF identification. Possible values are `"MHChisq"` (default) and `"logOR"`. See Details.
`correct`	logical: should the Mantel-Haenszel continuity correction be used? (default is TRUE).
`exact`	logical: should an exact test be computed? (default is `FALSE`).
`stdWeight`	character: the type of weights used for the standardized P-DIF statistic. Possible values are `"focal"` (default), `"reference"` and `"total"`. See Details.
`thrSTD`	numeric: the threshold (cut-score) for standardized P-DIF statistic (default is 0.10).
`BDstat`	character specifying the DIF statistic to be used. Possible values are `"BD"` (default) and `"trend"`. See Details.
`member.type`	character: either `"group"` (default) to specify that group membership is made of two groups, or `"cont"` to indicate that group membership is based on a continuous criterion. See Details.
`match`	specifies the type of matching criterion. Can be either `"score"` (default) to compute the test score, or any continuous or discrete variable with the same length as the number of rows of `Data`. See Details.
`type`	a character string specifying which DIF effects must be tested. Possible values are `"both"` (default), `"udif"` and `"nudif"`. See Details.
`criterion`	a character string specifying which DIF statistic is computed. Possible values are `"LRT"` (default) or `"Wald"`. See Details.
`model`	character: the IRT model to be fitted (either `"1PL"`, `"2PL"` or `"3PL"`). Default is `"2PL"`.
`c`	optional numeric value or vector giving the values of the constrained pseudo-guessing parameters. See Details.
`engine`	character: the engine for estimating the 1PL model, either `"ltm"` (default) or `"lme4"`.
`discr`	either `NULL` or a real positive value for the common discrimination parameter (default is 1). Used onlky if `model` is `"1PL"` and `engine` is `"ltm"`. See Details.
`irtParam`	matrix with 2J rows (where J is the number of items) and at most 9 columns containing item parameters estimates. See Details.
`same.scale`	logical: are the item parameters of the `irtParam` matrix on the same scale? (default is "TRUE"). See Details.
`signed`	logical: should the Raju's statistics be computed using the signed (`TRUE`) or unsigned (`FALSE`, default) area? See Details.
`purify`	logical: should the method be used iteratively to purify the set of anchor items? (default is FALSE).
`purType`	character: the type of purification process to be run. Possible values are `"IPP1"` (default), `"IPP2"` and `"IPP3"`. Ignored if `purify` is `FALSE` or `method` does not supply the `"TID"` method.
`nrIter`	numeric: the maximal number of iterations in the item purification process (default is 10).
`extreme`	character: the method used to modify the extreme proportions. Possible values are `"constraint"` (default) or `"add"`. Ignored if `method` is not `"TID"`.
`const.range`	numeric: a vector of two constraining proportions. Default values are 0.001 and 0.999. Ignored if `method` is not `"TID"` or if `extreme` is `"add"`.
`nrAdd`	integer: the number of successes and the number of failures to add to the data in order to adjust the proportions. Default value is 1. Ignored if `method` is not `"TID"` or if `extreme` is `"constraint"`.
`p.adjust.method`	either `NULL` (default) or the acronym of the method for p-value adjustment for multiple comparisons. See Details.
`save.output`	logical: should the output be saved into a text file? (Default is `FALSE`).
`output`	character: a vector of two components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value). See Details.
`x`	result from a `dichoDif` class object.
`...`	other generic parameters for the `print` function.

Details

dichoDif is a generic function which calls one or several DIF detection methods and summarize their output. The possible methods are:

"TID" for Transformed Item Difficulties (TID) method (Angoff and Ford, 1973),
"MH" for mantel-Haenszel (Holland and Thayer, 1988),
"Std" for standardization (Dorans and Kulick, 1986),
"BD" for Breslow-Day method (Penfield, 2003),
"Logistic" for logistic regression (Swaminathan and Rogers, 1990),
"SIBTEST" for SIBTEST (Shealy and Stout) and Crossing-SIBTEST (Chalmers, 2018; Li and Stout, 1996) methods,
"Lord" for Lord's chi-square test (Lord, 1980),
"Raju" for Raju's area method (Raju, 1990), and
"LRT" for likelihood-ratio test method (Thissen, Steinberg and Wainer, 1988).

If method has a single component, the output of dichoDif is exactly the one provided by the method itself. Otherwise, the main output is a matrix with one row per item and one column per method. For each specified method and related arguments, items detected as DIF and non-DIF are respectively encoded as "DIF" and "NoDIF". When printing the output an additional column is added, counting the number of times each item was detected as functioning differently (Note: this is just an informative summary, since the methods are obviously not independent for the detection of DIF items).

The Data is a matrix whose rows correspond to the subjects and columns to the items. In addition, Data can hold the vector of group membership. If so, group indicates the column of Data which corresponds to the group membership, either by specifying its name or by giving the column number. Otherwise, group must be a vector of same length as nrow(Data).

Missing values are allowed for item responses (not for group membership) but must be coded as NA values. They are discarded from either the computation of the sum-scores, the fitting of the logistic models or the IRT models (according to the method).

The vector of group membership must hold only two different values, either as numeric or character. The focal group is defined by the value of the argument focal.name.

For "MH", "Std", "Logistic" and "BD" methods, the matching criterion can be either the test score or any other continuous or discrete variable to be passed in the Logistik function. This is specified by the match argument. By default, it takes the value "score" and the test score (i.e. raw score) is computed. The second option is to assign to match a vector of continuous or discrete numeric values, which acts as the matching criterion. Note that for consistency this vector should not belong to the Data matrix.

For Lord and Raju methods, one can specify either the IRT model to be fitted (by means of model, c, engine and discr arguments), or the item parameter estimates with arguments irtParam and same.scale. See difLord and difRaju for further details.

The threshold for detecting DIF items depends on the method. For standardization it has to be fully specified (with the thr argument), as well as for the TID method (through the thrTID argument). For the other methods it is depending on the significance level set by alpha.

For Mantel-Haenszel method, the DIF statistic can be either the Mantel-Haenszel chi-square statistic or the log odds-ratio statistic. The method is specified by the argument MHstat, and the default value is "MHChisq" for the chi-square statistic. Moreover, the option correct specifies whether the continuity correction has to be applied to Mantel-Haenszel statistic. See difMH for further details.

By default, the asymptotic Mantel-Haenszel statistic is computed. However, the exact statistics and related P-values can be obtained by specifying the logical argument exact to TRUE. See Agresti (1990, 1992) for further details about exact inference.

The weights for computing the standardized P-DIF statistics are defined through the argument stdWeight, with possible values "focal" (default value), "reference" and "total". See stdPDIF for further details.

For Breslow-Day method, two test statistics are available: the usual Breslow-Day statistic for testing homogeneous association (Aguerri, Galibert, Attorresi and Maranon, 2009) and the trend test statistic for assessing some monotonic trend in the odss ratios (Penfield, 2003). The DIF statistic is supplied by the BDstat argument, with values "BD" (default) for the usual statistic and "trend" for the trend test statistic.

For logistic regression, the argument type permits to test either both uniform and nonuniform effects simultaneously (type="both"), only uniform DIF effect (type="udif") or only nonuniform DIF effect (type="nudif"). The criterion argument specifies the DIF statistic to be computed, either the likelihood ratio test statistic (by setting criterion="LRT") or the Wald test (by setting criterion="Wald"). Moreover, the group membership can be either a vector of two distinct values, one for the reference group and one for the focal group, or a continuous or discrete variable that acts as the "group" membership variable. In the former case, the member.type argument is set to "group" and the focal.name defines which value in the group variable stands for the focal group. In the latter case, member.type is set to "cont", focal.name is ignored and each value of the group represents one "group" of data (that is, the DIF effects are investigated among participants relying on different values of some discrete or continuous trait). See Logistik for further details.

The SIBTEST method (Shealy and Stout, 1993) and its modified version, the Crossing-SIBTEST (Chalmers, 2018; Li and Stout, 1996) are returned by the difSIBTEST function. SIBTEST method is returned when type argument is set to "udif", while Crossing-SIBTEST is set with "nudif" value for the type argument. Note that type takes the by-default value "both" which is not allowed within the difSIBTEST function; however, within this fucntion, keeping the by-default value yields selection of Crossing-SIBTEST.

The difSIBTEST function is a wrapper to the SIBTEST function from the mirt package (Chalmers, 2012) to fit within the difR framework (Magis et al., 2010). Therefore, if you are using this function for publication purposes please cite Chalmers (2018; 2012) and Magis et al. (2010).

For Raju's method, the type of area (signed or unsigned) is fixed by the logical signed argument, with default value FALSE (i.e. unsigned areas). See RajuZ for further details.

Item purification can be requested by specifying purify option to TRUE. Recall that item purification process is slightly different for IRT and for non-IRT based methods. See the corresponding methods for further information.

Adjustment for multiple comparisons is possible with the argument p.adjust.method. See the corresponding methods for further information.

A pre-specified set of anchor items can be provided through the anchor argument. For non-IRT methods, anchor items are used to compute the test score (as matching criterion). For IRT methods, anchor items are used to rescale the item parameters on a common metric. See the corresponding methods for further information. Note that anchor argument is not working with "LRT" method.

The output of the dichoDif function can be stored in a text file by fixing save.output and output appropriately. See the help file of selectDif function (or any other DIF method) for further information.

Value

Either the output of one of the DIF detection methods, or a list of class "dichoDif" with the following arguments:

`DIF`	a character matrix with one row per item and whose columns refer to the different specified detection methods. See Details.
`props`	the value of the `props` argument.
`thrTID`	the value of the `thrTID` argument.
`correct`	the value of `correct` argument.
`exact`	the value of `exact` argument.
`alpha`	the significance level `alpha`.
`MHstat`	the value of the `MHstat` argument.
`stdWeight`	the value of the `stdWeight` argument.
`thrSTD`	the value of `thrSTD` argument.
`BDstat`	the value of the `BDstat` argument.
`member.type`	the value of the `member.type` argument.
`match`	the value of the `match` argument.
`type`	the value of the `type` argument.
`criterion`	the value of the `criterion` argument.
`model`	the value of `model` argument.
`c`	the value of `c` argument.
`engine`	The value of the `engine` argument.
`discr`	the value of the `discr` argument.
`irtParam`	the value of `irtParam` argument.
`same.scale`	the value of `same.scale` argument.
`p.adjust.method`	the value of the `p.adjust.method` argument.
`purification`	the value of `purify` argument.
`nrPur`	an integer vector (of length equal to the number of methods) with the number of iterations in the purification process. Returned only if `purify` is TRUE.
`convergence`	a logical vector (of length equal to the number of methods) indicating whether the iterative purification process converged. Returned only if `purify` is TRUE.
`anchor.names`	the value of the `anchor` argument.
`save.output`	the value of the `save.output` argument.
`output`	the value of the `output` argument.

Author(s)

References

Agresti, A. (1990). Categorical data analysis. New York: Wiley.

Agresti, A. (1992). A survey of exact inference for contingency tables. Statistical Science, 7, 131-177. doi:10.1214/ss/1177011454

Angoff, W. H., and Ford, S. F. (1973). Item-race interaction on a test of scholastic aptitude. Journal of Educational Measurement, 2, 95-106. doi:10.1111/j.1745-3984.1973.tb00787.x

Chalmers, R. P. (2012). mirt: A Multidimensional item response theory package for the R environment. Journal of Statistical Software, 48(6), 1-29. doi:10.18637/jss.v048.i06

Chalmers, R. P. (2018). Improving the Crossing-SIBTEST statistic for detecting non-uniform DIF. Psychometrika, 83(2), 376–386. doi:10.1007/s11336-017-9583-8

Holland, P. W. and Thayer, D. T. (1988). Differential item performance and the Mantel-Haenszel procedure. In H. Wainer and H. I. Braun (Dirs.), Test validity. Hillsdale, NJ: Lawrence Erlbaum Associates.

Li, H.-H., and Stout, W. (1996). A new procedure for detection of crossing DIF. Psychometrika, 61, 647–677. doi:10.1007/BF02294041

Lord, F. (1980). Applications of item response theory to practical testing problems. Hillsdale, NJ: Lawrence Erlbaum Associates.

Penfield, R.D. (2003). Application of the Breslow-Day test of trend in odds ratio heterogeneity to the detection of nonuniform DIF. Alberta Journal of Educational Research, 49, 231-243.

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Excluding the "Anger" variable
 verbal <- verbal[colnames(verbal)!="Anger"]

 # Comparing TID, Mantel-Haenszel, standardization; logistic regression and SIBTEST
 # TID threshold 1.0
 # Standardization threshold 0.08
 # no continuity correction,
 # with item purification
 # both types of DIF effect for logistic regression
 # CSIBTEST method 
 dichoDif(verbal, group = 25, focal.name = 1, method = c("TID", "MH", "Std",
          "Logistic", "SIBTEST"), correct = FALSE, thrSTD = 0.08, thrTID = 1, purify = TRUE)

 # Same analysis, but using items 1 to 5 as anchor and saving the output into 
 # the 'dicho' file 
 dichoDif(verbal, group = 25, focal.name = 1, method = c("TID", "MH", "Std",
          "Logistic"), correct = FALSE, thrSTD = 0.08, thrTID = 1, purify = TRUE, 
          anchor = 1:5,save.output = TRUE, output = c("dicho", "default"))

 # Comparing Lord and Raju results with 2PL model and
 # with item purification 
 dichoDif(verbal, group = 25, focal.name = 1, method = c("Lord", "Raju"),
          model = "2PL", purify = TRUE)

## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Excluding the "Anger" variable
 verbal <- verbal[colnames(verbal)!="Anger"]

 # Comparing TID, Mantel-Haenszel, standardization; logistic regression and SIBTEST
 # TID threshold 1.0
 # Standardization threshold 0.08
 # no continuity correction,
 # with item purification
 # both types of DIF effect for logistic regression
 # CSIBTEST method 
 dichoDif(verbal, group = 25, focal.name = 1, method = c("TID", "MH", "Std",
          "Logistic", "SIBTEST"), correct = FALSE, thrSTD = 0.08, thrTID = 1, purify = TRUE)

 # Same analysis, but using items 1 to 5 as anchor and saving the output into 
 # the 'dicho' file 
 dichoDif(verbal, group = 25, focal.name = 1, method = c("TID", "MH", "Std",
          "Logistic"), correct = FALSE, thrSTD = 0.08, thrTID = 1, purify = TRUE, 
          anchor = 1:5,save.output = TRUE, output = c("dicho", "default"))

 # Comparing Lord and Raju results with 2PL model and
 # with item purification 
 dichoDif(verbal, group = 25, focal.name = 1, method = c("Lord", "Raju"),
          model = "2PL", purify = TRUE)

## End(Not run)

Breslow-Day DIF method

Description

Performs DIF detection using Breslow-Day method.

Usage

difBD(Data, group, focal.name, anchor = NULL, match = "score", BDstat = "BD", 
  	alpha = 0.05, purify = FALSE, nrIter = 10, p.adjust.method = NULL, 
  	save.output = FALSE, output = c("out", "default"))
## S3 method for class 'BD'
print(x, ...)
## S3 method for class 'BD'
plot(x, pch = 8, number = TRUE, col = "red", save.plot = FALSE, 
  	save.options = c("plot", "default", "pdf"), ...)
 difBD(Data, group, focal.name, anchor = NULL, match = "score", BDstat = "BD", 
  	alpha = 0.05, purify = FALSE, nrIter = 10, p.adjust.method = NULL, 
  	save.output = FALSE, output = c("out", "default"))
## S3 method for class 'BD'
print(x, ...)
## S3 method for class 'BD'
plot(x, pch = 8, number = TRUE, col = "red", save.plot = FALSE, 
  	save.options = c("plot", "default", "pdf"), ...)

Arguments

`Data`	numeric: either the data matrix only, or the data matrix plus the vector of group membership. See Details.
`group`	numeric or character: either the vector of group membership or the column indicator (within `Data`) of group membership. See Details.
`focal.name`	numeric or character indicating the level of `group` which corresponds to the focal group.
`anchor`	either `NULL` (default) or a vector of item names (or identifiers) to specify the anchor items. See Details.
`match`	specifies the type of matching criterion. Can be either `"score"` (default) to compute the test score, or any continuous or discrete variable with the same length as the number of rows of `Data`. See Details.
`BDstat`	character specifying the DIF statistic to be used. Possible values are `"BD"` (default) and `"trend"`. See Details.
`alpha`	numeric: significance level (default is 0.05).
`purify`	logical: should the method be used iteratively to purify the set of anchor items? (default is `FALSE`).
`nrIter`	numeric: the maximal number of iterations in the item purification process (default is 10).
`p.adjust.method`	either `NULL` (default) or the acronym of the method for p-value adjustment for multiple comparisons. See Details.
`save.output`	logical: should the output be saved into a text file? (Default is `FALSE`).
`output`	character: a vector of two components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value). See Details.
`x`	the result from a BD class object.
`pch`, `col`	type of usual `pch` and `col` graphical options.
`number`	logical: should the item number identification be printed (default is `TRUE`).
`save.plot`	logical: should the plot be saved into a separate file? (default is `FALSE`).
`save.options`	character: a vector of three components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value), and the third component is the file extension, either `"pdf"` (default) or `"jpeg"`. See Details.
`...`	other generic parameters for the `plot` or the `print` functions.

Details

The method of Breslow-Day (1980) allows for detecting non-uniform differential item functioning without requiring an item response model approach.

Missing values are allowed for item responses (not for group membership) but must be coded as NA values. They are discarded from sum-score computation.

The vector of group membership must hold only two different values, either as numeric or character. The focal group is defined by the value of the argument focal.name.

Two test statistics are available: the usual Breslow-Day statistic for testing homogeneous association (Aguerri, Galibert, Attorresi and Maranon, 2009) and the trend test statistic for assessing some monotonic trend in the odds ratios (Penfield, 2003). The DIF statistic is supplied by the BDstat argument, with values "BD" (default) for the usual statistic and "trend" for the trend test statistic.

The threshold (or cut-score) for classifying items as DIF is computed as the quantile of the chi-squared distribution with lower-tail probability of one minus alpha, and the degrees of freedom depend on the DIF statistic. With the usual Breslow-Day statistic (BDstat=="BD"), it is the number of partial tables taken into account (Aguerri et al., 2009). With the trend test statistic, the degrees of freedom are always equal to one (Penfield, 2003).

Item purification can be performed by setting purify to TRUE. Purification works as follows: if at least one item was detected as functioning differently at the first step of the process, then the data set of the next step consists in all items that are currently anchor (DIF free) items, plus the tested item (if necessary). The process stops when either two successive applications of the method yield the same classifications of the items (Clauser and Mazor, 1998), or when nrIter iterations are run without obtaining two successive identical classifications. In the latter case a warning message is printed.

Adjustment for multiple comparisons is possible with the argument p.adjust.method. The latter must be an acronym of one of the available adjustment methods of the p.adjust function. According to Kim and Oshima (2013), Holm and Benjamini-Hochberg adjustments (set respectively by "Holm" and "BH") perform best for DIF purposes. See p.adjust function for further details. Note that item purification is performed on original statistics and p-values; in case of adjustment for multiple comparisons this is performed after item purification.

A pre-specified set of anchor items can be provided through the anchor argument. It must be a vector of either item names (which must match exactly the column names of Data argument) or integer values (specifying the column numbers for item identification). In case anchor items are provided, they are used to compute the test score (matching criterion), including also the tested item. None of the anchor items are tested for DIF: the output separates anchor items and tested items and DIF results are returned only for the latter. Note also that item purification is not activated when anchor items are provided (even if purify is set to TRUE). By default it is NULL so that no anchor item is specified.

The output of the difBD, as displayed by the print.BD function, can be stored in a text file provided that save.output is set to TRUE (the default value FALSE does not execute the storage). In this case, the name of the text file must be given as a character string into the first component of the output argument (default name is "out"), and the path for saving the text file can be given through the second component of output. The default value is "default", meaning that the file will be saved in the current working directory. Any other path can be specified as a character string: see the Examples section for an illustration.

The plot.BD function displays the DIF statistics in a plot, with each item on the X axis. The type of point and the colour are fixed by the usual pch and col arguments. Option number permits to display the item numbers instead. Also, the plot can be stored in a figure file, either in PDF or JPEG format. Fixing save.plot to TRUE allows this process. The figure is defined through the components of save.options. The first two components perform similarly as those of the output argument. The third component is the figure format, with allowed values "pdf" (default) for PDF file and "jpeg" for JPEG file.

Value

A list of class "BD" with the following arguments:

`BD`	a matrix with one row per item and three columns: the first one contains the Breslow-Day statistic value, the second column indicates the degrees of freedom, and the last column displays the asymptotic p-values.
`p.value`	the vector of p-values for the BD statistics.
`alpha`	the significance level for DIF detection.
`DIFitems`	either the column indicators of the items which were detected as DIF items, or "No DIF item detected".
`BDstat`	the value of the `BDstat` argument.
`match`	a character string, either `"score"` or `"matching variable"` depending on the `match` argument.
`p.adjust.method`	the value of the `p.adjust.method` argument.
`adjusted.p`	either `NULL` or the vector of adjusted p-values for multiple comparisons.
`purification`	the value of `purify` option.
`nrPur`	the number of iterations in the item purification process. Returned only if `purify` is `TRUE`.
`difPur`	a binary matrix with one row per iteration in the item purification process and one column per item. Zeros and ones in the i-th row refer to items which were classified respectively as non-DIF and DIF items at the (i-1)-th step. The first row corresponds to the initial classification of the items. Returned only if `purify` is `TRUE`.
`convergence`	logical indicating whether the iterative item purification process stopped before the maximal number `nrIter` of allowed iterations. Returned only if `purify` is `TRUE`.
`names`	the names of the items.
`anchor.names`	the value of the `anchor` argument.
`save.output`	the value of the `save.output` argument.
`output`	the value of the `output` argument.

Author(s)

References

Clauser, B.E. and Mazor, K.M. (1998). Using statistical procedures to identify differential item functioning test items. Educational Measurement: Issues and Practice, 17, 31-44.

Kim, J., and Oshima, T. C. (2013). Effect of multiple testing adjustment in differential item functioning detection. Educational and Psychological Measurement, 73, 458–470. doi:10.1177/0013164412467033

Penfield, R.D. (2003). Application of the Breslow-Day test of trend in odds ratio heterogeneity to the detection of nonuniform DIF. Alberta Journal of Educational Research, 49, 231-243.

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)

 # Excluding the "Anger" variable
 verbal<-verbal[colnames(verbal) != "Anger"]

 # Three equivalent settings of the data matrix and the group membership
 difBD(verbal, group = 25, focal.name = 1)
 difBD(verbal, group = "Gender", focal.name = 1)
 difBD(verbal[,1:24], group = verbal[,25], focal.name = 1)

 # With the BD trend test statistic
 difBD(verbal, group = 25, focal.name = 1, BDstat = "trend")

 # Multiple comparisons adjustment using Benjamini-Hochberg method
 difBD(verbal, group = 25, focal.name = 1, p.adjust.method = "BH")

 # With item purification  
 difBD(verbal, group = "Gender", focal.name = 1, purify = TRUE)
 difBD(verbal, group = "Gender", focal.name = 1, purify = TRUE, nrIter = 5)

 # With items 1 to 5 set as anchor items
 difBD(verbal, group = "Gender", focal.name = 1, anchor = 1:5)
 difBD(verbal, group = "Gender", focal.name = 1, anchor = 1:5, purify = TRUE)

 # Saving the output into the "BDresults.txt" file (and default path)
 r <- difBD(verbal, group = 25, focal.name = 1, save.output = TRUE, 
            output = c("BDresults","default"))

 # Graphical devices
 plot(r)

 # Plotting results and saving it in a PDF figure
 plot(r, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)

 # Excluding the "Anger" variable
 verbal<-verbal[colnames(verbal) != "Anger"]

 # Three equivalent settings of the data matrix and the group membership
 difBD(verbal, group = 25, focal.name = 1)
 difBD(verbal, group = "Gender", focal.name = 1)
 difBD(verbal[,1:24], group = verbal[,25], focal.name = 1)

 # With the BD trend test statistic
 difBD(verbal, group = 25, focal.name = 1, BDstat = "trend")

 # Multiple comparisons adjustment using Benjamini-Hochberg method
 difBD(verbal, group = 25, focal.name = 1, p.adjust.method = "BH")

 # With item purification  
 difBD(verbal, group = "Gender", focal.name = 1, purify = TRUE)
 difBD(verbal, group = "Gender", focal.name = 1, purify = TRUE, nrIter = 5)

 # With items 1 to 5 set as anchor items
 difBD(verbal, group = "Gender", focal.name = 1, anchor = 1:5)
 difBD(verbal, group = "Gender", focal.name = 1, anchor = 1:5, purify = TRUE)

 # Saving the output into the "BDresults.txt" file (and default path)
 r <- difBD(verbal, group = 25, focal.name = 1, save.output = TRUE, 
            output = c("BDresults","default"))

 # Graphical devices
 plot(r)

 # Plotting results and saving it in a PDF figure
 plot(r, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

## End(Not run)

Generalized logistic regression DIF method

Description

Performs DIF detection among multiple groups using generalized logistic regression method.

Usage

difGenLogistic(Data, group, focal.names, anchor = NULL, match = "score", 
 	type = "both", criterion = "LRT", alpha = 0.05, purify = FALSE, nrIter = 10,
 	p.adjust.method = NULL, save.output = FALSE, output = c("out", "default"))
## S3 method for class 'genLogistic'
print(x, ...)
## S3 method for class 'genLogistic'
plot(x, plot = "lrStat", item = 1, itemFit = "best",pch = 8, number = TRUE,
  	col = "red", colIC = rep("black", length(x$focal.names)+1),
  	ltyIC = 1:(length(x$focal.names)+1), title = NULL, save.plot = FALSE, 
  	save.options = c("plot", "default", "pdf"), ref.name = NULL, ...)
 difGenLogistic(Data, group, focal.names, anchor = NULL, match = "score", 
 	type = "both", criterion = "LRT", alpha = 0.05, purify = FALSE, nrIter = 10,
 	p.adjust.method = NULL, save.output = FALSE, output = c("out", "default"))
## S3 method for class 'genLogistic'
print(x, ...)
## S3 method for class 'genLogistic'
plot(x, plot = "lrStat", item = 1, itemFit = "best",pch = 8, number = TRUE,
  	col = "red", colIC = rep("black", length(x$focal.names)+1),
  	ltyIC = 1:(length(x$focal.names)+1), title = NULL, save.plot = FALSE, 
  	save.options = c("plot", "default", "pdf"), ref.name = NULL, ...)

Arguments

`Data`	numeric: either the data matrix only, or the data matrix plus the vector of group membership. See Details.
`group`	numeric or character: either the vector of group membership or the column indicator (within `data`) of group membership. See Details.
`focal.names`	numeric or character vector indicating the levels of `group` which correspond to the focal groups.
`anchor`	either `NULL` (default) or a vector of item names (or identifiers) to specify the anchor items. Ignored if `match` is not `"score"`. See Details.
`match`	specifies the type of matching criterion. Can be either `"score"` (default) to compute the test score, or any continuous or discrete variable with the same length as the number of rows of `Data`. See Details.
`type`	a character string specifying which DIF effects must be tested. Possible values are `"both"` (default), `"udif"` and `"nudif"`. See Details.
`criterion`	character: the type of test statistic used to detect DIF items. Possible values are `"LRT"` (default) and `"Wald"`. See Details.
`alpha`	numeric: significance level (default is 0.05).
`purify`	logical: should the method be used iteratively to purify the set of anchor items? (default is FALSE).
`nrIter`	numeric: the maximal number of iterations in the item purification process (default is 10).
`p.adjust.method`	either `NULL` (default) or the acronym of the method for p-value adjustment for multiple comparisons. See Details.
`save.output`	logical: should the output be saved into a text file? (Default is `FALSE`).
`output`	character: a vector of two components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value). See Details.
`x`	the result from a `Logistik` class object.
`plot`	character: the type of plot, either `"lrStat"` or `"itemCurve"`. See Details.
`item`	numeric or character: either the number or the name of the item for which logistic curves are plotted. Use only when `plot="itemCurve"`.
`itemFit`	character: the model to be selected for drawing the item curves. Possible values are `"best"` (default) for drawing from the best of the two models, and `"null"` for using fitted parameters of the null model $M_0$ . Not used if `"plot"` is `"lrStat"`. See Details.
`pch`, `col`	type of usual `pch` and `col` graphical options.
`number`	logical: should the item number identification be printed (default is `TRUE`).
`colIC`, `ltyIC`	vectors of elements of the usual `col` and `lty` arguments for logistic curves. Used only when `plot="itemCurve"`.
`title`	either a character string with the title of the plot, or `NULL` (default), for which a specific title is automatically displayed.
`save.plot`	logical: should the plot be saved into a separate file? (default is `FALSE`).
`save.options`	character: a vector of three components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value), and the third component is the file extension, either `"pdf"` (default) or `"jpeg"`. See Details.
`ref.name`	either `NULL`(default) or a character string for the name of the reference group (to be used instead of "Reference" in the legend). Ignored if `plot` is `"lrStat"`.
`...`	other generic parameters for the `plot` or the `print` functions.

Details

The generalized logistic regression method (Magis, Raiche, Beland and Gerard, 2011) allows for detecting both uniform and non-uniform differential item functioning among multiple groups without requiring an item response model approach. It consists in fitting a logistic model with the matching criterion, the group membership and an interaction between both as covariates. The statistical significance of the parameters related to group membership and the group-score interaction is then evaluated by means of the usual likelihood-ratio test. The argument type permits to test either both uniform and nonuniform effects simultaneously (type="both"), only uniform DIF effect (type="udif") or only nonuniform DIF effect (type="nudif"). The identification of DIF items can be performed with either the Wald test or the likelihood ratio test, by setting the criterion argument to "Wald" or "LRT" respectively. See genLogistik for further details.

The matching criterion can be either the test score or any other continuous or discrete variable to be passed in the genLogistik function. This is specified by the match argument. By default, it takes the value "score" and the test score (i.e. raw score) is computed. The second option is to assign to match a vector of continuous or discrete numeric values, which acts as the matching criterion. Note that for consistency this vector should not belong to the Data matrix.

Missing values are allowed for item responses (not for group membership) but must be coded as NA values. They are discarded from the fitting of the logistic models (see glm for further details).

The vector of group membership must hold at least three values, either as numeric or character. The focal groups are defined by the values of the argument focal.names. If there is a unique focal group, then difGenLogistic returns the output of difLogistic.

The threshold (or cut-score) for classifying items as DIF is computed as the quantile of the chi-squared distribution with lower-tail probability of one minus alpha and with J (if type="udif" or type="nudif") or 2J (if type="both") degrees of freedom (J is the number of focal groups).

Item purification can be performed by setting purify to TRUE. Purification works as follows: if at least one item is detected as functioning differently at the first step of the process, then the data set of the next step consists in all items that are currently anchor (DIF free) items, plus the tested item (if necessary). The process stops when either two successive applications of the method yield the same classifications of the items (Clauser and Mazor, 1998), or when nrIter iterations are run without obtaining two successive identical classifications. In the latter case a warning message is printed.

A pre-specified set of anchor items can be provided through the anchor argument. It must be a vector of either item names (which must match exactly the column names of Data argument) or integer values (specifying the column numbers for item identification). In case anchor items are provided, they are used to compute the test score (matching criterion), including also the tested item. None of the anchor items are tested for DIF: the output separates anchor items and tested items and DIF results are returned only for the latter. By default it is NULL so that no anchor item is specified. Note also that item purification is not activated when anchor items are provided (even if purify is set to TRUE). Moreover, if the match argument is not set to "score", anchor items will not be taken into account even if anchor is not NULL.

The measures of effect size are provided by the difference $\Delta R^2$ between the $R^2$ coefficients of the two nested models (Nagelkerke, 1991; Gomez-Benito, Dolores Hidalgo and Padilla, 2009). The effect sizes are classified as "negligible", "moderate" or "large". Two scales are available, one from Zumbo and Thomas (1997) and one from Jodoin and Gierl (2001). The output displays the $\Delta R^2$ measures, together with the two classifications.

The output of the difGenLogistic, as displayed by the print.genLogistic function, can be stored in a text file provided that save.output is set to TRUE (the default value FALSE does not execute the storage). In this case, the name of the text file must be given as a character string into the first component of the output argument (default name is "out"), and the path for saving the text file can be given through the second component of output. The default value is "default", meaning that the file will be saved in the current working directory. Any other path can be specified as a character string: see the Examples section for an illustration.

Two types of plots are available. The first one is obtained by setting plot="lrStat" and it is the default option. The likelihood ratio statistics are displayed on the Y axis, for each item. The detection threshold is displayed by a horizontal line, and items flagged as DIF are printed with the color defined by argument col. By default, items are spotted with their number identification (number=TRUE); otherwise they are simply drawn as dots whose form is given by the option pch.

The other type of plot is obtained by setting plot="itemCurve". In this case, the fitted logistic curves are displayed for one specific item set by the argument item. The latter argument can hold either the name of the item or its number identification. If the argument itemFit takes the value "best", the curves are drawn according to the output of the best model among $M_0$ and $M_1$ . That is, two curves are drawn if the item is flagged as DIF, and only one if the item is flagged as non-DIF. If itemFit takes the value "null", then the two curves are drawn from the fitted parameters of the null model $M_0$ . See genLogistik for further details on the models. The colors and types of traits for these curves are defined by means of the arguments colIC and ltyIC respectively. These are set as vectors of length $J+1$ , the first element for the reference group and the others for the focal groups. Finally, the ref.name argument permits to display the name if the reference group (instead of "Reference") in the legend.

Both types of plots can be stored in a figure file, either in PDF or JPEG format. Fixing save.plot to TRUE allows this process. The figure is defined through the components of save.options. The first two components perform similarly as those of the output argument. The third component is the figure format, with allowed values "pdf" (default) for PDF file and "jpeg" for JPEG file.

Value

A list of class "genLogistic" with the following arguments:

`genLogistik`	the values of the generalized logistic regression statistics.
`p.value`	the vector of p-values for the generalized logistic regression statistics.
`logitPar`	a matrix with one row per item and $2+J*2$ columns, holding the fitted parameters of the best model (among the two tested models) for each item.
`parM0`	the matrix of fitted parameters of the null model $M_0$ , as returned by the `Logistik` command.
`covMat`	a 3-dimensional matrix of size p x p x K, where p is the number of estimated parameters and K is the number of items, holding the p x p covariance matrices of the estimated parameters (one matrix for each tested item).
`deltaR2`	the differences in Nagelkerke's $R^2$ coefficients. See Details.
`alpha`	the value of `alpha` argument.
`thr`	the threshold (cut-score) for DIF detection.
`DIFitems`	either the column indicators for the items which were detected as DIF items, or "No DIF item detected".
`type`	the value of `type` argument.
`p.adjust.method`	the value of the `p.adjust.method` argument.
`adjusted.p`	either `NULL` or the vector of adjusted p-values for multiple comparisons.
`purification`	the value of `purify` option.
`nrPur`	the number of iterations in the item purification process. Returned only if `purify` is `TRUE`.
`difPur`	a binary matrix with one row per iteration in the item purification process and one column per item. Zeros and ones in the i-th row refer to items which were classified respectively as non-DIF and DIF items at the (i-1)-th step. The first row corresponds to the initial classification of the items. Returned only if `purify` is `TRUE`.
`convergence`	logical indicating whether the iterative item purification process stopped before the maximal number of `nrItem` allowed iterations. Returned only if `purify` is `TRUE`.
`names`	the names of the items.
`anchor.names`	the value of the `anchor` argument.
`focal.names`	the value of `focal.names` argument.
`criterion`	the value of the `criterion` argument.
`save.output`	the value of the `save.output` argument.
`output`	the value of the `output` argument.

Author(s)

References

Clauser, B.E. and Mazor, K.M. (1998). Using statistical procedures to identify differential item functioning test items. Educational Measurement: Issues and Practice, 17, 31-44.

Gomez-Benito, J., Dolores Hidalgo, M. and Padilla, J.-L. (2009). Efficacy of effect size measures in logistic regression: an application for detecting DIF. Methodology, 5, 18-25. doi:10.1027/1614-2241.5.1.18

Hidalgo, M. D. and Lopez-Pina, J.A. (2004). Differential item functioning detection and effect size: a comparison between logistic regression and Mantel-Haenszel procedures. Educational and Psychological Measurement, 64, 903-915. doi:10.1177/0013164403261769

Jodoin, M. G. and Gierl, M. J. (2001). Evaluating Type I error and power rates using an effect size measure with logistic regression procedure for DIF detection. Applied Measurement in Education, 14, 329-349. doi:10.1207/S15324818AME1404_2

Nagelkerke, N. J. D. (1991). A note on a general definition of the coefficient of determination. Biometrika, 78, 691-692. doi:10.1093/biomet/78.3.691

Zumbo, B. D. and Thomas, D. R. (1997). A measure of effect size for a model-based approach for studying DIF. Prince George, Canada: University of Northern British Columbia, Edgeworth Laboratory for Quantitative Behavioral Science.

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Creating four groups according to gender ("Man" or "Woman") and
 # trait anger score ("Low" or "High")
 group <- rep("WomanLow", nrow(verbal))
 group[Anger>20 & Gender==0] <- "WomanHigh"
 group[Anger<=20 & Gender==1] <- "ManLow"
 group[Anger>20 & Gender==1] <- "ManHigh"

 # New data set
 Verbal <- cbind(verbal[,1:24], group)

 # Reference group: "WomanLow"
 names <- c("WomanHigh", "ManLow", "ManHigh")

 # Testing both types of DIF effects
 # Three equivalent settings of the data matrix and the group membership
 r <- difGenLogistic(Verbal, group = 25, focal.names = names)
 difGenLogistic(Verbal, group = "group", focal.name = names)
 difGenLogistic(Verbal[,1:24], group = Verbal[,25], focal.names = names)

 # Using the Wald test
 difGenLogistic(Verbal, group = 25, focal.names = names, criterion = "Wald")

 # Multiple comparisons adjustment using Benjamini-Hochberg method
difGenLogistic(Verbal, group = 25, focal.names = names, p.adjust.method = "BH")

 # With item purification
 difGenLogistic(Verbal, group = 25, focal.names = names, purify = TRUE)
 difGenLogistic(Verbal, group = 25, focal.names = names, purify = TRUE,
   nrIter = 5)

 # With items 1 to 5 set as anchor items
 difGenLogistic(Verbal, group = 25, focal.name = names, anchor = 1:5)

 # Testing for nonuniform DIF effect
 difGenLogistic(Verbal, group = 25, focal.names = names, type = "nudif")

 # Testing for uniform DIF effect
 difGenLogistic(Verbal, group = 25, focal.names = names, type = "udif")

 # User anger trait score as matching criterion
 anger <- verbal[,25]
 difGenLogistic(Verbal, group = 25, focal.names = names, match = anger)

 # Saving the output into the "GLresults.txt" file (and default path)
 r <- difGenLogistic(Verbal, group = 25, focal.name = names, 
                save.output = TRUE, output = c("GLresults","default"))

 # Graphical devices
 plot(r)
 plot(r, plot = "itemCurve", item = 1)
 plot(r, plot = "itemCurve", item = 1, itemFit = "best")
 plot(r, plot = "itemCurve", item = 6)
 plot(r, plot = "itemCurve", item = 6, itemFit = "best")

 # Plotting results and saving it in a PDF figure
 plot(r, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Creating four groups according to gender ("Man" or "Woman") and
 # trait anger score ("Low" or "High")
 group <- rep("WomanLow", nrow(verbal))
 group[Anger>20 & Gender==0] <- "WomanHigh"
 group[Anger<=20 & Gender==1] <- "ManLow"
 group[Anger>20 & Gender==1] <- "ManHigh"

 # New data set
 Verbal <- cbind(verbal[,1:24], group)

 # Reference group: "WomanLow"
 names <- c("WomanHigh", "ManLow", "ManHigh")

 # Testing both types of DIF effects
 # Three equivalent settings of the data matrix and the group membership
 r <- difGenLogistic(Verbal, group = 25, focal.names = names)
 difGenLogistic(Verbal, group = "group", focal.name = names)
 difGenLogistic(Verbal[,1:24], group = Verbal[,25], focal.names = names)

 # Using the Wald test
 difGenLogistic(Verbal, group = 25, focal.names = names, criterion = "Wald")

 # Multiple comparisons adjustment using Benjamini-Hochberg method
difGenLogistic(Verbal, group = 25, focal.names = names, p.adjust.method = "BH")

 # With item purification
 difGenLogistic(Verbal, group = 25, focal.names = names, purify = TRUE)
 difGenLogistic(Verbal, group = 25, focal.names = names, purify = TRUE,
   nrIter = 5)

 # With items 1 to 5 set as anchor items
 difGenLogistic(Verbal, group = 25, focal.name = names, anchor = 1:5)

 # Testing for nonuniform DIF effect
 difGenLogistic(Verbal, group = 25, focal.names = names, type = "nudif")

 # Testing for uniform DIF effect
 difGenLogistic(Verbal, group = 25, focal.names = names, type = "udif")

 # User anger trait score as matching criterion
 anger <- verbal[,25]
 difGenLogistic(Verbal, group = 25, focal.names = names, match = anger)

 # Saving the output into the "GLresults.txt" file (and default path)
 r <- difGenLogistic(Verbal, group = 25, focal.name = names, 
                save.output = TRUE, output = c("GLresults","default"))

 # Graphical devices
 plot(r)
 plot(r, plot = "itemCurve", item = 1)
 plot(r, plot = "itemCurve", item = 1, itemFit = "best")
 plot(r, plot = "itemCurve", item = 6)
 plot(r, plot = "itemCurve", item = 6, itemFit = "best")

 # Plotting results and saving it in a PDF figure
 plot(r, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

## End(Not run)

Generalized Lord's chi-squared DIF method

Description

Performs DIF detection among multiple groups using generalized Lord's chi-squared method.

Usage

difGenLord(Data, group, focal.names, model, c = NULL, engine = "ltm", 
 	discr = 1, irtParam = NULL, nrFocal = 2, same.scale = TRUE, anchor = NULL,
 	alpha = 0.05, purify = FALSE, nrIter = 10, p.adjust.method = NULL, 
 	save.output = FALSE, 	output = c("out", "default")) 
## S3 method for class 'GenLord'
print(x, ...)
## S3 method for class 'GenLord'
plot(x, plot = "lordStat", item = 1, pch = 8,
  	number = TRUE, col = "red", colIC = rep("black",
  	length(x$focal.names)+1), ltyIC = 1:(length(x$focal.names)
  	+ 1), save.plot = FALSE, save.options = c("plot", "default", "pdf"),
      ref.name = NULL, ...)
 difGenLord(Data, group, focal.names, model, c = NULL, engine = "ltm", 
 	discr = 1, irtParam = NULL, nrFocal = 2, same.scale = TRUE, anchor = NULL,
 	alpha = 0.05, purify = FALSE, nrIter = 10, p.adjust.method = NULL, 
 	save.output = FALSE, 	output = c("out", "default")) 
## S3 method for class 'GenLord'
print(x, ...)
## S3 method for class 'GenLord'
plot(x, plot = "lordStat", item = 1, pch = 8,
  	number = TRUE, col = "red", colIC = rep("black",
  	length(x$focal.names)+1), ltyIC = 1:(length(x$focal.names)
  	+ 1), save.plot = FALSE, save.options = c("plot", "default", "pdf"),
      ref.name = NULL, ...)

Arguments

`Data`	numeric: either the data matrix only, or the data matrix plus the vector of group membership. See Details.
`group`	numeric or character: either the vector of group membership or the column indicator (within `Data`) of group membership. See Details.
`focal.names`	numeric or character vector indicating the levels of `group` which correspond to the focal groups.
`model`	character: the IRT model to be fitted (either `"1PL"`, `"2PL"` or `"3PL"`).
`c`	optional numeric value or vector giving the values of the constrained pseudo-guessing parameters. See Details.
`engine`	character: the engine for estimating the 1PL model, either `"ltm"` (default) or `"lme4"`.
`discr`	either `NULL` or a real positive value for the common discrimination parameter (default is 1). Used onlky if `model` is `"1PL"` and `engine` is `"ltm"`. See Details.
`irtParam`	matrix with 2J rows (where J is the number of items) and at most 9 columns containing item parameters estimates. See Details.
`nrFocal`	numeric: the number of focal groups (default is 2).
`same.scale`	logical: are the item parameters of the `irtParam` matrix on the same scale? (default is `TRUE`). See Details.
`anchor`	either `NULL` (default) or a vector of item names (or identifiers) to specify the anchor items. See Details.
`alpha`	numeric: significance level (default is 0.05).
`purify`	logical: should the method be used iteratively to purify the set of anchor items? (default is FALSE).
`nrIter`	numeric: the maximal number of iterations in the item purification process (default is 10).
`p.adjust.method`	either `NULL` (default) or the acronym of the method for p-value adjustment for multiple comparisons. See Details.
`save.output`	logical: should the output be saved into a text file? (Default is `FALSE`).
`output`	character: a vector of two components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value). See Details.
`x`	the result from a `GenLord` class object.
`plot`	character: the type of plot, either `"lordStat"` or `"itemCurve"`. See Details.
`item`	numeric or character: either the number or the name of the item for which ICC curves are plotted. Used only when `plot="itemCurve"`.
`pch`, `col`	type of usual `pch` and `col` graphical options.
`number`	logical: should the item number identification be printed (default is `TRUE`).
`colIC`, `ltyIC`	vectors of elements of the usual `col` and `lty` arguments for ICC curves. Used only when `plot="itemCurve"`.
`save.plot`	logical: should the plot be saved into a separate file? (default is `FALSE`).
`save.options`	character: a vector of three components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value), and the third component is the file extension, either `"pdf"` (default) or `"jpeg"`. See Details.
`ref.name`	either `NULL`(default) or a character string for the name of the reference group (to be used instead of "Reference" in the legend). Ignored if `plot` is `"lordStat"`.
`...`	other generic parameters for the `plot` or the `print` functions.

Details

The generalized Lord's chi-squared method (Kim, Cohen and Park, 1995), also referred to as Qj statistic, allows for detecting uniform or non-uniform differential item functioning among multiple groups by setting an appropriate item response model. The input can be of two kinds: either by displaying the full data, the group membership, the focal groups and the model, or by giving the item parameter estimates (with the option irtParam). Both can be supplied, but in this case only the parameters in irtParam are used for computing generalized Lord's chi-squared statistic.

Missing values are allowed for item responses (not for group membership) but must be coded as NA values. They are discarded for item parameter estimation.

The vector of group membership must hold at least three different values, either as numeric or character. The focal groups are defined by the values of the argument focal.names.

With the "1PL" model and the "ltm" engine, the common discrimination parameter is set equal to 1 by default. It is possible to fix another value through the argumentdiscr. Alternatively, this common discrimination parameter can be estimated (though not returned) by fixing discr to NULL.

The 3PL model can be fitted either unconstrained (by setting c to NULL) or by fixing the pseudo-guessing values. In the latter case, the argument c is either a numeric vector of same length of the number of items, with one value per item pseudo-guessing parameter, or a single value which is duplicated for all the items. If c is different from NULL then the 3PL model is always fitted (whatever the value of model).

The irtParam matrix has a number of rows equal to the number of groups (reference and focal ones) times the number of items J. The first J rows refer to the item parameter estimates in the reference group, while the next sets of J rows correspond to the same items in each of the focal groups. The number of columns depends on the selected IRT model: 2 for the 1PL model, 5 for the 2PL model, 6 for the constrained 3PL model and 9 for the unconstrained 3PL model. The columns of irtParam have to follow the same structure as the output of itemParEst command (the latter can actually be used to create the irtParam matrix). The number of focal groups has to be specified with argument nrFocal (default value is 2).

In addition to the matrix of parameter estimates, one has to specify whether items in the focal groups were rescaled to those of the reference group. If not, rescaling is performed by equal means anchoring (Cook and Eignor, 1991). Argument same.scale is used for this choice (default option is TRUE and assumes therefore that the parameters are already placed on a same scale).

The threshold (or cut-score) for classifying items as DIF is computed as the quantile of the chi-squared distribution with lower-tail probability of one minus alpha and p degrees of freedom. The value of p is the product of the number of focal groups by the number of item parameters to be tested (1 for the 1PL model, 2 for the 2PL model or the constrained 3PL model, and 3 for the unconstrained 3PL model).

Item purification can be performed by setting purify to TRUE. In this case, the purification occurs in the equal means anchoring process: items detected as DIF are iteratively removed from the set of items used for equal means anchoring, and the procedure is repeated until either the same items are identified twice as functioning differently, or when nrIter iterations have been performed. In the latter case a warning message is printed. See Candell and Drasgow (1988) for further details.

A pre-specified set of anchor items can be provided through the anchor argument. It must be a vector of either item names (which must match exactly the column names of Data argument) or integer values (specifying the column numbers for item identification). In case anchor items are provided, they are used to rescale the item parameters on a common metric. None of the anchor items are tested for DIF: the output separates anchor items and tested items and DIF results are returned only for the latter. Note also that item purification is not activated when anchor items are provided (even if purify is set to TRUE). By default it is NULL so that no anchor item is specified. If item parameters are provided thorugh the irtParam argument and if they are on the same scale (i.e. if same.scale is TRUE), then anchor items are not used (even if they are specified).

The output of the difGenLord, as displayed by the print.GenLord function, can be stored in a text file provided that save.output is set to TRUE (the default value FALSE does not execute the storage). In this case, the name of the text file must be given as a character string into the first component of the output argument (default name is "out"), and the path for saving the text file can be given through the second component of output. The default value is "default", meaning that the file will be saved in the current working directory. Any other path can be specified as a character string: see the Examples section for an illustration.

Two types of plots are available. The first one is obtained by setting plot="lordStat" and it is the default option. The chi-squared statistics are displayed on the Y axis, for each item. The detection threshold is displayed by a horizontal line, and items flagged as DIF are printed with the color defined by argument col. By default, items are spotted with their number identification (number=TRUE); otherwise they are simply drawn as dots whose form is given by the option pch.

The other type of plot is obtained by setting plot="itemCurve". In this case, the fitted ICC curves are displayed for one specific item set by the argument item. The latter argument can hold either the name of the item or its number identification. The item parameters are extracted from the itemParFinal matrix if the output argument purification is TRUE, otherwise from the itemParInit matrix and after a rescaling of the item parameters using the itemRescale command. A legend is displayed in the upper left corner of the plot. The colors and types of traits for these curves are defined by means of the arguments colIC and ltyIC respectively. These are set as vectors of length 2, the first element for the reference group and the second for the focal group. Finally, the ref.name argument permits to display the name if the reference group (instead of "Reference") in the legend.

Value

A list of class "GenLord" with the following arguments:

`genLordChi`	the values of the generalized Lord's chi-squared statistics.
`p.value`	the vector of p-values for the generalized Lord's chi-square statistics.
`alpha`	the value of `alpha` argument.
`thr`	the threshold (cut-score) for DIF detection.
`df`	the degrees of freedom of the asymptotic null distribution of the statistics.
`DIFitems`	either the column indicators of the items which were detected as DIF items, or "No DIF item detected".
`p.adjust.method`	the value of the `p.adjust.method` argument.
`adjusted.p`	either `NULL` or the vector of adjusted p-values for multiple comparisons.
`purification`	the value of `purify` option.
`nrPur`	the number of iterations in the item purification process. Returned only if `purify` is `TRUE`.
`difPur`	a binary matrix with one row per iteration in the item purification process and one column per item. Zeros and ones in the i-th row refer to items which were classified respectively as non-DIF and DIF items at the (i-1)-th step. The first row corresponds to the initial classification of the items. Returned only if `purify` is `TRUE`.
`convergence`	logical indicating whether the iterative item purification process stopped before the maximal number `nrIter`of allowed iterations. Returned only if `purify` is `TRUE`.
`model`	the value of `model` argument.
`c`	The value of the `c` argument.
`engine`	The value of the `engine` argument.
`discr`	the value of the `discr` argument.
`itemParInit`	the matrix of initial parameter estimates, with the same format as `irtParam` either provided by the user (through `irtParam`) or estimated from the data (and displayed after rescaling).
`itemParFinal`	the matrix of final parameter estimates, with the same format as `irtParam`, obtained after item purification. Returned only if `purify` is `TRUE`.
`estPar`	a logical value indicating whether the item parameters were estimated (`TRUE`) or provided by the user (`FALSE`).
`names`	the names of the items.
`anchor.names`	the value of the `anchor` argument.
`focal.names`	the value of the `focal.names` argument.
`save.output`	the value of the `save.output` argument.
`output`	the value of the `output` argument.

Author(s)

References

Bates, D. and Maechler, M. (2009). lme4: Linear mixed-effects models using S4 classes. R package version 0.999375-31. http://CRAN.R-project.org/package=lme4

Candell, G.L. and Drasgow, F. (1988). An iterative procedure for linking metrics and assessing item bias in item response theory. Applied Psychological Measurement, 12, 253-260.

Cook, L. L. and Eignor, D. R. (1991). An NCME instructional module on IRT equating methods. Educational Measurement: Issues and Practice, 10, 37-45.

Rizopoulos, D. (2006). ltm: An R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17, 1–25. doi:10.18637/jss.v017.i05

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Creating four groups according to gender ("Man" or "Woman") and trait
 # anger score ("Low" or "High")
 group <- rep("WomanLow",nrow(verbal))
 group[Anger>20 & Gender==0] <- "WomanHigh"
 group[Anger<=20 & Gender==1] <- "ManLow"
 group[Anger>20 & Gender==1] <- "ManHigh"

 # New data set
 Verbal <- cbind(verbal[,1:24], group)

 # Reference group: "WomanLow"
 names <- c("WomanHigh", "ManLow", "ManHigh")

 # Three equivalent settings of the data matrix and the group membership
 # 1PL model, "ltm" engine 
 r <- difGenLord(Verbal, group = 25, focal.names = names, model = "1PL")
 difGenLord(Verbal, group = "group", focal.name = names, model = "1PL")
 difGenLord(Verbal[,1:24], group = Verbal[,25], focal.names = names, model = "1PL")

 # 1PL model, "ltm" engine, estimated common discrimination 
 r <- difGenLord(Verbal, group = 25, focal.names = names, model = "1PL", discr = NULL)

 # 1PL model, "lme4" engine 
 difGenLord(Verbal, group = "group", focal.name = names, model = "1PL", engine = "lme4")

 # With items 1 to 5 set as anchor items
 difGenLord(Verbal, group = 25, focal.names = names, model = "1PL", anchor = 1:5)

 # Multiple comparisons adjustment using Benjamini-Hochberg method
 difGenLord(Verbal, group = 25, focal.names = names, model = "1PL", p.adjust.method = "BH")

 # With item purification
 difGenLord(Verbal, group = 25, focal.names = names, model = "1PL", purify = TRUE)

 # Saving the output into the "GLresults.txt" file (and default path)
 r <- difGenLord(Verbal, group = 25, focal.names = names, model = "1PL", 
         save.output = TRUE, output = c("GLresults", "default"))

 # Splitting the data into the four subsets according to "group"
 data0<-data1<-data2<-data3<-NULL
 for (i in 1:nrow(verbal)){
  if (group[i]=="WomanLow") data0<-rbind(data0,as.numeric(verbal[i,1:24]))
  if (group[i]=="WomanHigh") data1<-rbind(data1,as.numeric(verbal[i,1:24]))
  if (group[i]=="ManLow") data2<-rbind(data2,as.numeric(verbal[i,1:24]))
  if (group[i]=="ManHigh") data3<-rbind(data3,as.numeric(verbal[i,1:24]))
  }

 # Estimation of the item parameters (1PL model)
 m0.1PL<-itemParEst(data0, model = "1PL")
 m1.1PL<-itemParEst(data1, model = "1PL")
 m2.1PL<-itemParEst(data2, model = "1PL")
 m3.1PL<-itemParEst(data3, model = "1PL")

 # Merging the item parameters WITHOUT rescaling
 irt.noscale<-rbind(m0.1PL,m1.1PL,m2.1PL,m3.1PL)
 rownames(irt.noscale)<-rep(colnames(verbal[,1:24]),4)

 # Merging the item parameters WITH rescaling
 irt.scale<-rbind(m0.1PL, itemRescale(m0.1PL,m1.1PL),
 itemRescale(m0.1PL,m2.1PL) ,itemRescale(m0.1PL,m3.1PL))
 rownames(irt.scale)<-rep(colnames(verbal[,1:24]),4)

 # Equivalent calculations
 difGenLord(irtParam = irt.noscale, nrFocal = 3, same.scale = FALSE)
 difGenLord(irtParam = irt.scale, nrFocal = 3, same.scale = TRUE)

 # With item purification
 difGenLord(irtParam = irt.noscale, nrFocal = 3, same.scale = FALSE, purify = TRUE)

 # Graphical devices
 plot(r)
 plot(r, plot = "itemCurve", item = 1)
 plot(r, plot = "itemCurve", item = 6)
 plot(r, plot = "itemCurve", item = 6, ref.name = "WomanHigh")

 # Plotting results and saving it in a PDF figure
 plot(r, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Creating four groups according to gender ("Man" or "Woman") and trait
 # anger score ("Low" or "High")
 group <- rep("WomanLow",nrow(verbal))
 group[Anger>20 & Gender==0] <- "WomanHigh"
 group[Anger<=20 & Gender==1] <- "ManLow"
 group[Anger>20 & Gender==1] <- "ManHigh"

 # New data set
 Verbal <- cbind(verbal[,1:24], group)

 # Reference group: "WomanLow"
 names <- c("WomanHigh", "ManLow", "ManHigh")

 # Three equivalent settings of the data matrix and the group membership
 # 1PL model, "ltm" engine 
 r <- difGenLord(Verbal, group = 25, focal.names = names, model = "1PL")
 difGenLord(Verbal, group = "group", focal.name = names, model = "1PL")
 difGenLord(Verbal[,1:24], group = Verbal[,25], focal.names = names, model = "1PL")

 # 1PL model, "ltm" engine, estimated common discrimination 
 r <- difGenLord(Verbal, group = 25, focal.names = names, model = "1PL", discr = NULL)

 # 1PL model, "lme4" engine 
 difGenLord(Verbal, group = "group", focal.name = names, model = "1PL", engine = "lme4")

 # With items 1 to 5 set as anchor items
 difGenLord(Verbal, group = 25, focal.names = names, model = "1PL", anchor = 1:5)

 # Multiple comparisons adjustment using Benjamini-Hochberg method
 difGenLord(Verbal, group = 25, focal.names = names, model = "1PL", p.adjust.method = "BH")

 # With item purification
 difGenLord(Verbal, group = 25, focal.names = names, model = "1PL", purify = TRUE)

 # Saving the output into the "GLresults.txt" file (and default path)
 r <- difGenLord(Verbal, group = 25, focal.names = names, model = "1PL", 
         save.output = TRUE, output = c("GLresults", "default"))

 # Splitting the data into the four subsets according to "group"
 data0<-data1<-data2<-data3<-NULL
 for (i in 1:nrow(verbal)){
  if (group[i]=="WomanLow") data0<-rbind(data0,as.numeric(verbal[i,1:24]))
  if (group[i]=="WomanHigh") data1<-rbind(data1,as.numeric(verbal[i,1:24]))
  if (group[i]=="ManLow") data2<-rbind(data2,as.numeric(verbal[i,1:24]))
  if (group[i]=="ManHigh") data3<-rbind(data3,as.numeric(verbal[i,1:24]))
  }

 # Estimation of the item parameters (1PL model)
 m0.1PL<-itemParEst(data0, model = "1PL")
 m1.1PL<-itemParEst(data1, model = "1PL")
 m2.1PL<-itemParEst(data2, model = "1PL")
 m3.1PL<-itemParEst(data3, model = "1PL")

 # Merging the item parameters WITHOUT rescaling
 irt.noscale<-rbind(m0.1PL,m1.1PL,m2.1PL,m3.1PL)
 rownames(irt.noscale)<-rep(colnames(verbal[,1:24]),4)

 # Merging the item parameters WITH rescaling
 irt.scale<-rbind(m0.1PL, itemRescale(m0.1PL,m1.1PL),
 itemRescale(m0.1PL,m2.1PL) ,itemRescale(m0.1PL,m3.1PL))
 rownames(irt.scale)<-rep(colnames(verbal[,1:24]),4)

 # Equivalent calculations
 difGenLord(irtParam = irt.noscale, nrFocal = 3, same.scale = FALSE)
 difGenLord(irtParam = irt.scale, nrFocal = 3, same.scale = TRUE)

 # With item purification
 difGenLord(irtParam = irt.noscale, nrFocal = 3, same.scale = FALSE, purify = TRUE)

 # Graphical devices
 plot(r)
 plot(r, plot = "itemCurve", item = 1)
 plot(r, plot = "itemCurve", item = 6)
 plot(r, plot = "itemCurve", item = 6, ref.name = "WomanHigh")

 # Plotting results and saving it in a PDF figure
 plot(r, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

## End(Not run)

Generalized Mantel-Haenszel DIF method

Description

Performs DIF detection among multiple groups using the generalized Mantel-Haenszel method.

Usage

difGMH(Data, group, focal.names, anchor = NULL, match = "score", alpha = 0.05, 
  	purify = FALSE, nrIter = 10, p.adjust.method = NULL, save.output = FALSE, 
  	output = c("out", "default"))
## S3 method for class 'GMH'
print(x, ...)
## S3 method for class 'GMH'
plot(x, pch = 8, number = TRUE, col = "red", save.plot = FALSE, 
  	save.options = c("plot", "default", "pdf"), ...)
difGMH(Data, group, focal.names, anchor = NULL, match = "score", alpha = 0.05, 
  	purify = FALSE, nrIter = 10, p.adjust.method = NULL, save.output = FALSE, 
  	output = c("out", "default"))
## S3 method for class 'GMH'
print(x, ...)
## S3 method for class 'GMH'
plot(x, pch = 8, number = TRUE, col = "red", save.plot = FALSE, 
  	save.options = c("plot", "default", "pdf"), ...)

Arguments

`Data`	numeric: either the data matrix only, or the data matrix plus the vector of group membership. See Details.
`group`	numeric or character: either the vector of group membership or the column indicator (within `Data`) of group membership. See Details.
`focal.names`	numeric or character vector indicating the levels of `group` which correspond to the focal groups.
`anchor`	either `NULL` (default) or a vector of item names (or identifiers) to specify the anchor items. See Details.
`match`	specifies the type of matching criterion. Can be either `"score"` (default) to compute the test score, or any continuous or discrete variable with the same length as the number of rows of `Data`. See Details.
`alpha`	numeric: significance level (default is 0.05).
`purify`	logical: should the method be used iteratively to purify the set of anchor items? (default is FALSE).
`nrIter`	numeric: the maximal number of iterations in the item purification process (default is 10).
`p.adjust.method`	either `NULL` (default) or the acronym of the method for p-value adjustment for multiple comparisons. See Details.
`save.output`	logical: should the output be saved into a text file? (Default is `FALSE`).
`output`	character: a vector of two components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value). See Details.
`x`	the result from a `GMH` class object.
`pch`, `col`	type of usual `pch` and `col` graphical options.
`number`	logical: should the item number identification be printed (default is `TRUE`).
`save.plot`	logical: should the plot be saved into a separate file? (default is `FALSE`).
`save.options`	character: a vector of three components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value), and the third component is the file extension, either `"pdf"` (default) or `"jpeg"`. See Details.
`...`	other generic parameters for the `plot` or the `print` functions.

Details

The generalized Mantel-Haenszel statistic (Somes, 1986) can be used to detect uniform differential item functioning among multiple groups, without requiring an item response model approach (Penfield, 2001).

Missing values are allowed for item responses (not for group membership) but must be coded as NA values. They are discarded from sum-score computation.

The vector of group membership must hold at least three value, either as numeric or character. The focal groups are defined by the values of the argument focal.names. If there is a unique focal group, then difGMH returns the output of difMH (without continuity correction).

The threshold (or cut-score) for classifying items as DIF is computed as the quantile of the chi-squared distribution with lower-tail probability of one minus alpha and with as many degrees of freedom as the number of focal groups.

The matching criterion can be either the test score or any other continuous or discrete variable to be passed in the genMantelHaenszel function. This is specified by the match argument. By default, it takes the value "score" and the test score (i.e. raw score) is computed. The second option is to assign to match a vector of continuous or discrete numeric values, which acts as the matching criterion. Note that for consistency this vector should not belong to the Data matrix.

Item purification can be performed by setting purify to TRUE. Purification works as follows: if at least one item detected as functioning differently at the first step of the process, then the data set of the next step consists in all items that are currently anchor (DIF free) items, plus the tested item (if necessary). The process stops when either two successive applications of the method yield the same classifications of the items (Clauser and Mazor, 1998), or when nrIter iterations are run without obtaining two successive identical classifications. In the latter case a warning message is printed.

The output of the difGMH, as displayed by the print.GMH function, can be stored in a text file provided that save.output is set to TRUE (the default value FALSE does not execute the storage). In this case, the name of the text file must be given as a character string into the first component of the output argument (default name is "out"), and the path for saving the text file can be given through the second component of output. The default value is "default", meaning that the file will be saved in the current working directory. Any other path can be specified as a character string: see the Examples section for an illustration.

The plot.GMH function displays the DIF statistics in a plot, with each item on the X axis. The type of point and the colour are fixed by the usual pch and col arguments. Option number permits to display the item numbers instead. Also, the plot can be stored in a figure file, either in PDF or JPEG format. Fixing save.plot to TRUE allows this process. The figure is defined through the components of save.options. The first two components perform similarly as those of the output argument. The third component is the figure format, with allowed values "pdf" (default) for PDF file and "jpeg" for JPEG file.

Value

A list of class "GMH" with the following arguments:

`GMH`	the values of the generalized Mantel-Haenszel statistics.
`p.value`	the vector of p-values for the generalized Mantel-Haenszel statistics.
`alpha`	the value of `alpha` argument.
`thr`	the threshold (cut-score) for DIF detection.
`DIFitems`	either the items which were detected as DIF items, or "No DIF item detected".
`match`	a character string, either `"score"` or `"matching variable"` depending on the `match` argument.
`p.adjust.method`	the value of the `p.adjust.method` argument.
`adjusted.p`	either `NULL` or the vector of adjusted p-values for multiple comparisons.
`purification`	the value of `purify` option.
`nrPur`	the number of iterations in the item purification process. Returned only if `purify` is `TRUE`.
`difPur`	a binary matrix with one row per iteration in the item purification process and one column per item. Zeros and ones in the i-th row refer to items which were classified respectively as non-DIF and DIF items at the (i-1)-th step. The first row corresponds to the initial classification of the items. Returned only if `purify` is `TRUE`.
`convergence`	logical indicating whether the iterative item purification process stopped before the maximal number `nrIter` of allowed iterations. Returned only if `purify` is `TRUE`.
`names`	the names of the items.
`anchor.names`	the value of the `anchor` argument.
`focal.names`	the value of `focal.names` argument.
`save.output`	the value of the `save.output` argument.
`output`	the value of the `output` argument.

Author(s)

References

Clauser, B. E. and Mazor, K. M. (1998). Using statistical procedures to identify differential item functioning test items. Educational Measurement: Issues and Practice, 17, 31-44.

Somes, G. W. (1986). The generalized Mantel-Haenszel statistic. The American Statistician, 40, 106-108. doi:10.2307/2684866

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Creating four groups according to gender ("Man" or "Woman") and
 # trait anger score ("Low" or "High")
 group <- rep("WomanLow",nrow(verbal))
 group[Anger>20 & Gender==0] <- "WomanHigh"
 group[Anger<=20 & Gender==1] <- "ManLow"
 group[Anger>20 & Gender==1] <- "ManHigh"

 # New data set
 Verbal <- cbind(verbal[,1:24], group)

 # Reference group: "WomanLow"
 names <- c("WomanHigh", "ManLow", "ManHigh")

 # Three equivalent settings of the data matrix and the group membership
 difGMH(Verbal, group = 25, focal.names = names)
 difGMH(Verbal, group = "group", focal.name = names)
 difGMH(Verbal[,1:24], group = Verbal[,25], focal.names = names)

 # Multiple comparisons adjustment using Benjamini-Hochberg method
 difGMH(Verbal, group = 25, focal.names = names, p.adjust.method = "BH")

 # With item purification 
 difGMH(Verbal, group = 25, focal.names = names, purify = TRUE)
 difGMH(Verbal, group = 25, focal.names = names, purify = TRUE, nrIter = 5)

 # With items 1 to 5 set as anchor items
 difMH(Verbal, group = 25, focal.name = names, anchor = 1:5)
 difMH(Verbal, group = 25, focal.name = names, anchor = 1:5, purify = TRUE)


 # Saving the output into the "GMHresults.txt" file (and default path)
 r <- difGMH(Verbal, group = 25, focal.name = names, save.output = TRUE, 
            output = c("GMHresults","default"))

 # Graphical devices
 plot(r)

 # Plotting results and saving it in a PDF figure
 plot(r, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

## End(Not run)
## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Creating four groups according to gender ("Man" or "Woman") and
 # trait anger score ("Low" or "High")
 group <- rep("WomanLow",nrow(verbal))
 group[Anger>20 & Gender==0] <- "WomanHigh"
 group[Anger<=20 & Gender==1] <- "ManLow"
 group[Anger>20 & Gender==1] <- "ManHigh"

 # New data set
 Verbal <- cbind(verbal[,1:24], group)

 # Reference group: "WomanLow"
 names <- c("WomanHigh", "ManLow", "ManHigh")

 # Three equivalent settings of the data matrix and the group membership
 difGMH(Verbal, group = 25, focal.names = names)
 difGMH(Verbal, group = "group", focal.name = names)
 difGMH(Verbal[,1:24], group = Verbal[,25], focal.names = names)

 # Multiple comparisons adjustment using Benjamini-Hochberg method
 difGMH(Verbal, group = 25, focal.names = names, p.adjust.method = "BH")

 # With item purification 
 difGMH(Verbal, group = 25, focal.names = names, purify = TRUE)
 difGMH(Verbal, group = 25, focal.names = names, purify = TRUE, nrIter = 5)

 # With items 1 to 5 set as anchor items
 difMH(Verbal, group = 25, focal.name = names, anchor = 1:5)
 difMH(Verbal, group = 25, focal.name = names, anchor = 1:5, purify = TRUE)


 # Saving the output into the "GMHresults.txt" file (and default path)
 r <- difGMH(Verbal, group = 25, focal.name = names, save.output = TRUE, 
            output = c("GMHresults","default"))

 # Graphical devices
 plot(r)

 # Plotting results and saving it in a PDF figure
 plot(r, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

## End(Not run)

Logistic regression DIF method

Description

Performs DIF detection using logistic regression method.

Usage

difLogistic(Data, group, focal.name, anchor = NULL, member.type = "group", 
 	match = "score", type = "both", criterion = "LRT", alpha = 0.05, 
 	all.cov = FALSE, purify = FALSE, nrIter = 10, p.adjust.method = NULL, 
 	save.output = FALSE, output = c("out", "default"))
## S3 method for class 'Logistic'
print(x, ...)
## S3 method for class 'Logistic'
plot(x, plot="lrStat", item = 1, itemFit = "best", pch = 8, number = TRUE,
 	col = "red", colIC = rep("black", 2), ltyIC = c(1, 2), save.plot = FALSE,
 	save.options = c("plot", "default", "pdf"), group.names = NULL, ...)
 difLogistic(Data, group, focal.name, anchor = NULL, member.type = "group", 
 	match = "score", type = "both", criterion = "LRT", alpha = 0.05, 
 	all.cov = FALSE, purify = FALSE, nrIter = 10, p.adjust.method = NULL, 
 	save.output = FALSE, output = c("out", "default"))
## S3 method for class 'Logistic'
print(x, ...)
## S3 method for class 'Logistic'
plot(x, plot="lrStat", item = 1, itemFit = "best", pch = 8, number = TRUE,
 	col = "red", colIC = rep("black", 2), ltyIC = c(1, 2), save.plot = FALSE,
 	save.options = c("plot", "default", "pdf"), group.names = NULL, ...)

Arguments

`Data`	numeric: either the data matrix only, or the data matrix plus the vector of group membership. See Details.
`group`	numeric or character: either the vector of group membership or the column indicator (within `data`) of group membership. See Details.
`focal.name`	numeric or character indicating the level of `group` which corresponds to the focal group. Ignored if `member.type` is not `"group"`.
`anchor`	either `NULL` (default) or a vector of item names (or identifiers) to specify the anchor items. Ignored if `match` is not `"score"`. See Details.
`member.type`	character: either `"group"` (default) to specify that group membership is made of two groups, or `"cont"` to indicate that group membership is based on a continuous criterion. See Details.
`match`	specifies the type of matching criterion. Can be either `"score"` (default) to compute the test score, or any continuous or discrete variable with the same length as the number of rows of `Data`. See Details.
`type`	a character string specifying which DIF effects must be tested. Possible values are `"both"` (default), `"udif"` and `"nudif"`. See Details.
`criterion`	a character string specifying which DIF statistic is computed. Possible values are `"LRT"` (default) or `"Wald"`. See Details.
`alpha`	numeric: significance level (default is 0.05).
`all.cov`	logical: should all covariance matrices of model parameter estimates be returned (as lists) for both nested models and all items? (default is `FALSE`.
`purify`	logical: should the method be used iteratively to purify the set of anchor items? (default is FALSE). Ignored if `match` is not `"score"`.
`nrIter`	numeric: the maximal number of iterations in the item purification process. (default is 10).
`p.adjust.method`	either `NULL` (default) or the acronym of the method for p-value adjustment for multiple comparisons. See Details.
`save.output`	logical: should the output be saved into a text file? (Default is `FALSE`).
`output`	character: a vector of two components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value). See Details.
`x`	the result from a `Logistik` class object.
`plot`	character: the type of plot, either `"lrStat"` (default) or `"itemCurve"`. See Details.
`item`	numeric or character: either the number or the name of the item for which logistic curves are plotted. Used only when `plot="itemCurve"`.
`itemFit`	character: the model to be selected for drawing the item curves. Possible values are `"best"` (default) for drawing from the best of the two models, and `"null"` for using fitted parameters of the null model $M_0$ . Not used if `"plot"` is `"lrStat"`. See Details.
`pch`, `col`	type of usual `pch` and `col` graphical options.
`number`	logical: should the item number identification be printed (default is `TRUE`).
`colIC`, `ltyIC`	vectors of two elements of the usual `col` and `lty` arguments for logistic curves. Used only when `plot="itemCurve"`.
`save.plot`	logical: should the plot be saved into a separate file? (default is `FALSE`).
`save.options`	character: a vector of three components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value), and the third component is the file extension, either `"pdf"` (default) or `"jpeg"`. See Details.
`group.names`	either `NULL` (default) or a vector of two character strings giving the names of the reference group and the focal group (in this order) for display in the legend. Ignored if `plot` is `"lrStat"`.
`...`	other generic parameters for the `plot` or the `print` functions.

Details

The logistic regression method (Swaminathan and Rogers, 1990) allows for detecting both uniform and non-uniform differential item functioning without requiring an item response model approach. It consists in fitting a logistic model with the matching criterion, the group membership and an interaction between both as covariates. The statistical significance of the parameters related to group membership and the group-score interaction is then evaluated by means of either the likelihood-ratio test or the Wald test. The argument type permits to test either both uniform and nonuniform effects simultaneously (type="both"), only uniform DIF effect (type="udif") or only nonuniform DIF effect (type="nudif"). The argument criterion permits to select either the likelihood ratio test (criterion=="LRT") or the Wald test (criterion=="Wald"). See Logistik for further details.

The group membership can be either a vector of two distinct values, one for the reference group and one for the focal group, or a continuous or discrete variable that acts as the "group" membership variable. In the former case, the member.type argument is set to "group" and the focal.name defines which value in the group variable stands for the focal group. In the latter case, member.type is set to "cont", focal.name is ignored and each value of the group represents one "group" of data (that is, the DIF effects are investigated among participants relying on different values of some discrete or continuous trait). See Logistik for further details.

The matching criterion can be either the test score or any other continuous or discrete variable to be passed in the Logistik function. This is specified by the match argument. By default, it takes the value "score" and the test score (i.e. raw score) is computed. The second option is to assign to match a vector of continuous or discrete numeric values, which acts as the matching criterion. Note that for consistency this vector should not belong to the Data matrix.

Missing values are allowed for item responses (not for group membership) but must be coded as NA values. They are discarded from the fitting of the logistic models (see glm for further details).

The threshold (or cut-score) for classifying items as DIF is computed as the quantile of the chi-squared distribution with lower-tail probability of one minus alpha and with one (if type="udif" or type="nudif") or two (if type="both") degrees of freedom.

A pre-specified set of anchor items can be provided through the anchor argument. It must be a vector of either item names (which must match exactly the column names of Data argument) or integer values (specifying the column numbers for item identification). In case anchor items are provided, they are used to compute the test score (matching criterion), including also the tested item. None of the anchor items are tested for DIF: the output separates anchor items and tested items and DIF results are returned only for the latter. By default it is NULL so that no anchor item is specified. Note also that item purification is not activated when anchor items are provided (even if purify is set to TRUE). Moreover, if the match argument is not set to "score", anchor items will not be taken into account even if anchor is not NULL.

The output of the difLogistic, as displayed by the print.Logistic function, can be stored in a text file provided that save.output is set to TRUE (the default value FALSE does not execute the storage). In this case, the name of the text file must be given as a character string into the first component of the output argument (default name is "out"), and the path for saving the text file can be given through the second component of output. The default value is "default", meaning that the file will be saved in the current working directory. Any other path can be specified as a character string: see the Examples section for an illustration.

The other type of plot is obtained by setting plot="itemCurve". In this case, the fitted logistic curves are displayed for one specific item set by the argument item. The latter argument can hold either the name of the item or its number identification. If the argument itemFit takes the value "best", the curves are drawn according to the output of the best model among $M_0$ and $M_1$ . That is, two curves are drawn if the item is flagged as DIF, and only one if the item is flagged as non-DIF. If itemFit takes the value "null", then the two curves are drawn from the fitted parameters of the null model $M_0$ . See Logistik for further details on the models. The colors and types of traits for these curves are defined by means of the arguments colIC and ltyIC respectively. These are set as vectors of length 2, the first element for the reference group and the second for the focal group. Finally, the argument group.names permits to display the names of the reference and focal groups (instead of "Reference" and "Focal") in the legend.

Value

A list of class "Logistic" with the following arguments:

`Logistik`	the values of the logistic regression statistics.
`p.value`	the vector of p-values for the logistic regression statistics.
`logitPar`	a matrix with one row per item and four columns, holding the fitted parameters of the best model (among the two tested models) for each item.
`logitSe`	a matrix with one row per item and four columns, holding the standard errors of the fitted parameters of the best model (among the two tested models) for each item.
`parM0`	the matrix of fitted parameters of the null model $M_0$ , as returned by the `Logistik` command.
`seM0`	the matrix of standard error of fitted parameters of the null model $M_0$ , as returned by the `Logistik` command.
`cov.M0`	either `NULL` (if `all.cov` argument is `FALSE`) or a list of covariance matrices of parameter estimates of the "full" model ( $M_0$ ) for each item (if `all.cov` argument is `TRUE`).
`cov.M1`	either `NULL` (if `all.cov` argument is `FALSE`) or a list of covariance matrices of parameter estimates of the "reduced" model ( $M_1$ ) for each item (if `all.cov` argument is `TRUE`).
`deltaR2`	the differences in Nagelkerke's $R^2$ coefficients. See Details.
`alpha`	the value of `alpha` argument.
`thr`	the threshold (cut-score) for DIF detection.
`DIFitems`	either the column indicators for the items which were detected as DIF items, or "No DIF item detected".
`member.type`	the value of the `member.type` argument.
`match`	a character string, either `"score"` or `"matching variable"` depending on the `match` argument.
`type`	the value of `type` argument.
`p.adjust.method`	the value of the `p.adjust.method` argument.
`adjusted.p`	either `NULL` or the vector of adjusted p-values for multiple comparisons.
`purification`	the value of `purify` option.
`nrPur`	the number of iterations in the item purification process. Returned only if `purify` is `TRUE`.
`difPur`	a binary matrix with one row per iteration in the item purification process and one column per item. Zeros and ones in the i-th row refer to items which were classified respectively as non-DIF and DIF items at the (i-1)-th step. The first row corresponds to the initial classification of the items. Returned only if `purify` is `TRUE`.
`convergence`	logical indicating whether the iterative item purification process stopped before the maximal number of `nrItem` allowed iterations. Returned only if `purify` is `TRUE`.
`names`	the names of the items.
`anchor.names`	the value of the `anchor` argument.
`criterion`	the value of the `criterion` argument.
`save.output`	the value of the `save.output` argument.
`output`	the value of the `output` argument.

Author(s)

References

Clauser, B.E. and Mazor, K.M. (1998). Using statistical procedures to identify differential item functioning test items. Educational Measurement: Issues and Practice, 17, 31-44.

Finch, W.H. and French, B. (2007). Detection of crossing differential item functioning: a comparison of four methods. Educational and Psychological Measurement, 67, 565-582. doi:10.1177/0013164406296975

Nagelkerke, N. J. D. (1991). A note on a general definition of the coefficient of determination. Biometrika, 78, 691-692. doi:10.1093/biomet/78.3.691

Zumbo, B.D. (1999). A handbook on the theory and methods of differential item functioning (DIF): logistic regression modelling as a unitary framework for binary and Likert-type (ordinal) item scores. Ottawa, ON: Directorate of Human Resources Research and Evaluation, Department of National Defense.

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)

 # Excluding the "Anger" variable
 anger <- verbal[,colnames(verbal)=="Anger"]
 verbal <- verbal[,colnames(verbal)!="Anger"]

 # Testing both DIF effects simultaneously
 # Three equivalent settings of the data matrix and the group membership
 r <- difLogistic(verbal, group=25, focal.name = 1)
 difLogistic(verbal, group = "Gender", focal.name = 1)
 difLogistic(verbal[,1:24], group = verbal[,25], focal.name = 1)

 # Returning all covariance matrices of model parameters
 difLogistic(verbal, group=25, focal.name = 1, all.cov = TRUE)

 # Testing both DIF effects with the Wald test
 r2 <- difLogistic(verbal, group = 25, focal.name = 1, criterion = "Wald")

 # Testing nonuniform DIF effect
 difLogistic(verbal, group = 25, focal.name = 1, type = "nudif")

 # Testing uniform DIF effect
 difLogistic(verbal, group = 25, focal.name = 1, type = "udif")

 # Multiple comparisons adjustment using Benjamini-Hochberg method
 difLogistic(verbal, group=25, focal.name = 1, p.adjust.method = "BH")

 # With item purification
 difLogistic(verbal, group = "Gender", focal.name = 1, purify = TRUE)
 difLogistic(verbal, group = "Gender", focal.name = 1, purify = TRUE, nrIter = 5)

 # With items 1 to 5 set as anchor items
 difLogistic(verbal, group = 25, focal.name = 1, anchor = 1:5)

 # Using anger trait score as the matching criterion
 difLogistic(verbal,group = 25, focal.name = 1,match = anger)

 # Using trait anger score as the group variable (i.e. testing
 # for DIF with respect to trait anger score)
 difLogistic(verbal[,1:24],group = anger,member.type = "cont")

 # Saving the output into the "Lresults.txt" file (and default path)
 r <- difLogistic(verbal, group = 25, focal.name = 1, save.output = TRUE, 
           output = c("Lresults", "default"))

 # Graphical devices
 plot(r)
 plot(r2)
 plot(r, plot = "itemCurve", item = 1)
 plot(r, plot = "itemCurve", item = 1, itemFit = "null")
 plot(r, plot = "itemCurve", item = 6)
 plot(r, plot = "itemCurve", item = 6, itemFit = "null")

 # Plotting results and saving it in a PDF figure
 plot(r, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)

 # Excluding the "Anger" variable
 anger <- verbal[,colnames(verbal)=="Anger"]
 verbal <- verbal[,colnames(verbal)!="Anger"]

 # Testing both DIF effects simultaneously
 # Three equivalent settings of the data matrix and the group membership
 r <- difLogistic(verbal, group=25, focal.name = 1)
 difLogistic(verbal, group = "Gender", focal.name = 1)
 difLogistic(verbal[,1:24], group = verbal[,25], focal.name = 1)

 # Returning all covariance matrices of model parameters
 difLogistic(verbal, group=25, focal.name = 1, all.cov = TRUE)

 # Testing both DIF effects with the Wald test
 r2 <- difLogistic(verbal, group = 25, focal.name = 1, criterion = "Wald")

 # Testing nonuniform DIF effect
 difLogistic(verbal, group = 25, focal.name = 1, type = "nudif")

 # Testing uniform DIF effect
 difLogistic(verbal, group = 25, focal.name = 1, type = "udif")

 # Multiple comparisons adjustment using Benjamini-Hochberg method
 difLogistic(verbal, group=25, focal.name = 1, p.adjust.method = "BH")

 # With item purification
 difLogistic(verbal, group = "Gender", focal.name = 1, purify = TRUE)
 difLogistic(verbal, group = "Gender", focal.name = 1, purify = TRUE, nrIter = 5)

 # With items 1 to 5 set as anchor items
 difLogistic(verbal, group = 25, focal.name = 1, anchor = 1:5)

 # Using anger trait score as the matching criterion
 difLogistic(verbal,group = 25, focal.name = 1,match = anger)

 # Using trait anger score as the group variable (i.e. testing
 # for DIF with respect to trait anger score)
 difLogistic(verbal[,1:24],group = anger,member.type = "cont")

 # Saving the output into the "Lresults.txt" file (and default path)
 r <- difLogistic(verbal, group = 25, focal.name = 1, save.output = TRUE, 
           output = c("Lresults", "default"))

 # Graphical devices
 plot(r)
 plot(r2)
 plot(r, plot = "itemCurve", item = 1)
 plot(r, plot = "itemCurve", item = 1, itemFit = "null")
 plot(r, plot = "itemCurve", item = 6)
 plot(r, plot = "itemCurve", item = 6, itemFit = "null")

 # Plotting results and saving it in a PDF figure
 plot(r, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

## End(Not run)

General logistic regression DIF method

Description

Performs DIF detection using logistic regression method with either two groups, more than two groups, or a continuous group variable.

Usage

difLogReg(Data, group, focal.name, anchor = NULL, group.type = "group", 
 	match = "score", type = "both", criterion = "LRT", alpha = 0.05, 
 	purify = FALSE, nrIter = 10, p.adjust.method = NULL, save.output = FALSE, 
 	output = c("out", "default"))
 difLogReg(Data, group, focal.name, anchor = NULL, group.type = "group", 
 	match = "score", type = "both", criterion = "LRT", alpha = 0.05, 
 	purify = FALSE, nrIter = 10, p.adjust.method = NULL, save.output = FALSE, 
 	output = c("out", "default"))

Arguments

`Data`	numeric: either the data matrix only, or the data matrix plus the vector of group membership. See Details.
`group`	numeric or character: either the vector of group membership or the column indicator (within `data`) of group membership. See Details.
`focal.name`	numeric or character indicating the level(s) of `group` which corresponds to the focal group(s). Ignored if `group.type` is not `"group"`.
`anchor`	either `NULL` (default) or a vector of item names (or identifiers) to specify the anchor items. See Details.
`group.type`	character: either `"group"` (default) to specify that group membership is made of two (or more than two) groups, or `"cont"` to indicate that group membership is based on a continuous criterion. See Details.
`match`	specifies the type of matching criterion. Can be either `"score"` (default) to compute the test score, or any continuous or discrete variable with the same length as the number of rows of `Data`. See Details.
`type`	a character string specifying which DIF effects must be tested. Possible values are `"both"` (default), `"udif"` and `"nudif"`. See Details.
`criterion`	a character string specifying which DIF statistic is computed. Possible values are `"LRT"` (default) or `"Wald"`. See Details.
`alpha`	numeric: significance level (default is 0.05).
`purify`	logical: should the method be used iteratively to purify the set of anchor items? (default is FALSE). Ignored if `match` is not `"score"`.
`nrIter`	numeric: the maximal number of iterations in the item purification process. (default is 10).
`p.adjust.method`	either `NULL` (default) or the acronym of the method for p-value adjustment for multiple comparisons. See Details.
`save.output`	logical: should the output be saved into a text file? (Default is `FALSE`).
`output`	character: a vector of two components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value). See Details.

Details

The difLogReg function is a meta-function for logistic regression DIF analysis. It encompasses all possible cases that are currently implemented in difR and makes appropriate calls to the function difLogistic or difGenLogistic.

Three situations are embedded in this function.

The group membership is defined by two distinct groups. In this case, group.type must be "group" and focal.name must be a single value, referring to the name or label of the focal group.
The group membership is defined by a finite, yet larger than two, number of groups. In this case, group.type must be "group" and focal.name must be a vector with the names or labels of all focal groups.
The group membership is a continuous or discrete (but treated as continuous) variable. In this case, DIF is tested with respect to this "membership" variable. Furthermore, group.type must be "cont" and focal.name is ignored (though some value must be specified, for instance NULL).

The specification of the data, the options for item purification, DIF statistic selection, and output saving, are identical to the options arising from the difLogistic and difGenLogistic functions.

Value

A list of class "Logistic" (if group.type is "cont" or with the length of focal.name is one) or "genLogistic", with related arguments (see difLogistic and difGenLogistic).

Author(s)

References

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Few examples
 difLogReg(Data=verbal[,1:24], group=verbal[,26], focal.name=1)
 difLogReg(Data = verbal[,1:24], group = verbal[,26], focal.name = 1, match = verbal[,25])
 difLogReg(Data = verbal[,1:24], group = verbal[,25], focal.name = 1, group.type = "cont")

 group<-rep("WomanLow",nrow(verbal))
 group[Anger>20 & Gender==0] <- "WomanHigh"
 group[Anger<=20 & Gender==1] <- "ManLow"
 group[Anger>20 & Gender==1] <- "ManHigh"
 names <- c("WomanHigh", "ManLow", "ManHigh")

 difLogReg(Data = verbal[,1:24], group = group, focal.name = names)
 
## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Few examples
 difLogReg(Data=verbal[,1:24], group=verbal[,26], focal.name=1)
 difLogReg(Data = verbal[,1:24], group = verbal[,26], focal.name = 1, match = verbal[,25])
 difLogReg(Data = verbal[,1:24], group = verbal[,25], focal.name = 1, group.type = "cont")

 group<-rep("WomanLow",nrow(verbal))
 group[Anger>20 & Gender==0] <- "WomanHigh"
 group[Anger<=20 & Gender==1] <- "ManLow"
 group[Anger>20 & Gender==1] <- "ManHigh"
 names <- c("WomanHigh", "ManLow", "ManHigh")

 difLogReg(Data = verbal[,1:24], group = group, focal.name = names)
 
## End(Not run)

Lord's chi-squared DIF method

Description

Performs DIF detection using Lord's chi-squared method.

Usage

difLord(Data, group, focal.name, model, c = NULL, engine = "ltm", discr = 1,
 	irtParam = NULL, same.scale = TRUE, anchor = NULL, alpha = 0.05,
 	purify = FALSE, nrIter = 10, p.adjust.method = NULL, save.output = FALSE, 
 	output = c("out", "default"))  	
## S3 method for class 'Lord'
print(x, ...)
## S3 method for class 'Lord'
plot(x, plot = "lordStat", item = 1, pch = 8, number = TRUE, col = "red", 
  	colIC = rep("black", 2), ltyIC = c(1, 2), save.plot = FALSE, 
  	save.options = c("plot", "default", "pdf"), group.names = NULL, ...)  	
 difLord(Data, group, focal.name, model, c = NULL, engine = "ltm", discr = 1,
 	irtParam = NULL, same.scale = TRUE, anchor = NULL, alpha = 0.05,
 	purify = FALSE, nrIter = 10, p.adjust.method = NULL, save.output = FALSE, 
 	output = c("out", "default"))  	
## S3 method for class 'Lord'
print(x, ...)
## S3 method for class 'Lord'
plot(x, plot = "lordStat", item = 1, pch = 8, number = TRUE, col = "red", 
  	colIC = rep("black", 2), ltyIC = c(1, 2), save.plot = FALSE, 
  	save.options = c("plot", "default", "pdf"), group.names = NULL, ...)

Arguments

`Data`	numeric: either the data matrix only, or the data matrix plus the vector of group membership. See Details.
`group`	numeric or character: either the vector of group membership or the column indicator (within `data`) of group membership. See Details.
`focal.name`	numeric or character indicating the level of `group` which corresponds to the focal group.
`model`	character: the IRT model to be fitted (either `"1PL"`, `"2PL"` or `"3PL"`).
`c`	optional numeric value or vector giving the values of the constrained pseudo-guessing parameters. See Details.
`engine`	character: the engine for estimating the 1PL model, either `"ltm"` (default) or `"lme4"`.
`discr`	either `NULL` or a real positive value for the common discrimination parameter (default is 1). Used onlky if `model` is `"1PL"` and `engine` is `"ltm"`. See Details.
`irtParam`	matrix with 2J rows (where J is the number of items) and at most 9 columns containing item parameters estimates. See Details.
`same.scale`	logical: are the item parameters of the `irtParam` matrix on the same scale? (default is "TRUE"). See Details.
`anchor`	either `NULL` (default) or a vector of item names (or identifiers) to specify the anchor items. See Details.
`alpha`	numeric: significance level (default is 0.05).
`purify`	logical: should the method be used iteratively to purify the set of anchor items? (default is FALSE).
`nrIter`	numeric: the maximal number of iterations in the item purification process (default is 10).
`p.adjust.method`	either `NULL` (default) or the acronym of the method for p-value adjustment for multiple comparisons. See Details.
`save.output`	logical: should the output be saved into a text file? (Default is `FALSE`).
`output`	character: a vector of two components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value). See Details.
`x`	the result from a `Lord` class object.
`plot`	character: the type of plot, either `"lordStat"` or `"itemCurve"`. See Details.
`item`	numeric or character: either the number or the name of the item for which ICC curves are plotted. Used only when `plot="itemCurve"`.
`pch`, `col`	type of usual `pch` and `col` graphical options.
`number`	logical: should the item number identification be printed (default is `TRUE`).
`colIC`, `ltyIC`	vectors of two elements of the usual `col` and `lty` arguments for ICC curves. Used only when `plot="itemCurve"`.
`save.plot`	logical: should the plot be saved into a separate file? (default is `FALSE`).
`save.options`	character: a vector of three components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value), and the third component is the file extension, either `"pdf"` (default) or `"jpeg"`. See Details.
`group.names`	either `NULL` (default) or a vector of two character strings giving the names of the reference group and the focal group (in this order) for display in the legend. Ignored if `plot` is `"lordStat"`.
`...`	other generic parameters for the `plot` or the `print` functions.

Details

Lord's chi-squared method (Lord, 1980) allows for detecting uniform or non-uniform differential item functioning by setting an appropriate item response model. The input can be of two kinds: either by displaying the full data, the group membership and the model, or by giving the item parameter estimates (through the option irtParam). Both can be supplied, but in this case only the parameters in irtParam are used for computing Lord's chi-squared statistic.

Missing values are allowed for item responses (not for group membership) but must be coded as NA values. They are discarded for item parameter estimation.

The vector of group membership must hold only two different values, either as numeric or character. The focal group is defined by the value of the argument focal.name.

The 3PL model can be fitted either unconstrained (by setting c to NULL) or by fixing the pseudo-guessing values. In the latter case, the argument c holds either a numeric vector of same length of the number of items, with one value per item pseudo-guessing parameter, or a single value which is duplicated for all the items. If c is different from NULL then the 3PL model is always fitted (whatever the value of model).

The irtParam matrix has a number of rows equal to twice the number of items in the data set. The first J rows refer to the item parameter estimates in the reference group, while the last J ones correspond to the same items in the focal group. The number of columns depends on the selected IRT model: 2 for the 1PL model, 5 for the 2PL model, 6 for the constrained 3PL model and 9 for the unconstrained 3PL model. The columns of irtParam have to follow the same structure as the output of itemParEst command (the latter can actually be used to create the irtParam matrix).

In addition to the matrix of parameter estimates, one has to specify whether items in the focal group were rescaled to those of the reference group. If not, rescaling is performed by equal means anchoring (Cook and Eignor, 1991). Argument same.scale is used for this choice (default option is TRUE and assumes therefore that the parameters are already placed on the same scale).

The threshold (or cut-score) for classifying items as DIF is computed as the quantile of the chi-squared distribution with lower-tail probability of one minus alpha and p degrees of freedom (p=1 for the 1PL model, p=2 for the 2PL model or the 3PL model with constrained pseudo-guessing parameters, and p=3 for the unconstrained 3PL model).

Item purification can be performed by setting purify to TRUE. In this case, the purification occurs in the equal means anchoring process. Items detected as DIF are iteratively removed from the set of items used for equal means anchoring, and the procedure is repeated until either the same items are identified twice as functioning differently, or when nrIter iterations have been performed. In the latter case a warning message is printed. See Candell and Drasgow (1988) for further details. Note that item purification is performed on original statistics and p-values; in case of adjustment for multiple comparisons this is performed after item purification.

A pre-specified set of anchor items can be provided through the anchor argument. It must be a vector of either item names (which must match exactly the column names of Data argument) or integer values (specifying the column numbers for item identification). In case anchor items are provided, they are used to rescale the item parameters on a common metric. None of the anchor items are tested for DIF: the output separates anchor items and tested items and DIF results are returned only for the latter. Note also that item purification is not activated when anchor items are provided (even if purify is set to TRUE). By default it is NULL so that no anchor item is specified. If item parameters are provided thorugh the irtParam argument and if they are on the same scale (i.e. if same.scale is TRUE), then anchor items are not used (even if they are specified).

Under the 1PL model, the displayed output also proposes an effect size measure, which is -2.35 times the difference between item difficulties of the reference group and the focal group (Penfield and Camilli, 2007, p. 138). This effect size is similar Mantel-Haenszel's $\Delta_{MH}$ effect size, and the ETS delta scale is used to classify the effect sizes (Holland and Thayer, 1985).

The output of the difLord, as displayed by the print.Lord function, can be stored in a text file provided that save.output is set to TRUE (the default value FALSE does not execute the storage). In this case, the name of the text file must be given as a character string into the first component of the output argument (default name is "out"), and the path for saving the text file can be given through the second component of output. The default value is "default", meaning that the file will be saved in the current working directory. Any other path can be specified as a character string: see the Examples section for an illustration.

The other type of plot is obtained by setting plot="itemCurve". In this case, the fitted ICC curves are displayed for one specific item set by the argument item. The latter argument can hold either the name of the item or its number identification. The item parameters are extracted from the itemParFinal matrix if the output argument purification is TRUE, otherwise from the itemParInit matrix and after a rescaling of the item parameters using the itemRescale command. A legend is displayed in the upper left corner of the plot. The colors and types of traits for these curves are defined by means of the arguments colIC and ltyIC respectively. These are set as vectors of length 2, the first element for the reference group and the second for the focal group. Finally, the argument group.names permits to display the names of the reference and focal groups (instead of "Reference" and "Focal") in the legend.

Value

A list of class "Lord" with the following arguments:

`LordChi`	the values of the Lord's chi-square statistics.
`p.value`	the vector of p-values for the Lord's chi-square statistics.
`alpha`	the value of `alpha` argument.
`thr`	the threshold (cut-score) for DIF detection.
`DIFitems`	either the column indicators of the items which were detected as DIF items, or "No DIF item detected".
`purification`	the value of `purify` option.
`nrPur`	the number of iterations in the item purification process. Returned only if `purify` is `TRUE`.
`difPur`	a binary matrix with one row per iteration in the item purification process and one column per item. Zeros and ones in the i-th row refer to items which were classified respectively as non-DIF and DIF items at the (i-1)-th step. The first row corresponds to the initial classification of the items. Returned only if `purify` is `TRUE`.
`convergence`	logical indicating whether the iterative item purification process stopped before the maximal number `nrIter`of allowed iterations. Returned only if `purify` is `TRUE`.
`model`	the value of `model` argument.
`c`	The value of the `c` argument.
`engine`	The value of the `engine` argument.
`discr`	the value of the `discr` argument.
`p.adjust.method`	the value of the `p.adjust.method` argument.
`adjusted.p`	either `NULL` or the vector of adjusted p-values for multiple comparisons.
`itemParInit`	the matrix of initial parameter estimates,with the same format as `irtParam` either provided by the user (through `irtParam`) or estimated from the data (and displayed without rescaling).
`itemParFinal`	the matrix of final parameter estimates, with the same format as `irtParam`, obtained after item purification. Returned only if `purify` is `TRUE`.
`estPar`	a logical value indicating whether the item parameters were estimated (`TRUE`) or provided by the user (`FALSE`).
`names`	the names of the items.
`anchor.names`	the value of the `anchor` argument.
`save.output`	the value of the `save.output` argument.
`output`	the value of the `output` argument.

Author(s)

References

Bates, D. and Maechler, M. (2009). lme4: Linear mixed-effects models using S4 classes. R package version 0.999375-31. http://CRAN.R-project.org/package=lme4

Candell, G.L. and Drasgow, F. (1988). An iterative procedure for linking metrics and assessing item bias in item response theory. Applied Psychological Measurement, 12, 253–260. doi:10.1177/014662168801200304

Cook, L. L. and Eignor, D. R. (1991). An NCME instructional module on IRT equating methods. Educational Measurement: Issues and Practice, 10, 37–45.

Holland, P. W. and Thayer, D. T. (1985). An alternative definition of the ETS delta scale of item difficulty. Research Report RR-85-43. Princeton, New-Jersey: Educational Testing Service.

Lord, F. (1980). Applications of item response theory to practical testing problems. Hillsdale, NJ: Lawrence Erlbaum Associates.

Penfield, R. D., and Camilli, G. (2007). Differential item functioning and item bias. In C. R. Rao and S. Sinharray (Eds.), Handbook of Statistics 26: Psychometrics (pp. 125-167). Amsterdam, The Netherlands: Elsevier.

Rizopoulos, D. (2006). ltm: An R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17, 1-25. doi:10.18637/jss.v017.i05

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Excluding the "Anger" variable
 verbal <- verbal[colnames(verbal)!="Anger"]

 # Three equivalent settings of the data matrix and the group membership
 # (1PL model, "ltm" engine) 
 r <- difLord(verbal, group = 25, focal.name = 1, model = "1PL")
 difLord(verbal, group = "Gender", focal.name = 1, model = "1PL")
 difLord(verbal[,1:24], group = verbal[,25], focal.name = 1, model = "1PL")

 # With items 1 to 5 set as anchor items
 difLord(verbal, group = 25, focal.name = 1, model = "1PL", anchor = 1:5)

 # Multiple comparisons adjustment of p-values with Benjamini-Hochberg method
 difLord(verbal, group = 25, focal.name = 1, model = "1PL", anchor = 1:5, p.adjust.method = "BH")


 # 1PL model, "lme4" engine 
 difLord(verbal, group = 25, focal.name = 1, model = "1PL", engine = "lme4")

 # 2PL model   
 difLord(verbal, group = "Gender", focal.name = 1, model = "2PL")

 # 3PL model with all pseudo-guessing parameters constrained to 0.05
 difLord(verbal, group = "Gender", focal.name = 1, model = "3PL", c = 0.05)

 # Same models, with item purification 
 difLord(verbal, group = 25, focal.name = 1, model = "1PL", purify = TRUE)
 difLord(verbal, group = "Gender", focal.name = 1, model = "2PL", purify = TRUE)
 difLord(verbal, group = "Gender", focal.name = 1, model = "3PL", c = 0.05,
 purify = TRUE)

 # Saving the output into the "LordResults.txt" file (and default path)
 r <- difLord(verbal, group = 25, focal.name = 1, model = "1PL",
 	    save.output = TRUE, output = c("LordResults","default"))

 # Splitting the data into reference and focal groups
 nF<-sum(Gender)
 nR<-nrow(verbal)-nF
 data.ref<-verbal[,1:24][order(Gender),][1:nR,]
 data.focal<-verbal[,1:24][order(Gender),][(nR+1):(nR+nF),]

 ## Pre-estimation of the item parameters (1PL model, "ltm" engine)
 item.1PL<-rbind(itemParEst(data.ref, model = "1PL"),
 itemParEst(data.focal, model = "1PL"))
 difLord(irtParam = item.1PL, same.scale = FALSE)

 ## Pre-estimation of the item parameters (1PL model, "lme4" engine)
 item.1PL<-rbind(itemParEst(data.ref, model = "1PL", engine = "lme4"),
 itemParEst(data.focal, model = "1PL", engine = "lme4"))
 difLord(irtParam = item.1PL, same.scale = FALSE)

 ## Pre-estimation of the item parameters (2PL model) 
 item.2PL<-rbind(itemParEst(data.ref, model = "2PL"),
 itemParEst(data.focal, model = "2PL"))
 difLord(irtParam = item.2PL, same.scale = FALSE)

 ## Pre-estimation of the item parameters (constrained 3PL model)
 item.3PL<-rbind(itemParEst(data.ref, model = "3PL", c = 0.05),
 itemParEst(data.focal, model = "3PL", c = 0.05))
 difLord(irtParam = item.3PL, same.scale = FALSE)

 # Graphical devices
 plot(r)
 plot(r, plot = "itemCurve", item = 1)
 plot(r, plot = "itemCurve", item = 6)

 # Plotting results and saving it in a PDF figure
 plot(r, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Excluding the "Anger" variable
 verbal <- verbal[colnames(verbal)!="Anger"]

 # Three equivalent settings of the data matrix and the group membership
 # (1PL model, "ltm" engine) 
 r <- difLord(verbal, group = 25, focal.name = 1, model = "1PL")
 difLord(verbal, group = "Gender", focal.name = 1, model = "1PL")
 difLord(verbal[,1:24], group = verbal[,25], focal.name = 1, model = "1PL")

 # With items 1 to 5 set as anchor items
 difLord(verbal, group = 25, focal.name = 1, model = "1PL", anchor = 1:5)

 # Multiple comparisons adjustment of p-values with Benjamini-Hochberg method
 difLord(verbal, group = 25, focal.name = 1, model = "1PL", anchor = 1:5, p.adjust.method = "BH")


 # 1PL model, "lme4" engine 
 difLord(verbal, group = 25, focal.name = 1, model = "1PL", engine = "lme4")

 # 2PL model   
 difLord(verbal, group = "Gender", focal.name = 1, model = "2PL")

 # 3PL model with all pseudo-guessing parameters constrained to 0.05
 difLord(verbal, group = "Gender", focal.name = 1, model = "3PL", c = 0.05)

 # Same models, with item purification 
 difLord(verbal, group = 25, focal.name = 1, model = "1PL", purify = TRUE)
 difLord(verbal, group = "Gender", focal.name = 1, model = "2PL", purify = TRUE)
 difLord(verbal, group = "Gender", focal.name = 1, model = "3PL", c = 0.05,
 purify = TRUE)

 # Saving the output into the "LordResults.txt" file (and default path)
 r <- difLord(verbal, group = 25, focal.name = 1, model = "1PL",
 	    save.output = TRUE, output = c("LordResults","default"))

 # Splitting the data into reference and focal groups
 nF<-sum(Gender)
 nR<-nrow(verbal)-nF
 data.ref<-verbal[,1:24][order(Gender),][1:nR,]
 data.focal<-verbal[,1:24][order(Gender),][(nR+1):(nR+nF),]

 ## Pre-estimation of the item parameters (1PL model, "ltm" engine)
 item.1PL<-rbind(itemParEst(data.ref, model = "1PL"),
 itemParEst(data.focal, model = "1PL"))
 difLord(irtParam = item.1PL, same.scale = FALSE)

 ## Pre-estimation of the item parameters (1PL model, "lme4" engine)
 item.1PL<-rbind(itemParEst(data.ref, model = "1PL", engine = "lme4"),
 itemParEst(data.focal, model = "1PL", engine = "lme4"))
 difLord(irtParam = item.1PL, same.scale = FALSE)

 ## Pre-estimation of the item parameters (2PL model) 
 item.2PL<-rbind(itemParEst(data.ref, model = "2PL"),
 itemParEst(data.focal, model = "2PL"))
 difLord(irtParam = item.2PL, same.scale = FALSE)

 ## Pre-estimation of the item parameters (constrained 3PL model)
 item.3PL<-rbind(itemParEst(data.ref, model = "3PL", c = 0.05),
 itemParEst(data.focal, model = "3PL", c = 0.05))
 difLord(irtParam = item.3PL, same.scale = FALSE)

 # Graphical devices
 plot(r)
 plot(r, plot = "itemCurve", item = 1)
 plot(r, plot = "itemCurve", item = 6)

 # Plotting results and saving it in a PDF figure
 plot(r, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

## End(Not run)

Likelihood-Ratio Test DIF method

Description

Performs DIF detection using Likelihood Ratio Test (LRT) method.

Usage

difLRT(Data, group, focal.name, alpha = 0.05, purify = FALSE, nrIter = 10, 
 	p.adjust.method = NULL, save.output = FALSE, output = c("out", "default")) 
## S3 method for class 'LRT'
print(x, ...)
## S3 method for class 'LRT'
plot(x, pch = 8, number = TRUE, col = "red", save.plot = FALSE, 
 	save.options = c("plot", "default", "pdf"), ...)

difLRT(Data, group, focal.name, alpha = 0.05, purify = FALSE, nrIter = 10, 
 	p.adjust.method = NULL, save.output = FALSE, output = c("out", "default")) 
## S3 method for class 'LRT'
print(x, ...)
## S3 method for class 'LRT'
plot(x, pch = 8, number = TRUE, col = "red", save.plot = FALSE, 
 	save.options = c("plot", "default", "pdf"), ...)

Arguments

`Data`	numeric: either the data matrix only, or the data matrix plus the vector of group membership. See Details.
`group`	numeric or character: either the vector of group membership or the column indicator (within `data`) of group membership. See Details.
`focal.name`	numeric or character indicating the level of `group` which corresponds to the focal group.
`alpha`	numeric: significance level (default is 0.05).
`purify`	logical: should the method be used iteratively to purify the set of anchor items? (default is FALSE).
`nrIter`	numeric: the maximal number of iterations in the item purification process (default is 10).
`p.adjust.method`	either `NULL` (default) or the acronym of the method for p-value adjustment for multiple comparisons. See Details.
`save.output`	logical: should the output be saved into a text file? (Default is `FALSE`).
`output`	character: a vector of two components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value). See Details.
`x`	the result from a `LRT` class object.
`pch`, `col`	type of usual `pch` and `col` graphical options.
`number`	logical: should the item number identification be printed (default is `TRUE`).
`save.plot`	logical: should the plot be saved into a separate file? (default is `FALSE`).
`save.options`	character: a vector of three components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value), and the third component is the file extension, either `"pdf"` (default) or `"jpeg"`. See Details.
`...`	other generic parameters for the `plot` or the `print` functions.

Details

The likelihood-ratio test method (Thissen, Steinberg and Wainer, 1988) allows for detecting uniform differential item functioning by fitting a closed-form Rasch model and by testing for extra interactions between group membership and item response. Currently only the Rasch model can be used, so only uniform DIF can be detected. Moreover, items are tested one by one and the other items act as anchor items.

The Data is a matrix whose rows correspond to the subjects and columns to the items. Missing values are allowed but must be coded as NA values. In addition, Data can hold the vector of group membership. If so, group indicates the column of Data which corresponds to the group membership, either by specifying its name or by giving the column number. Otherwise, group must be a vector of same length as nrow(Data).

The vector of group membership must hold only two different values, either as numeric or character. The focal group is defined by the value of the argument focal.name.

The function glmer from package lme4 (Bates and Maechler, 2009) is used to fit the closed-form Rasch model. More precisely, the probability that response $Y_{ijg}$ of subject i from group g (focal or reference) to item j is modeled as

$logit (Pr(Y_{ijg}=1) = \theta_{ig} + \gamma_g - \beta_j$

where $\theta_i$ is subject's ability, $\beta_j$ is the item difficulty and $\gamma_g$ is the difference mean ability level between the focal and the reference groups. Subject abilities are treated as random effects, while item difficulties and $\gamma_g$ are treated as fixed effects. Each item is tested by incorporating an interaction term, $\delta_{gj}$ , and by testing its statistical significance using the traditional likelihood-ratio test.

The threshold (or cut-score) for classifying items as DIF is computed as the quantile of the chi-squared distribution with lower-tail probability of one minus alpha and one degree of freedom.

Item purification can be performed by setting purify to TRUE. In this case, items detected as DIF are iteratively removed from the set of tested items, and the procedure is repeated (using the remaining items) until no additional item is identified as functioning differently. The process stops when either there is no new item detected as DIF, or when nrIter iterations are run and new DIF items are nevertheless detected. In the latter case a warning message is printed.

The output of the difLRT, as displayed by the print.LRT function, can be stored in a text file provided that save.output is set to TRUE (the default value FALSE does not execute the storage). In this case, the name of the text file must be given as a character string into the first component of the output argument (default name is "out"), and the path for saving the text file can be given through the second component of output. The default value is "default", meaning that the file will be saved in the current working directory. Any other path can be specified as a character string: see the Examples section for an illustration.

The plot.LRT function displays the DIF statistics in a plot, with each item on the X axis. The type of point and the color are fixed by the usual pch and col arguments. Option number permits to display the item numbers instead. Also, the plot can be stored in a figure file, either in PDF or JPEG format. Fixing save.plot to TRUE allows this process. The figure is defined through the components of save.options. The first two components perform similarly as those of the output argument. The third component is the figure format, with allowed values "pdf" (default) for PDF file and "jpeg" for JPEG file.

Value

A list of class "LRT" with the following arguments:

`LRT`	the values of the likelihood-ratio statistics.
`p.value`	the vector of p-values for the likelihood-ratio statistics.
`alpha`	the value of `alpha` argument.
`thr`	the threshold (cut-score) for DIF detection.
`DIFitems`	either the items which were detected as DIF items, or "No DIF item detected".
`p.adjust.method`	the value of the `p.adjust.method` argument.
`adjusted.p`	either `NULL` or the vector of adjusted p-values for multiple comparisons.
`purification`	the value of `purify` option.
`nrPur`	the number of iterations in the item purification process. Returned only if `purify` is `TRUE`.
`convergence`	logical indicating whether the iterative item purification process stopped before the maximal number of allowed iterations (10 by default). Returned only if `purify` is `TRUE`.
`names`	the names of the items.
`save.output`	the value of the `save.output` argument.
`output`	the value of the `output` argument.

Note

Because of the fitting of the modified Rasch model with glmer, the process can be very time consuming.

Author(s)

References

Bates, D. and Maechler, M. (2009). lme4: Linear mixed-effects models using S4 classes. R package version 0.999375-31. http://CRAN.R-project.org/package=lme4

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Excluding the "Anger" variable
 verbal <- verbal[colnames(verbal)!="Anger"]

 # Keeping the first 5 items and the first 50 subjects
 # (this is an artificial simplification to reduce the computational time)
 verbal <- verbal[1:50, c(1:5, 25)]

 # Three equivalent settings of the data matrix and the group membership
 r <- difLRT(verbal, group = 6, focal.name = 1)
 difLRT(verbal, group = "Gender", focal.name = 1)
 difLRT(verbal[,1:5], group = verbal[,6], focal.name = 1)

 # Multiple comparisons adjustment using Benjamini-Hochberg method
 difLRT(verbal, group = 6, focal.name = 1, p.adjust.method = "BH")

 # With item purification
 difLRT(verbal, group = 6, focal.name = 1, purify = TRUE)

 # Saving the output into the "LRTresults.txt" file (and default path)
 r <- difLRT(verbal, group = 6, focal.name = 1, save.output = TRUE, 
            output = c("LRTresults", "default"))

 # Graphical devices
 plot(r)

 # Plotting results and saving it in a PDF figure
 plot(r, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

 # WARNING: do not trust the results above since they are based on a selected 
 # subset of the verbal data set!
 
## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Excluding the "Anger" variable
 verbal <- verbal[colnames(verbal)!="Anger"]

 # Keeping the first 5 items and the first 50 subjects
 # (this is an artificial simplification to reduce the computational time)
 verbal <- verbal[1:50, c(1:5, 25)]

 # Three equivalent settings of the data matrix and the group membership
 r <- difLRT(verbal, group = 6, focal.name = 1)
 difLRT(verbal, group = "Gender", focal.name = 1)
 difLRT(verbal[,1:5], group = verbal[,6], focal.name = 1)

 # Multiple comparisons adjustment using Benjamini-Hochberg method
 difLRT(verbal, group = 6, focal.name = 1, p.adjust.method = "BH")

 # With item purification
 difLRT(verbal, group = 6, focal.name = 1, purify = TRUE)

 # Saving the output into the "LRTresults.txt" file (and default path)
 r <- difLRT(verbal, group = 6, focal.name = 1, save.output = TRUE, 
            output = c("LRTresults", "default"))

 # Graphical devices
 plot(r)

 # Plotting results and saving it in a PDF figure
 plot(r, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

 # WARNING: do not trust the results above since they are based on a selected 
 # subset of the verbal data set!
 
## End(Not run)

Mantel-Haenszel DIF method

Description

Performs DIF detection using Mantel-Haenszel method.

Usage

difMH(Data, group, focal.name , anchor = NULL, match = "score", MHstat = "MHChisq", 
  	correct = TRUE, exact = FALSE, alpha = 0.05, purify = FALSE, nrIter = 10, 
  	p.adjust.method = NULL, save.output = FALSE, output = c("out", "default")) 
## S3 method for class 'MH'
print(x, ...)
## S3 method for class 'MH'
plot(x, pch = 8, number = TRUE, col = "red", save.plot = FALSE, 
  	save.options = c("plot", "default", "pdf"), ...)
 difMH(Data, group, focal.name , anchor = NULL, match = "score", MHstat = "MHChisq", 
  	correct = TRUE, exact = FALSE, alpha = 0.05, purify = FALSE, nrIter = 10, 
  	p.adjust.method = NULL, save.output = FALSE, output = c("out", "default")) 
## S3 method for class 'MH'
print(x, ...)
## S3 method for class 'MH'
plot(x, pch = 8, number = TRUE, col = "red", save.plot = FALSE, 
  	save.options = c("plot", "default", "pdf"), ...)

Arguments

`Data`	numeric: either the data matrix only, or the data matrix plus the vector of group membership. See Details.
`group`	numeric or character: either the vector of group membership or the column indicator (within `data`) of group membership. See Details.
`focal.name`	numeric or character indicating the level of `group` which corresponds to the focal group.
`anchor`	either `NULL` (default) or a vector of item names (or identifiers) to specify the anchor items. See Details.
`match`	specifies the type of matching criterion. Can be either `"score"` (default) to compute the test score, or any continuous or discrete variable with the same length as the number of rows of `Data`. See Details.
`MHstat`	character: specifies the DIF statistic to be used for DIF identification. Possible values are `"MHChisq"` (default) and `"logOR"`. See Details .
`correct`	logical: should the continuity correction be used? (default is `TRUE`)
`exact`	logical: should an exact test be computed? (default is `FALSE`).
`alpha`	numeric: significance level (default is 0.05).
`purify`	logical: should the method be used iteratively to purify the set of anchor items? (default is FALSE).
`nrIter`	numeric: the maximal number of iterations in the item purification process (default is 10).
`p.adjust.method`	either `NULL` (default) or the acronym of the method for p-value adjustment for multiple comparisons. See Details.
`save.output`	logical: should the output be saved into a text file? (Default is `FALSE`).
`output`	character: a vector of two components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value). See Details.
`x`	the result from a `MH` class object.
`pch`, `col`	type of usual `pch` and `col` graphical options.
`number`	logical: should the item number identification be printed (default is `TRUE`).
`save.plot`	logical: should the plot be saved into a separate file? (default is `FALSE`).
`save.options`	character: a vector of three components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value), and the third component is the file extension, either `"pdf"` (default) or `"jpeg"`. See Details.
`...`	other generic parameters for the `plot` or the `print` functions.

Details

The method of Mantel-Haenszel (1959) allows for detecting uniform differential item functioning without requiring an item response model approach.

Missing values are allowed for item responses (not for group membership) but must be coded as NA values. They are discarded from sum-score computation.

The vector of group membership must hold only two different values, either as numeric or character. The focal group is defined by the value of the argument focal.name.

The matching criterion can be either the test score or any other continuous or discrete variable to be passed in the mantelHaenszel function. This is specified by the match argument. By default, it takes the value "score" and the test score (i.e. raw score) is computed. The second option is to assign to match a vector of continuous or discrete numeric values, which acts as the matching criterion. Note that for consistency this vector should not belong to the Data matrix.

The DIF statistic is specified by the MHstat argument. By default, MHstat takes the value "MHChisq" and the Mantel-Haenszel chi-square statistic is used. The other optional value is "logOR", and the log odds-ratio statistic (that is, the log of alphaMH divided by the square root of varLambda) is used. See Penfield and Camilli (2007), Philips and Holland (1987) and mantelHaenszel help file.

The threshold (or cut-score) for classifying items as DIF depends on the DIF statistic. With the Mantel-Haenszel chi-squared statistic (MHstat=="MHChisq"), it is computed as the quantile of the chi-square distribution with lower-tail probability of one minus alpha and with one degree of freedom. With the log odds-ratio statistic (MHstat=="logOR"), it is computed as the quantile of the standard normal distribution with lower-tail probability of 1-alpha/2. With exact inference, it is simply the alpha level since exact P-values are returned.

By default, the continuity correction factor -0.5 is used (Holland and Thayer, 1988). One can nevertheless remove it by specifying correct=FALSE.

In addition, the Mantel-Haenszel estimates of the common odds ratios $\alpha_{MH}$ are used to measure the effect sizes of the items. These are obtained by $\Delta_{MH} = -2.35 \log \alpha_{MH}$ (Holland and Thayer, 1985). According to the ETS delta scale, the effect size of an item is classified as negligible if $|\Delta_{MH}| \leq 1$ , moderate if $1 \leq |\Delta_{MH}| \leq 1.5$ , and large if $|\Delta_{MH}| \geq 1.5$ . The values of the effect sizes, together with the ETS classification, are printed with the output. Note that this is returned only for asymptotic tests, i.e. when exact is FALSE.

Item purification can be performed by setting purify to TRUE. Purification works as follows: if at least one item was detected as functioning differently at some step of the process, then the data set of the next step consists in all items that are currently anchor (DIF free) items, plus the tested item (if necessary). The process stops when either two successive applications of the method yield the same classifications of the items (Clauser and Mazor, 1998), or when nrIter iterations are run without obtaining two successive identical classifications. In the latter case a warning message is printed.

The output of the difMH, as displayed by the print.MH function, can be stored in a text file provided that save.output is set to TRUE (the default value FALSE does not execute the storage). In this case, the name of the text file must be given as a character string into the first component of the output argument (default name is "out"), and the path for saving the text file can be given through the second component of output. The default value is "default", meaning that the file will be saved in the current working directory. Any other path can be specified as a character string: see the Examples section for an illustration.

The plot.MH function displays the DIF statistics in a plot, with each item on the X axis. The type of point and the color are fixed by the usual pch and col arguments. Option number permits to display the item numbers instead. Also, the plot can be stored in a figure file, either in PDF or JPEG format. Fixing save.plot to TRUE allows this process. The figure is defined through the components of save.options. The first two components perform similarly as those of the output argument. The third component is the figure format, with allowed values "pdf" (default) for PDF file and "jpeg" for JPEG file. Note that no plot is returned for exact inference.

Value

A list of class "MH" with the following arguments:

`MH`	the values of the Mantel-Haenszel DIF statistics (either exact or asymptotic).
`p.value`	the p-values for the Mantel-Haenszel statistics (either exact or asymptotic).
`alphaMH`	the values of the mantel-Haenszel estimates of common odds ratios. Returned only if `exact` is `FALSE`.
`varLambda`	the values of the variances of the log odds-ratio statistics. Returned only if `exact` is `FALSE`.
`MHstat`	the value of the `MHstat` argument. Returned only if `exact` is `FALSE`.
`alpha`	the value of `alpha` argument.
`thr`	the threshold (cut-score) for DIF detection. Returned only if `exact` is `FALSE`.
`DIFitems`	either the column indicators of the items which were detected as DIF items, or "No DIF item detected".
`correct`	the value of `correct` option.
`exact`	the value of `exact` option.
`match`	a character string, either `"score"` or `"matching variable"` depending on the `match` argument.
`p.adjust.method`	the value of the `p.adjust.method` argument.
`adjusted.p`	either `NULL` or the vector of adjusted p-values for multiple comparisons.
`purification`	the value of `purify` option.
`nrPur`	the number of iterations in the item purification process. Returned only if `purify` is `TRUE`.
`difPur`	a binary matrix with one row per iteration in the item purification process and one column per item. Zeros and ones in the i-th row refer to items which were classified respectively as non-DIF and DIF items at the (i-1)-th step. The first row corresponds to the initial classification of the items. Returned only if `purify` is `TRUE`.
`convergence`	logical indicating whether the iterative item purification process stopped before the maximal number `nrIter` of allowed iterations. Returned only if `purify` is `TRUE`.
`names`	the names of the items.
`anchor.names`	the value of the `anchor` argument.
`save.output`	the value of the `save.output` argument.
`output`	the value of the `output` argument.

Author(s)

References

Agresti, A. (1990). Categorical data analysis. New York: Wiley.

Agresti, A. (1992). A survey of exact inference for contingency tables. Statistical Science, 7, 131-177. doi:10.1214/ss/1177011454

Holland, P. W. and Thayer, D. T. (1985). An alternative definition of the ETS delta scale of item difficulty. Research Report RR-85-43. Princeton, NJ: Educational Testing Service.

Holland, P. W. and Thayer, D. T. (1988). Differential item performance and the Mantel-Haenszel procedure. In H. Wainer and H. I. Braun (Ed.), Test validity. Hillsdale, NJ: Lawrence Erlbaum Associates.

Mantel, N. and Haenszel, W. (1959). Statistical aspects of the analysis of data from retrospective studies of disease. Journal of the National Cancer Institute, 22, 719-748.

Philips, A., and Holland, P. W. (1987). Estimators of the Mantel-Haenszel log odds-ratio estimate. Biometrics, 43, 425-431. doi:10.2307/2531824

Raju, N. S., Bode, R. K. and Larsen, V. S. (1989). An empirical assessment of the Mantel-Haenszel statistic to detect differential item functioning. Applied Measurement in Education, 2, 1-13. doi:10.1207/s15324818ame0201_1

Uttaro, T. and Millsap, R. E. (1994). Factors influencing the Mantel-Haenszel procedure in the detection of differential item functioning. Applied Psychological Measurement, 18, 15-25. doi:10.1177/014662169401800102

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)

 # Excluding the "Anger" variable
 verbal <- verbal[colnames(verbal) != "Anger"]

 # Three equivalent settings of the data matrix and the group membership
 r <- difMH(verbal, group = 25, focal.name = 1)
 difMH(verbal, group = "Gender", focal.name = 1)
 difMH(verbal[,1:24], group = verbal[,25], focal.name = 1)

 # With log odds-ratio statistic
 r2 <- difMH(verbal, group = 25, focal.name = 1, MHstat = "logOR")

 # With exact inference
 difMH(verbal, group = 25, focal.name = 1, exact = TRUE)

# Multiple comparisons adjustment using Benjamini-Hochberg method
 difMH(verbal, group = 25, focal.name = 1, p.adjust.method = "BH")

 # With item purification
 difMH(verbal, group = "Gender", focal.name = 1, purify = TRUE)
 difMH(verbal, group = "Gender", focal.name = 1, purify = TRUE, nrIter = 5)

 # Without continuity correction and with 0.01 significance level
 difMH(verbal, group = "Gender", focal.name = 1, alpha = 0.01, correct = FALSE)

 # With items 1 to 5 set as anchor items
 difMH(verbal, group = "Gender", focal.name = 1, anchor = 1:5)
 difMH(verbal, group = "Gender", focal.name = 1, anchor = 1:5, purify = TRUE)

 # Saving the output into the "MHresults.txt" file (and default path)
 r <- difMH(verbal, group = 25, focal.name = 1, save.output = TRUE, 
            output = c("MHresults","default"))

 # Graphical devices
 plot(r)
 plot(r2)

 # Plotting results and saving it in a PDF figure
 plot(r, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)

 # Excluding the "Anger" variable
 verbal <- verbal[colnames(verbal) != "Anger"]

 # Three equivalent settings of the data matrix and the group membership
 r <- difMH(verbal, group = 25, focal.name = 1)
 difMH(verbal, group = "Gender", focal.name = 1)
 difMH(verbal[,1:24], group = verbal[,25], focal.name = 1)

 # With log odds-ratio statistic
 r2 <- difMH(verbal, group = 25, focal.name = 1, MHstat = "logOR")

 # With exact inference
 difMH(verbal, group = 25, focal.name = 1, exact = TRUE)

# Multiple comparisons adjustment using Benjamini-Hochberg method
 difMH(verbal, group = 25, focal.name = 1, p.adjust.method = "BH")

 # With item purification
 difMH(verbal, group = "Gender", focal.name = 1, purify = TRUE)
 difMH(verbal, group = "Gender", focal.name = 1, purify = TRUE, nrIter = 5)

 # Without continuity correction and with 0.01 significance level
 difMH(verbal, group = "Gender", focal.name = 1, alpha = 0.01, correct = FALSE)

 # With items 1 to 5 set as anchor items
 difMH(verbal, group = "Gender", focal.name = 1, anchor = 1:5)
 difMH(verbal, group = "Gender", focal.name = 1, anchor = 1:5, purify = TRUE)

 # Saving the output into the "MHresults.txt" file (and default path)
 r <- difMH(verbal, group = 25, focal.name = 1, save.output = TRUE, 
            output = c("MHresults","default"))

 # Graphical devices
 plot(r)
 plot(r2)

 # Plotting results and saving it in a PDF figure
 plot(r, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

## End(Not run)

Raju's area DIF method

Description

Performs DIF detection using Raju's area method.

Usage

difRaju(Data, group, focal.name, model, c = NULL, engine = "ltm", discr = 1, 
 	irtParam = NULL,  same.scale = TRUE, anchor = NULL, alpha = 0.05, 
 	signed = FALSE, purify = FALSE, nrIter = 10, p.adjust.method = NULL, 
 	save.output = FALSE, output = c("out","default"))   	
## S3 method for class 'Raj'
print(x, ...)
## S3 method for class 'Raj'
plot(x, pch = 8, number = TRUE, col = "red", save.plot = FALSE, 
 	save.options = c("plot","default","pdf"), ...)
 difRaju(Data, group, focal.name, model, c = NULL, engine = "ltm", discr = 1, 
 	irtParam = NULL,  same.scale = TRUE, anchor = NULL, alpha = 0.05, 
 	signed = FALSE, purify = FALSE, nrIter = 10, p.adjust.method = NULL, 
 	save.output = FALSE, output = c("out","default"))   	
## S3 method for class 'Raj'
print(x, ...)
## S3 method for class 'Raj'
plot(x, pch = 8, number = TRUE, col = "red", save.plot = FALSE, 
 	save.options = c("plot","default","pdf"), ...)

Arguments

`Data`	numeric: either the data matrix only, or the data matrix plus the vector of group membership. See Details.
`group`	numeric or character: either the vector of group membership or the column indicator (within `data`) of group membership. See Details.
`focal.name`	numeric or character indicating the level of `group` which corresponds to the focal group.
`model`	character: the IRT model to be fitted (either `"1PL"`, `"2PL"` or `"3PL"`).
`c`	optional numeric value or vector giving the values of the constrained pseudo-guessing parameters. See Details.
`engine`	character: the engine for estimating the 1PL model, either `"ltm"` (default) or `"lme4"`.
`discr`	either `NULL` or a real positive value for the common discrimination parameter (default is 1). Used only if `model` is `"1PL"` and `engine` is `"ltm"`. See Details.
`irtParam`	matrix with 2J rows (where J is the number of items) and at most 9 columns containing item parameters estimates. See Details.
`same.scale`	logical: are the item parameters of the `irtParam` matrix on the same scale? (default is "TRUE"). See Details.
`anchor`	either `NULL` (default) or a vector of item names (or identifiers) to specify the anchor items. See Details.
`alpha`	numeric: significance level (default is 0.05).
`signed`	logical: should the Raju's statistics be computed using the signed (`TRUE`) or unsigned (`FALSE`, default) area? See Details.
`purify`	logical: should the method be used iteratively to purify the set of anchor items? (default is FALSE).
`nrIter`	numeric: the maximal number of iterations in the item purification process (default is 10).
`p.adjust.method`	either `NULL` (default) or the acronym of the method for p-value adjustment for multiple comparisons. See Details.
`save.output`	logical: should the output be saved into a text file? (Default is `FALSE`).
`output`	character: a vector of two components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value). See Details.
`x`	the result from a `Raj` class object.
`pch`, `col`	type of usual `pch` and `col` graphical options.
`number`	logical: should the item number identification be printed (default is `TRUE`).
`save.plot`	logical: should the plot be saved into a separate file? (default is `FALSE`).
`save.options`	character: a vector of three components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value), and the third component is the file extension, either `"pdf"` (default) or `"jpeg"`. See Details.
`...`	other generic parameters for the `plot` or the `print` functions.

Details

Raju's area method (Raju, 1988, 1990) allows for detecting uniform or non-uniform differential item functioning by setting an appropriate item response model. The input can be of two kinds: either by displaying the full data, the group membership and the model, or by giving the item parameter estimates (with the option irtParam). Both can be supplied, but in this case only the parameters in irtParam are used for computing Raju's statistic.

By default, the Raju's Z statistics are obtained by using the unsigned areas between the ICCs. However, these statistics can also be computed using the signed areas, by setting the argument signed to TRUE (default value is FALSE). See RajuZ for further details.

Missing values are allowed for item responses (not for group membership) but must be coded as NA values. They are discarded for item parameter estimation.

The vector of group membership must hold only two different values, either as numeric or character. The focal group is defined by the value of the argument focal.name.

The threshold (or cut-score) for classifying items as DIF is computed as the quantile of the standard normal distribution with lower-tail probability of 1-alpha/2.

A pre-specified set of anchor items can be provided through the anchor argument. It must be a vector of either item names (which must match exactly the column names of Data argument) or integer values (specifying the column numbers for item identification). In case anchor items are provided, they are used to rescale the item parameters on a common metric. None of the anchor items are tested for DIF: the output separates anchor items and tested items and DIF results are returned only for the latter. Note also that item purification is not activated when anchor items are provided (even if purify is set to TRUE). By default it is NULL so that no anchor item is specified. If item parameters are provided thorugh the irtParam argument and if they are on the same scale (i.e. if same.scale is TRUE), then anchor items are not used (even if they are specified).

The output of the difRaju, as displayed by the print.Raj function, can be stored in a text file provided that save.output is set to TRUE (the default value FALSE does not execute the storage). In this case, the name of the text file must be given as a character string into the first component of the output argument (default name is "out"), and the path for saving the text file can be given through the second component of output. The default value is "default", meaning that the file will be saved in the current working directory. Any other path can be specified as a character string: see the Examples section for an illustration.

The plot.Raj function displays the DIF statistics in a plot, with each item on the X axis. The type of point and the color are fixed by the usual pch and col arguments. Option number permits to display the item numbers instead. Also, the plot can be stored in a figure file, either in PDF or JPEG format. Fixing save.plot to TRUE allows this process. The figure is defined through the components of save.options. The first two components perform similarly as those of the output argument. The third component is the figure format, with allowed values "pdf" (default) for PDF file and "jpeg" for JPEG file.

Value

A list of class "Raj" with the following arguments:

`RajuZ`	the values of the Raju's statistics.
`p.value`	the p-values for the Raju's statistics.
`alpha`	the value of `alpha` argument.
`thr`	the threshold (cut-score) for DIF detection.
`DIFitems`	either the column indicators of the items which were detected as DIF items, or "No DIF item detected".
`signed`	the value of the `signed` argument.
`p.adjust.method`	the value of the `p.adjust.method` argument.
`adjusted.p`	either `NULL` or the vector of adjusted p-values for multiple comparisons.
`purification`	the value of `purify` option.
`nrPur`	the number of iterations in the item purification process. Returned only if `purify` is `TRUE`.
`difPur`	a binary matrix with one row per iteration in the item purification process and one column per item. Zeros and ones in the i-th row refer to items which were classified respectively as non-DIF and DIF items at the (i-1)-th step. The first row corresponds to the initial classification of the items. Returned only if `purify` is `TRUE`.
`convergence`	logical indicating whether the iterative item purification process stopped before the maximal number `nrIter`of allowed iterations. Returned only if `purify` is `TRUE`.
`model`	the value of `model` argument.
`c`	The value of the `c` argument.
`engine`	The value of the `engine` argument.
`discr`	the value of the `discr` argument.
`itemParInit`	the matrix of initial parameter estimates,with the same format as `irtParam` either provided by the user (through `irtParam`) or estimated from the data (and displayed without rescaling).
`itemParFinal`	the matrix of final parameter estimates, with the same format as `irtParam`, obtained after item purification. Returned only if `purify` is `TRUE`.
`estPar`	a logical value indicating whether the item parameters were estimated (`TRUE`) or provided by the user (`FALSE`).
`names`	the names of the items.
`anchor.names`	the value of the `anchor` argument.
`save.output`	the value of the `save.output` argument.
`output`	the value of the `output` argument.

Author(s)

References

Bates, D. and Maechler, M. (2009). lme4: Linear mixed-effects models using S4 classes. R package version 0.999375-31. http://CRAN.R-project.org/package=lme4

Cook, L. L. and Eignor, D. R. (1991). An NCME instructional module on IRT equating methods. Educational Measurement: Issues and Practice, 10, 37-45.

Holland, P. W. and Thayer, D. T. (1985). An alternative definition of the ETS delta scale of item difficulty. Research Report RR-85-43. Princeton, NJ: Educational Testing Service.

Raju, N.S. (1988). The area between two item characteristic curves. Psychometrika, 53, 495-502. doi:10.1007/BF02294403

Rizopoulos, D. (2006). ltm: An R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17, 1-25. doi:10.18637/jss.v017.i05

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Excluding the "Anger" variable
 verbal<-verbal[colnames(verbal)!="Anger"]

 # Three equivalent settings of the data matrix and the group membership
 # (1PL model, "ltm" engine) 
 difRaju(verbal, group = 25, focal.name = 1, model = "1PL")
 difRaju(verbal, group = "Gender", focal.name = 1, model = "1PL")
 difRaju(verbal[,1:24], group = verbal[,25], focal.name = 1, model = "1PL")

 # Multiple comparisons adjustment using Benjamini-Hochberg method
 difRaju(verbal, group = 25, focal.name = 1, model = "1PL", p.adjust.method = "BH")

 # With signed areas
 difRaju(verbal, group = 25, focal.name = 1, model = "1PL", signed = TRUE)

 # With items 1 to 5 set as anchor items
 difRaju(verbal, group = 25, focal.name = 1, model = "1PL", anchor = 1:5)

 # (1PL model, "lme4" engine) 
 difRaju(verbal, group = "Gender", focal.name = 1, model = "1PL",
 engine = "lme4")

 # 2PL model, signed and unsigned areas
 difRaju(verbal, group = "Gender", focal.name = 1, model = "2PL")
 difRaju(verbal, group = "Gender", focal.name = 1, model = "2PL", signed = TRUE)

 # 3PL model with all pseudo-guessing parameters constrained to 0.05
 # Signed and unsigned areas
 difRaju(verbal, group = "Gender", focal.name = 1, model = "3PL", c = 0.05)
 difRaju(verbal, group = "Gender", focal.name = 1, model = "3PL", c = 0.05,
   signed = TRUE)
 
 # Same models, with item purification
 difRaju(verbal, group = "Gender", focal.name = 1, model = "1PL", purify = TRUE)
 difRaju(verbal, group = "Gender", focal.name = 1, model = "2PL", purify = TRUE)
 difRaju(verbal, group = "Gender", focal.name = 1, model = "3PL", c = 0.05,
 purify = TRUE)

 # With signed areas
 difRaju(verbal, group = "Gender", focal.name = 1, model = "1PL", purify = TRUE,
   signed = TRUE)
 difRaju(verbal, group = "Gender", focal.name = 1, model = "2PL", purify = TRUE,
   signed = TRUE)
 difRaju(verbal, group = "Gender", focal.name = 1, model = "3PL", c = 0.05,
 purify = TRUE, signed = TRUE)

 ## Splitting the data into reference and focal groups
 nF<-sum(Gender)
 nR<-nrow(verbal)-nF
 data.ref<-verbal[,1:24][order(Gender),][1:nR,]
 data.focal<-verbal[,1:24][order(Gender),][(nR+1):(nR+nF),]

 ## Pre-estimation of the item parameters (1PL model, "ltm" engine)
 item.1PL<-rbind(itemParEst(data.ref,model = "1PL"),
 itemParEst(data.focal,model = "1PL"))
 difRaju(irtParam = item.1PL,same.scale = FALSE)

 ## Pre-estimation of the item parameters (1PL model, "lme4" engine)
 item.1PL<-rbind(itemParEst(data.ref, model = "1PL", engine = "lme4"),
 itemParEst(data.focal, model = "1PL", engine = "lme4"))
 difRaju(irtParam = item.1PL, same.scale = FALSE)

 ## Pre-estimation of the item parameters (2PL model)
 item.2PL<-rbind(itemParEst(data.ref, model = "2PL"),
 itemParEst(data.focal, model = "2PL"))
 difRaju(irtParam = item.2PL, same.scale = FALSE)

 ## Pre-estimation of the item parameters (constrained 3PL model)
 item.3PL<-rbind(itemParEst(data.ref, model = "3PL", c = 0.05),
 itemParEst(data.focal, model = "3PL", c = 0.05))
 difRaju(irtParam = item.3PL, same.scale = FALSE)

 # Saving the output into the "RAJUresults.txt" file (and default path)
 r <- difRaju(verbal, group = 25, focal.name = 1, model = "1PL",
          save.output = TRUE, output = c("RAJUresults","default"))

 # Graphical devices
 plot(r)

 # Plotting results and saving it in a PDF figure
 plot(r, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Excluding the "Anger" variable
 verbal<-verbal[colnames(verbal)!="Anger"]

 # Three equivalent settings of the data matrix and the group membership
 # (1PL model, "ltm" engine) 
 difRaju(verbal, group = 25, focal.name = 1, model = "1PL")
 difRaju(verbal, group = "Gender", focal.name = 1, model = "1PL")
 difRaju(verbal[,1:24], group = verbal[,25], focal.name = 1, model = "1PL")

 # Multiple comparisons adjustment using Benjamini-Hochberg method
 difRaju(verbal, group = 25, focal.name = 1, model = "1PL", p.adjust.method = "BH")

 # With signed areas
 difRaju(verbal, group = 25, focal.name = 1, model = "1PL", signed = TRUE)

 # With items 1 to 5 set as anchor items
 difRaju(verbal, group = 25, focal.name = 1, model = "1PL", anchor = 1:5)

 # (1PL model, "lme4" engine) 
 difRaju(verbal, group = "Gender", focal.name = 1, model = "1PL",
 engine = "lme4")

 # 2PL model, signed and unsigned areas
 difRaju(verbal, group = "Gender", focal.name = 1, model = "2PL")
 difRaju(verbal, group = "Gender", focal.name = 1, model = "2PL", signed = TRUE)

 # 3PL model with all pseudo-guessing parameters constrained to 0.05
 # Signed and unsigned areas
 difRaju(verbal, group = "Gender", focal.name = 1, model = "3PL", c = 0.05)
 difRaju(verbal, group = "Gender", focal.name = 1, model = "3PL", c = 0.05,
   signed = TRUE)
 
 # Same models, with item purification
 difRaju(verbal, group = "Gender", focal.name = 1, model = "1PL", purify = TRUE)
 difRaju(verbal, group = "Gender", focal.name = 1, model = "2PL", purify = TRUE)
 difRaju(verbal, group = "Gender", focal.name = 1, model = "3PL", c = 0.05,
 purify = TRUE)

 # With signed areas
 difRaju(verbal, group = "Gender", focal.name = 1, model = "1PL", purify = TRUE,
   signed = TRUE)
 difRaju(verbal, group = "Gender", focal.name = 1, model = "2PL", purify = TRUE,
   signed = TRUE)
 difRaju(verbal, group = "Gender", focal.name = 1, model = "3PL", c = 0.05,
 purify = TRUE, signed = TRUE)

 ## Splitting the data into reference and focal groups
 nF<-sum(Gender)
 nR<-nrow(verbal)-nF
 data.ref<-verbal[,1:24][order(Gender),][1:nR,]
 data.focal<-verbal[,1:24][order(Gender),][(nR+1):(nR+nF),]

 ## Pre-estimation of the item parameters (1PL model, "ltm" engine)
 item.1PL<-rbind(itemParEst(data.ref,model = "1PL"),
 itemParEst(data.focal,model = "1PL"))
 difRaju(irtParam = item.1PL,same.scale = FALSE)

 ## Pre-estimation of the item parameters (1PL model, "lme4" engine)
 item.1PL<-rbind(itemParEst(data.ref, model = "1PL", engine = "lme4"),
 itemParEst(data.focal, model = "1PL", engine = "lme4"))
 difRaju(irtParam = item.1PL, same.scale = FALSE)

 ## Pre-estimation of the item parameters (2PL model)
 item.2PL<-rbind(itemParEst(data.ref, model = "2PL"),
 itemParEst(data.focal, model = "2PL"))
 difRaju(irtParam = item.2PL, same.scale = FALSE)

 ## Pre-estimation of the item parameters (constrained 3PL model)
 item.3PL<-rbind(itemParEst(data.ref, model = "3PL", c = 0.05),
 itemParEst(data.focal, model = "3PL", c = 0.05))
 difRaju(irtParam = item.3PL, same.scale = FALSE)

 # Saving the output into the "RAJUresults.txt" file (and default path)
 r <- difRaju(verbal, group = 25, focal.name = 1, model = "1PL",
          save.output = TRUE, output = c("RAJUresults","default"))

 # Graphical devices
 plot(r)

 # Plotting results and saving it in a PDF figure
 plot(r, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

## End(Not run)

SIBTEST and Crossing-SIBTEST DIF method

Description

Performs DIF detection using SIBTEST (Shealy and Stout, 1993) or the modified Crossing-SIBTEST method (Chalmers, 2018).

Usage

difSIBTEST(Data, group, focal.name, type = "udif", anchor = NULL, alpha = 0.05,
  	purify = FALSE, nrIter = 10, p.adjust.method = NULL,
  	save.output = FALSE, output = c("out", "default"))
## S3 method for class 'SIBTEST'
print(x, ...)
## S3 method for class 'SIBTEST'
plot(x, pch = 8, number = TRUE, col = "red", save.plot = FALSE,
  	save.options = c("plot", "default", "pdf"), ...)
 difSIBTEST(Data, group, focal.name, type = "udif", anchor = NULL, alpha = 0.05,
  	purify = FALSE, nrIter = 10, p.adjust.method = NULL,
  	save.output = FALSE, output = c("out", "default"))
## S3 method for class 'SIBTEST'
print(x, ...)
## S3 method for class 'SIBTEST'
plot(x, pch = 8, number = TRUE, col = "red", save.plot = FALSE,
  	save.options = c("plot", "default", "pdf"), ...)

Arguments

`Data`	numeric: either the data matrix only, or the data matrix plus the vector of group membership. See Details.
`group`	numeric or character: either the vector of group membership or the column indicator (within `data`) of group membership. See Details.
`focal.name`	numeric or character indicating the level of `group` which corresponds to the focal group.
`type`	character: the type of DIF effect to test. Possible values are `"udif"` (default) for uniform DIF using SIBTEST, or `"nudif"` for nonuniform DIF using Crossing-SIBTEST (CSIBTEST).
`anchor`	either `NULL` (default) or a vector of item names (or identifiers) to specify the anchor items. See Details.
`alpha`	numeric: significance level (default is 0.05).
`purify`	logical: should the method be used iteratively to purify the set of anchor items? (default is FALSE).
`nrIter`	numeric: the maximal number of iterations in the item purification process (default is 10).
`p.adjust.method`	either `NULL` (default) or the acronym of the method for p-value adjustment for multiple comparisons. See Details.
`save.output`	logical: should the output be saved into a text file? (Default is `FALSE`).
`output`	character: a vector of two components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value). See Details.
`x`	the result from a `SIBTEST` class object.
`pch`, `col`	type of usual `pch` and `col` graphical options.
`number`	logical: should the item number identification be printed (default is `TRUE`).
`save.plot`	logical: should the plot be saved into a separate file? (default is `FALSE`).
`save.options`	character: a vector of three components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value), and the third component is the file extension, either `"pdf"` (default) or `"jpeg"`. See Details.
`...`	other generic parameters for the `plot` or the `print` functions.

Details

The SIBTEST method (Shealy and Stout, 1993) allows for detecting uniform differential item functioning without requiring an item response model approach. Its modified version, the Crossing-SIBTEST (Chalmers, 2018; Li and Stout, 1996), focuses on nonuniform DIF instead. This function provides a wrapper to the SIBTEST function from the mirt package (Chalmers, 2012) to fit within the difR framework (Magis et al., 2010). Therefore, if you are using this function for publication purposes please cite Chalmers (2018; 2012) and Magis et al. (2010).

The vector of group membership must hold only two different values, either as numeric or character. The focal group is defined by the value of the argument focal.name.

The type of DIF effect, uniform through SIBTEST or nonuniform through Crossing-SIBTEST, is determined by the type argument. By default it is "udif" for uniform DIF, and may take the value "nudif" for nonuniform DIF.

The threshold (or cut-score) for classifying items as DIF is computed as the quantile of the chi-square distribution with lower-tail probability of one minus alpha and with one degree of freedom. Note that the degrees of freedom are also returned by the df argument.

The output of the difSIBTEST, as displayed by the print.SIBTEST function, can be stored in a text file provided that save.output is set to TRUE (the default value FALSE does not execute the storage). In this case, the name of the text file must be given as a character string into the first component of the output argument (default name is "out"), and the path for saving the text file can be given through the second component of output. The default value is "default", meaning that the file will be saved in the current working directory. Any other path can be specified as a character string: see the Examples section for an illustration.

The plot.SIBTEST function displays the DIF statistics in a plot, with each item on the X axis. The type of point and the color are fixed by the usual pch and col arguments. Option number permits to display the item numbers instead. Also, the plot can be stored in a figure file, either in PDF or JPEG format. Fixing save.plot to TRUE allows this process. The figure is defined through the components of save.options. The first two components perform similarly as those of the output argument. The third component is the figure format, with allowed values "pdf" (default) for PDF file and "jpeg" for JPEG file. Note that no plot is returned for exact inference.

Value

A list of class "SIBTEST" with the following arguments:

`Beta`	the values of the SIBTEST Beta values.
`SE`	the standard errors of the Beta values.
`X2`	the values of the SIBTEST or Crossing-SITBTEST chi-square statistics.
`df`	the degrees of freedom for `X2` statistics.
`p.value`	the p-values for the SIBTEST or Crossing-SIBTEST statistics.
`type`	the value of the `type` argument.
`alpha`	the value of `alpha` argument.
`DIFitems`	either the column indicators of the items which were detected as DIF items, or "No DIF item detected".
`p.adjust.method`	the value of the `p.adjust.method` argument.
`adjusted.p`	either `NULL` or the vector of adjusted p-values for multiple comparisons.
`purification`	the value of `purify` option.
`nrPur`	the number of iterations in the item purification process. Returned only if `purify` is `TRUE`.
`difPur`	a binary matrix with one row per iteration in the item purification process and one column per item. Zeros and ones in the i-th row refer to items which were classified respectively as non-DIF and DIF items at the (i-1)-th step. The first row corresponds to the initial classification of the items. Returned only if `purify` is `TRUE`.
`convergence`	logical indicating whether the iterative item purification process stopped before the maximal number `nrIter` of allowed iterations. Returned only if `purify` is `TRUE`.
`names`	the names of the items or `NULL` if the items have no name.
`anchor.names`	the value of the `anchor` argument.
`save.output`	the value of the `save.output` argument.
`output`	the value of the `output` argument.

Author(s)

David Magis
Department of Psychology, University of Liege
Research Group of Quantitative Psychology and Individual Differences, KU Leuven
[email protected], http://ppw.kuleuven.be/okp/home/

References

Chalmers, R. P. (2012). mirt: A Multidimensional item response theory package for the R environment. Journal of Statistical Software, 48(6), 1-29. doi:10.18637/jss.v048.i06

Chalmers, R. P. (2018). Improving the Crossing-SIBTEST statistic for detecting non-uniform DIF. Psychometrika, 83(2), 376–386. doi:10.1007/s11336-017-9583-8

Li, H.-H., and Stout, W. (1996). A new procedure for detection of crossing DIF. Psychometrika, 61, 647–677. doi:10.1007/BF02294041

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)

 # Excluding the "Anger" variable
 verbal <- verbal[colnames(verbal) != "Anger"]

 # Three equivalent settings of the data matrix and the group membership
 r <- difSIBTEST(verbal, group = 25, focal.name = 1)
 difSIBTEST(verbal, group = "Gender", focal.name = 1)
 difSIBTEST(verbal[,1:24], group = verbal[,25], focal.name = 1)

 # Test for nonuniform DIF
 difSIBTEST(verbal, group = 25, focal.name = 1, type = "nudif")

 # Multiple comparisons adjustment using Benjamini-Hochberg method
 difSIBTEST(verbal, group = 25, focal.name = 1, p.adjust.method = "BH")

 # With item purification
 difSIBTEST(verbal, group = 25, focal.name = 1, purify = TRUE)
 r2 <- difSIBTEST(verbal, group = 25, focal.name = 1, purify = TRUE, nrIter = 5)

 # With items 1 to 5 set as anchor items
 difSIBTEST(verbal, group = "Gender", focal.name = 1, anchor = 1:5)
 difSIBTEST(verbal, group = "Gender", focal.name = 1, anchor = 1:5, purify = TRUE)

 # Saving the output into the "SIBresults.txt" file (and default path)
 r <- difSIBTEST(verbal, group = 25, focal.name = 1, save.output = TRUE,
            output = c("SIBresults","default"))

 # Graphical devices
 plot(r)
 plot(r2)

 # Plotting results and saving it in a PDF figure
 plot(r, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)

 # Excluding the "Anger" variable
 verbal <- verbal[colnames(verbal) != "Anger"]

 # Three equivalent settings of the data matrix and the group membership
 r <- difSIBTEST(verbal, group = 25, focal.name = 1)
 difSIBTEST(verbal, group = "Gender", focal.name = 1)
 difSIBTEST(verbal[,1:24], group = verbal[,25], focal.name = 1)

 # Test for nonuniform DIF
 difSIBTEST(verbal, group = 25, focal.name = 1, type = "nudif")

 # Multiple comparisons adjustment using Benjamini-Hochberg method
 difSIBTEST(verbal, group = 25, focal.name = 1, p.adjust.method = "BH")

 # With item purification
 difSIBTEST(verbal, group = 25, focal.name = 1, purify = TRUE)
 r2 <- difSIBTEST(verbal, group = 25, focal.name = 1, purify = TRUE, nrIter = 5)

 # With items 1 to 5 set as anchor items
 difSIBTEST(verbal, group = "Gender", focal.name = 1, anchor = 1:5)
 difSIBTEST(verbal, group = "Gender", focal.name = 1, anchor = 1:5, purify = TRUE)

 # Saving the output into the "SIBresults.txt" file (and default path)
 r <- difSIBTEST(verbal, group = 25, focal.name = 1, save.output = TRUE,
            output = c("SIBresults","default"))

 # Graphical devices
 plot(r)
 plot(r2)

 # Plotting results and saving it in a PDF figure
 plot(r, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

## End(Not run)

Standardization DIF method

Description

Performs DIF detection using standardization method.

Usage

difStd(Data, group, focal.name, anchor = NULL, match = "score", 
  	stdWeight = "focal", thrSTD = 0.1, purify = FALSE, nrIter = 10, 
  	save.output = FALSE, output = c("out", "default"))
## S3 method for class 'PDIF'
print(x, ...)
## S3 method for class 'PDIF'
plot(x, pch = 8, number = TRUE, col = "red", save.plot = FALSE, 
  	save.options = c("plot", "default", "pdf"), ...)
 difStd(Data, group, focal.name, anchor = NULL, match = "score", 
  	stdWeight = "focal", thrSTD = 0.1, purify = FALSE, nrIter = 10, 
  	save.output = FALSE, output = c("out", "default"))
## S3 method for class 'PDIF'
print(x, ...)
## S3 method for class 'PDIF'
plot(x, pch = 8, number = TRUE, col = "red", save.plot = FALSE, 
  	save.options = c("plot", "default", "pdf"), ...)

Arguments

`Data`	numeric: either the data matrix only, or the data matrix plus the vector of group membership. See Details.
`group`	numeric or character: either the vector of group membership or the column indicator (within `data`) of group membership. See Details.
`focal.name`	numeric or character indicating the level of `group` which corresponds to the focal group.
`anchor`	either `NULL` (default) or a vector of item names (or identifiers) to specify the anchor items. See Details.
`match`	specifies the type of matching criterion. Can be either `"score"` (default) to compute the test score, or any continuous or discrete variable with the same length as the number of rows of `Data`. See Details.
`stdWeight`	character: the type of weights used for the standardized P-DIF statistic. Possible values are `"focal"` (default), `"reference"` and `"total"`. See Details.
`thrSTD`	numeric: the threshold (cut-score) for standardized P-DIF statistic (default is 0.10).
`purify`	logical: should the method be used iteratively to purify the set of anchor items? (default is `FALSE`).
`nrIter`	numeric: the maximal number of iterations in the item purification process (default is 10).
`save.output`	logical: should the output be saved into a text file? (Default is `FALSE`).
`output`	character: a vector of two components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value). See Details.
`x`	the result from a `PDIF` class object.
`pch`, `col`	type of usual `pch` and `col` graphical options.
`number`	logical: should the item number identification be printed (default is `TRUE`).
`save.plot`	logical: should the plot be saved into a separate file? (default is `FALSE`).
`save.options`	character: a vector of three components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value), and the third component is the file extension, either `"pdf"` (default) or `"jpeg"`. See Details.
`...`	other generic parameters for the `plot` or the `print` functions.

Details

The method of standardization (Dorans and Kulick, 1986) allows for detecting uniform differential item functioning without requiring an item response model approach.

Missing values are allowed for item responses (not for group membership) but must be coded as NA values. They are discarded from sum-score computation.

The vector of group membership must hold only two different values, either as numeric or character. The focal group is defined by the value of the argument focal.name.

The matching criterion can be either the test score or any other continuous or discrete variable to be passed in the stdPDIF function. This is specified by the match argument. By default, it takes the value "score" and the test score (i.e. raw score) is computed. The second option is to assign to match a vector of continuous or discrete numeric values, which acts as the matching criterion. Note that for consistency this vector should not belong to the Data matrix.

The threshold (or cut-score) for classifying items as DIF has to be set by the user by the argument thrSTD. Default value is 0.10 but Dorans (1989) also recommends value 0.05. For this reason it is not possible to provide asymptotic p-values.

In addition, two types of effect sizes are displayed. The first one is obtained from the standardized P-DIF statistic itself. According to Dorans, Schmitt and Bleistein (1992), the effect size of an item is classified as negligible if $|St-P-DIF| \leq 0.05$ , moderate if $0.05 \leq |St-P-DIF| \leq 0.10$ , and large if if $|St-P-DIF| \geq 0.10$ . The second one is based on the transformation to the ETS Delta Scale (Holland and Thayer, 1985) of the standardized 'alpha' values (Dorans, 1989; Holland, 1985). The values of the effect sizes, together with the Dorans, Schmitt and Bleistein (DSB) and the ETS Delta scale (ETS) classification, are printed with the output.

The output of the difStd, as displayed by the print.PDIF function, can be stored in a text file provided that save.output is set to TRUE (the default value FALSE does not execute the storage). In this case, the name of the text file must be given as a character string into the first component of the output argument (default name is "out"), and the path for saving the text file can be given through the second component of output. The default value is "default", meaning that the file will be saved in the current working directory. Any other path can be specified as a character string: see the Examples section for an illustration.

The plot.PDIF function displays the DIF statistics in a plot, with each item on the X axis. The type of point and the color are fixed by the usual pch and col arguments. Option number permits to display the item numbers instead. Also, the plot can be stored in a figure file, either in PDF or JPEG format. Fixing save.plot to TRUE allows this process. The figure is defined through the components of save.options. The first two components perform similarly as those of the output argument. The third component is the figure format, with allowed values "pdf" (default) for PDF file and "jpeg" for JPEG file.

Value

A list of class "PDIF" with the following arguments:

`PDIF`	the values of the standardized P-DIF statistics.
`stdAlpha`	the values of the standardized alpha values (for effect sizes computation).
`alpha`	the value of `alpha` argument.
`thr`	the value of the `thrSTD` argument.
`DIFitems`	either the column indicators of the items which were detected as DIF items, or "No DIF item detected".
`match`	a character string, either `"score"` or `"matching variable"` depending on the `match` argument.
`purification`	the value of `purify` option.
`nrPur`	the number of iterations in the item purification process. Returned only if `purify` is `TRUE`.
`difPur`	a binary matrix with one row per iteration in the item purification process and one column per item. Zeros and ones in the i-th row refer to items which were classified respectively as non-DIF and DIF items at the (i-1)-th step. The first row corresponds to the initial classification of the items. Returned only if `purify` is `TRUE`.
`convergence`	logical indicating whether the iterative item purification process stopped before the maximal number `nrIter` of allowed iterations. Returned only if `purify` is `TRUE`.
`names`	the names of the items.
`anchor.names`	the value of the `anchor` argument.
`stdWeight`	the value of the `stdWeight` argument.
`save.output`	the value of the `save.output` argument.
`output`	the value of the `output` argument.

Author(s)

References

Clauser, B.E. and Mazor, K.M. (1998). Using statistical procedures to identify differential item functioning test items. Educational Measurement: Issues and Practice, 17, 31-44.

Dorans, N. J. (1989). Two new approaches to assessing differential item functioning. Standardization and the Mantel-Haenszel method. Applied Measurement in Education, 2, 217-233. doi:10.1207/s15324818ame0203_3

Dorans, N. J., Schmitt, A. P. and Bleistein, C. A. (1992). The standardization approach to assessing comprehensive differential item functioning. Journal of Educational Measurement, 29, 309-319. doi:10.1111/j.1745-3984.1992.tb00379.x

Holland, P. W. (1985, October). On the study of differential item performance without IRT. Paper presented at the meeting of Military Testing Association, San Diego (CA).

Holland, P. W. and Thayer, D. T. (1985). An alternative definition of the ETS delta scale of item difficulty. Research Report RR-85-43. Princeton, NJ: Educational Testing Service.

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)

 # Excluding the "Anger" variable
 verbal<-verbal[colnames(verbal) != "Anger"]

 # Three equivalent settings of the data matrix and the group membership
 difStd(verbal, group = 25, focal.name = 1)
 difStd(verbal, group = "Gender", focal.name = 1)
 difStd(verbal[,1:24], group = verbal[,25], focal.name = 1)

 # With other weights
 difStd(verbal, group = "Gender", focal.name = 1, stdWeight = "reference")
 difStd(verbal, group = "Gender", focal.name = 1, stdWeight = "total")
 
 # With item purification
 difStd(verbal, group = "Gender", focal.name = 1, purify = TRUE)
 difStd(verbal, group = "Gender", focal.name = 1, purify = TRUE, nrIter = 5)

 # With items 1 to 5 set as anchor items
 difStd(verbal, group = "Gender", focal.name = 1, anchor = 1:5)
 difStd(verbal, group = "Gender", focal.name = 1, anchor = 1:5, purify = TRUE)


 # With detection threshold of 0.05
 difStd(verbal, group = "Gender", focal.name = 1, thrSTD = 0.05)

 # Saving the output into the "STDresults.txt" file (and default path)
 r <- difStd(verbal, group = 25, focal.name = 1, save.output  =  TRUE, 
            output = c("STDresults","default"))

 # Graphical devices
 plot(r)

 # Plotting results and saving it in a PDF figure
 plot(r, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)

 # Excluding the "Anger" variable
 verbal<-verbal[colnames(verbal) != "Anger"]

 # Three equivalent settings of the data matrix and the group membership
 difStd(verbal, group = 25, focal.name = 1)
 difStd(verbal, group = "Gender", focal.name = 1)
 difStd(verbal[,1:24], group = verbal[,25], focal.name = 1)

 # With other weights
 difStd(verbal, group = "Gender", focal.name = 1, stdWeight = "reference")
 difStd(verbal, group = "Gender", focal.name = 1, stdWeight = "total")
 
 # With item purification
 difStd(verbal, group = "Gender", focal.name = 1, purify = TRUE)
 difStd(verbal, group = "Gender", focal.name = 1, purify = TRUE, nrIter = 5)

 # With items 1 to 5 set as anchor items
 difStd(verbal, group = "Gender", focal.name = 1, anchor = 1:5)
 difStd(verbal, group = "Gender", focal.name = 1, anchor = 1:5, purify = TRUE)


 # With detection threshold of 0.05
 difStd(verbal, group = "Gender", focal.name = 1, thrSTD = 0.05)

 # Saving the output into the "STDresults.txt" file (and default path)
 r <- difStd(verbal, group = 25, focal.name = 1, save.output  =  TRUE, 
            output = c("STDresults","default"))

 # Graphical devices
 plot(r)

 # Plotting results and saving it in a PDF figure
 plot(r, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

## End(Not run)

Transformed Item Difficulties (TID) DIF method

Description

Performs DIF detection using Transformed Item Difficulties (TID) method.

Usage

difTID(Data, group, focal.name, thrTID = 1.5, purify = FALSE, purType = "IPP1", 
  	nrIter = 10, alpha = 0.05, extreme = "constraint", 
  	const.range = c(0.001, 0.999), nrAdd = 1, save.output = FALSE, 
  	output = c("out", "default"))  
## S3 method for class 'TID'
print(x, only.final = TRUE, ...)
## S3 method for class 'TID'
plot(x, plot = "dist",pch = 2, pch.mult = 17, axis.draw = TRUE, 
  	thr.draw = FALSE, dif.draw = c(1, 3), print.corr = FALSE, xlim = NULL, 
  	ylim = NULL, xlab = NULL, ylab = NULL, main = NULL, col = "red", 
  	number = TRUE, save.plot = FALSE, save.options = c("plot", 
  	"default", "pdf"), ...)
 difTID(Data, group, focal.name, thrTID = 1.5, purify = FALSE, purType = "IPP1", 
  	nrIter = 10, alpha = 0.05, extreme = "constraint", 
  	const.range = c(0.001, 0.999), nrAdd = 1, save.output = FALSE, 
  	output = c("out", "default"))  
## S3 method for class 'TID'
print(x, only.final = TRUE, ...)
## S3 method for class 'TID'
plot(x, plot = "dist",pch = 2, pch.mult = 17, axis.draw = TRUE, 
  	thr.draw = FALSE, dif.draw = c(1, 3), print.corr = FALSE, xlim = NULL, 
  	ylim = NULL, xlab = NULL, ylab = NULL, main = NULL, col = "red", 
  	number = TRUE, save.plot = FALSE, save.options = c("plot", 
  	"default", "pdf"), ...)

Arguments

`Data`	numeric: either the data matrix only, or the data matrix plus the vector of group membership. See Details.
`group`	numeric or character: either the vector of group membership or the column indicator (within `Data`) of group membership. See Details.
`focal.name`	numeric or character indicating the level of `group` which corresponds to the focal group.
`thrTID`	either the threshold for detecting DIF items (default is 1.5) or `"norm"`.
`purify`	logical: should the method be used iteratively to purify the set of anchor items? (default is FALSE).
`purType`	character: the type of purification process to be run. Possible values are `"IPP1"` (default), `"IPP2"` and `"IPP3"`. Ignored if `purify` is `FALSE`. See Details.
`nrIter`	numeric: the maximal number of iterations in the item purification process (default is 10).
`alpha`	numeric: the significance level for calculating the detection threshold (default is 0.05). Ignored if `thrTID` is numeric.
`extreme`	character: the method used to modify the extreme proportions. Possible values are `"constraint"` (default) or `"add"`. See Details.
`const.range`	numeric: a vector of two constraining proportions. Default values are 0.001 and 0.999. Ignored if `extreme` is `"add"`.
`nrAdd`	integer: the number of successes and the number of failures to add to the data in order to adjust the proportions. Default value is 1. Ignored if `extreme` is `"constraint"`.
`save.output`	logical: should the output be saved into a text file? (Default is `FALSE`).
`output`	character: a vector of two components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value). See Details.
`x`	the result from a `TID` class object.
`only.final`	logical: should only the first and last steps of the purification process be printed? (default is `TRUE`. If `FALSE` all perpendicular distances, parameters of the major axis, and detection thresholds are printed additionally. Ignored if `purify` is `FALSE`.
`plot`	character: either `"dist"` (default) to display the perpendicular distances, or `"delta"` for the Delta plot. See Details.
`pch`	integer: the usual point character type for point display. Default value is 2, that is, Delta points are drawn as empty triangles.
`pch.mult`	integer: the type of point to be used for superposing onto Delta points that correspond to several items. Default value is 17, that is, full black traingles are drawn onto existing Delta plots wherein multiple items are located.
`axis.draw`	logical: should the major axis be drawn? (default is `TRUE`). If so, it will be drawn as a solid line.
`thr.draw`	logical: should the upper and lower bounds for DIF detection be drawn? (default is `FALSE`). If `TRUE`, they will be drawn as dashed lines.
`dif.draw`	numeric: a vector of two integer values to specify how the DIF items should be displayed. The first component of `dif.draw` is the type of point (i.e. the usual `pch` argument) and the second component determines the point size (i.e. the usual `cex` argument). Default values are 1 and 3, meaning that empty circles of three times the usual size are drawn around the Delta points of items flagged as DIF.
`print.corr`	logical: should the sample correlation of Delta scores be printed? (default is `FALSE`). If `TRUE`, it is printed in upper-left corner of the plot.
`xlim`, `ylim`, `xlab`, `ylab`, `main`	either the usual plot arguments `xlim`, `ylim`, `xlab`, `ylab` and `main`, or `NULL` (default value for all arguments). If `NULL`, the X and Y axis limits are computed from the range of Delta scores, the X and Y axis labels are `"Reference group"` and `"Focal group"` respectively, and no main title is produced.
`col`	character: the color type for the items. Used only when `plot` is `"dist"`.
`number`	logical: should the item number identification be printed (default is `TRUE`).
`save.plot`	logical: should the plot be saved into a separate file? (default is `FALSE`).
`save.options`	character: a vector of three components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value), and the third component is the file extension, either `"pdf"` (default) or `"jpeg"`. See Details.
`...`	other generic parameters for the `plot` or the `print` functions.

Details

The Transformed Item Difficulties (TID) method, also known as Angoff's Delta method (Angoff, 1982; Angoff and Ford, 1973) allows for detecting uniform differential item functioning without requiring an item response model approach. The presnt implementation relies on the deltaPlot and diagPlot functions from packagedeltaPlotR (Magis and Facon, 2014).

Missing values are allowed for item responses (not for group membership) but must be coded as NA values. They are discarded from the computation of proportions of success.

The vector of group membership must hold only two different values, either as numeric or character. The focal group is defined by the value of the argument focal.name.

The threshold for flaging items as DIF can be of two types and is specified by the thr argument.

It can be fixed to some arbitrary positive value by the user, for instance 1.5 (Angoff and Ford, 1973). In this case, thr takes the required numeric threshold value.
Alternatively, it can be derived from the bivariate normal approximation of the Delta points (Magis and Facon, 2012). In this case, thr must be given the character value "norm" (which is the default value). This threshold equals

$\Phi^{-1}(1-\alpha/2) \; \sqrt{\frac{b^2\,{s_0}^2-2\,b\,s_{01}+{s_1}^2}{b^2+1}}$

where $\Phi$ is the density of the standard normal distribution, $\alpha$ is the significance level (set by the argument alpha with default value 0.05), $b$ is the slope parameter of the major axis, $s_0$ and $s_1$ are the sample standard deviations of the Delta scores in the reference group and the focal group, respecively, and $s_{01}$ is the sample covariance of the Delta scores (see Magis and Facon, 2012, for further details).

Item purification can be performed by setting the argument purify to TRUE (by default it is FALSE so no purification is performed). The item purification process (IPP) starts when at least one item was flagged as DIF after the first run of the Delta plot, and proceeds as follows.

The intercept and slope parameters of the major axis are re-calculated by removing all DIF that are currently flagged as DIF. This yields updated values $a^*$ , $b^*$ , $s_0^*$ , $s_1^*$ and $s_{01}^*$ of the intercept and slope parameters, sample stanbdard deviations and sample covariance of the Delta scores.
Perpendicular distances (for all items) are updated with respect to the updated major axis.
Detection threshold is also updated. Three possible updates are possible: see below.
All items are now tested for the presence of DIF, given the updated perpendicular distances and major axis.
If the set of items flagged as DIF is the same as the one from the previous loop, stop the process. Otherwise go back to step 1.

Unlike traditional DIF methods, the detection threshold may also be updated since it depends on the sample estimates (when the normal approximation is considered). Three approaches are currently implemented and are specified by the purType argument.

Method 1 (purType=="IPP1"): the same threshold is used throughout the purification process, it is not iteratively updated. The threshold is the one obtained after the first run of the Delta plot.
Method 2 (purType=="IPP2"): only the slope parameter is updated in the threshold formula. By this way, one keeps the full data structure (i.e. neither the sample variances nor the sample covariance of the Delta scores are modified) but only the slope parameter is adjusted to lessen the impact of DIF items.
Method 3 (purType=="IPP3"): all adjusted parameters are plugged in the threshold formula. This approach completely discards the effect of items flagged as DIF from the computation of the threshold.

See Magis and Facon (2013) for further details. Note that purification can also be performed with fixed threshold (i.e. specified by the user), but then only IPP1 process is performed.

In order to avoid possible infinite loops in the purification process, a maximal number of iterations must be specified through the argument maxIter. The default maximal number of iterations is 10.

The output contains all input information, the Delta scores and perpendicular distances, the parameter of the major axis and the items flagged as DIF (if none, a character sentence is returned). In addition, the detection threshold and the type of threshold (fixed or normal approximation) is provided.

If item purification was run, several additional elements are returned: the number of iterations, a logical indicator whether the convergence was reached (or not, meaning that the process stopped because of reaching the maximal number of allowed iterations), a matrix with indicators of which items were flagged as DIF at each iteration, and the type of item purification process. Moreover, perpendicular distances are returned in a matrix format (one column per iteration), as well as successive major axis parameters (one row per iteration) and successive thresholds (as a vector).

The output is managed and printed in a more user-friendly way. When item purification is performed, only the first and last steps are displayed. Specifying the argument only.final to FALSE prints in addition all intermediate steps of the process (successive perpendicular distances, parameters of the major axis, and detection thresholds).

The output of the difTID, as displayed by the print.TID function, can be stored in a text file provided that save.output is set to TRUE (the default value FALSE does not execute the storage). In this case, the name of the text file must be given as a character string into the first component of the output argument (default name is "out"), and the path for saving the text file can be given through the second component of output. The default value is "default", meaning that the file will be saved in the current working directory. Any other path can be specified as a character string: see the Examples section for an illustration.

Two types of plots are available through the plot.TID function. If the argument plot is set to "dist" (the default value), then the perpendicular distances are represented on the Y axis of a scatter plot, with each item on the X axis. If plot is set to "delta", the Delta plot is returned. In the latter, all particular options can be found from the diagPlot function. Also, the plot can be stored in a figure file, either in PDF or JPEG format. Fixing save.plot to TRUE allows this process. The figure is defined through the components of save.options. The first two components perform similarly as those of the output argument. The third component is the figure format, with allowed values "pdf" (default) for PDF file and "jpeg" for JPEG file.

Value

A list of class "TID" with the following arguments:

`Props`	the matrix of proportions of correct responses, or `NA` if `type` is `"delta"`.
`adjProps`	the restricted proportions, in the same format as the output `Props` matrix, or `NA` if `type` is `"delta"`.
`Deltas`	the matrix of Delta scores.
`Dist`	a matrix with perpendicular distances, one row per item and one column per run of the Delta plot. If `purify` is `FALSE`, only a single column is returned.
`axis.par`	a matrix with two columns, holding respectively the intercepts and the slope parameters of the major axis. Each row refers to one step of the purification process. If `purify` is `FALSE`, only a single row is returned.
`nrIter`	the number of iterations invloved in the purification process. Returned only if `purify` is `TRUE`.
`maxIter`	the value of the `maxIter` argument. Returned only if `purify` is `TRUE`.
`convergence`	a logical value indicating whether convergence was reached in the purification process. Returned only if `purify` is `TRUE`.
`difPur`	a matrix with one column per item and one row per iteration in the purification process, holding zeros and ones to indicate which items were flagged as DIF or not at each step of the process. Returned only if `purify` is `TRUE`.
`thr`	a vector of successive threshold values used during the purification process. If `purify` is `FALSE`, a single value is returned.
`rule`	a character value indicating whether the threshold was `"fixed"` by the user (i.e. by setting `thr` to a numeric value) or whether it was computed by normal approximation (i.e. by setting `thr` to `"norm"`).
`purType`	the value of the `purType` argument. Returned only if `purify` is `TRUE`.
`DIFitems`	either `"No DIF item detected"` or an integer vector with the items that were flagged as DIF.
`adjust.extreme`	the value of the `extreme` argument.
`const.range`	the value of the `const.range` argument.
`nrAdd`	the value of the `nrAdd` argument.
`purify`	the value of the `purify` argument.
`alpha`	the value of the `alpha` argument.
`save.output`	the value of the `save.output` argument.
`output`	the value of the `output` argument.
`names`	either the names of the items (defined by the column names of the `Data` matrix) or the series of integers from one to the number of items.
`number`	a boolean value, being `TRUE` if the item names are simply their number in the `Data` matrix, or `FALSE` if real item names are available in the `names` element.

Author(s)

David Magis
Department of Psychology, University of Liege
Research Group of Quantitative Psychology and Individual Differences, KU Leuven
[email protected], http://ppw.kuleuven.be/okp/home/

References

Angoff, W. H. (1982). Use of difficulty and discrimination indices for detecting item bias. In R. A. Berck (Ed.), Handbook of methods for detecting item bias (pp. 96-116). Baltimore, MD: Johns Hopkins University Press.

Angoff, W. H., and Ford, S. F. (1973). Item-race interaction on a test of scholastic aptitude. Journal of Educational Measurement, 2, 95-106. doi:10.1111/j.1745-3984.1973.tb00787.x

Magis, D., and Facon, B. (2012). Angoff's Delta method revisited: improving the DIF detection under small samples. British Journal of Mathematical and Statistical Psychology, 65, 302-321. doi:10.1111/j.2044-8317.2011.02025.x

Magis, D., and Facon, B. (2013). Item purification does not always improve DIF detection: a counter-example with Angoff's Delta plot. Educational and Psychological Measurement, 73, 293-311. doi:10.1177/0013164412451903

Magis, D. and Facon, B. (2014). deltaPlotR: An R Package for Differential Item Functioning Analysis with Angoff's Delta Plot. Journal of Statistical Software, Code Snippets, 59(1), 1-19. doi:10.18637/jss.v059.c01

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)

 # Excluding the "Anger" variable
 verbal <- verbal[colnames(verbal) != "Anger"]

 # Three equivalent settings of the data matrix and the group membership
 r <- difTID(verbal, group = 25, focal.name = 1)
 difTID(verbal, group = "Gender", focal.name = 1)
 difTID(verbal[,1:24], group = verbal[,25], focal.name = 1)

 # With item purification and threshold 1
 r2 <- difTID(verbal, group = "Gender", focal.name = 1, purify = TRUE, thrTID = 1)

 # Saving the output into the "TIDresults.txt" file (and default path)
 difTID(verbal, group = 25, focal.name = 1, save.output = TRUE, 
   output = c("TIDresults", "default"))

 # Graphical devices
 plot(r2)
 plot(r2, plot = "delta")

 # Plotting results and saving it in a PDF figure
 plot(r2, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r2, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)

 # Excluding the "Anger" variable
 verbal <- verbal[colnames(verbal) != "Anger"]

 # Three equivalent settings of the data matrix and the group membership
 r <- difTID(verbal, group = 25, focal.name = 1)
 difTID(verbal, group = "Gender", focal.name = 1)
 difTID(verbal[,1:24], group = verbal[,25], focal.name = 1)

 # With item purification and threshold 1
 r2 <- difTID(verbal, group = "Gender", focal.name = 1, purify = TRUE, thrTID = 1)

 # Saving the output into the "TIDresults.txt" file (and default path)
 difTID(verbal, group = 25, focal.name = 1, save.output = TRUE, 
   output = c("TIDresults", "default"))

 # Graphical devices
 plot(r2)
 plot(r2, plot = "delta")

 # Plotting results and saving it in a PDF figure
 plot(r2, save.plot = TRUE, save.options = c("plot", "default", "pdf"))

 # Changing the path, JPEG figure
 path <- "c:/Program Files/"
 plot(r2, save.plot = TRUE, save.options = c("plot", path, "jpeg"))

## End(Not run)

Comparison of DIF detection methods among multiple groups

Description

This function compares the specified DIF detection methods among multiple groups, with respect to the detected items.

Usage

genDichoDif(Data, group, focal.names, method, anchor = NULL, match = "score", 
 	type = "both", criterion = "LRT", alpha = 0.05, model = "2PL", c = NULL, 
 	engine = "ltm", discr = 1, irtParam = NULL, nrFocal = 2, same.scale = TRUE, 
 	purify = FALSE, nrIter = 10, p.adjust.method = NULL, save.output = FALSE, 
 	output = c("out", "default")) 
## S3 method for class 'genDichoDif'
print(x, ...)
 genDichoDif(Data, group, focal.names, method, anchor = NULL, match = "score", 
 	type = "both", criterion = "LRT", alpha = 0.05, model = "2PL", c = NULL, 
 	engine = "ltm", discr = 1, irtParam = NULL, nrFocal = 2, same.scale = TRUE, 
 	purify = FALSE, nrIter = 10, p.adjust.method = NULL, save.output = FALSE, 
 	output = c("out", "default")) 
## S3 method for class 'genDichoDif'
print(x, ...)

Arguments

`Data`	numeric: either the data matrix only, or the data matrix plus the vector of group membership. See Details.
`group`	numeric or character: either the vector of group membership or the column indicator (within data) of group membership. See Details.
`focal.names`	numeric or character vector indicating the levels of `group` which correspond to the focal groups.
`method`	character: the name of the selected methods. See Details.
`anchor`	either `NULL` (default) or a vector of item names (or identifiers) to specify the anchor items. See Details.
`match`	specifies the type of matching criterion. Can be either `"score"` (default) to compute the test score, or any continuous or discrete variable with the same length as the number of rows of `Data`. See Details.
`type`	a character string specifying which DIF effects must be tested (default is `"both"`). See Details.
`criterion`	character: the type of test statistic used to detect DIF items with generalized logistic regression. Possible values are `"LRT"` (default) and `"Wald"`. See Details.
`alpha`	numeric: significance level (default is 0.05).
`model`	character: the IRT model to be fitted (either `"1PL"`, `"2PL"` or `"3PL"`). Default is `"2PL"`.
`c`	optional numeric value or vector giving the values of the constrained pseudo-guessing parameters. See Details.
`engine`	character: the engine for estimating the 1PL model, either `"ltm"` (default) or `"lme4"`.
`discr`	either `NULL` or a real positive value for the common discrimination parameter (default is 1). Used onlky if `model` is `"1PL"` and `engine` is `"ltm"`. See Details.
`irtParam`	matrix with 2J rows (where J is the number of items) and at most 9 columns containing item parameters estimates. See Details.
`nrFocal`	numeric: the number of focal groups (default is 2).
`same.scale`	logical: are the item parameters of the `irtParam` matrix on the same scale? (default is "TRUE"). See Details.
`purify`	logical: should the method be used iteratively to purify the set of anchor items? (default is FALSE).
`nrIter`	numeric: the maximal number of iterations in the item purification process (default is 10).
`p.adjust.method`	either `NULL` (default) or the acronym of the method for p-value adjustment for multiple comparisons. See Details.
`save.output`	logical: should the output be saved into a text file? (Default is `FALSE`).
`output`	character: a vector of two components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value). See Details.
`x`	result from a `genDichoDif` class object.
`...`	other generic parameters for the `print` function.

Details

genDichoDif is a generic function which calls one or several DIF detection methods among multiple groups, and summarize their output. The possible methods are: "GMH" for Generalized Mantel-Haenszel (Penfield, 2001), "genLogistic" for generalized logistic regression (Magis, Raiche Beland and Gerard, 2011) and "genLord" for generalized Lord's chi-square test (Kim, Cohen and Park, 1995).

If method has a single component, the output of genDichoDif is exactly the one provided by the method itself. Otherwise, the main output is a matrix with one row per item and one column per method. For each specified method and related arguments, items detected as DIF and non-DIF are respectively encoded as "DIF" and "NoDIF". When printing the output an additional column is added, counting the number of times each item was detected as functioning differently (Note: this is just an informative summary, since the methods are obviously not independent for the detection of DIF items).

The vector of group membership must hold at least three different values, either as numeric or character. The focal groups are defined by the values of the argument focal.names.

For generalized Mantel-Haenszel and generalized logistic methods, the matching criterion can be either the test score or any other continuous or discrete variable to be passed in the DIF function. This is specified by the match argument. By default, it takes the value "score" and the test score (i.e. raw score) is computed. The second option is to assign to match a vector of continuous or discrete numeric values, which acts as the matching criterion. Note that for consistency this vector should not belong to the Data matrix.

For the generalized logistic regression method, the argument type permits to test either both uniform and nonuniform effects simultaneously (with type="both"), only uniform DIF effect (with type="udif") or only nonuniform DIF effect (with type="nudif"). Furthermore, the argument criterion defines which test must be used, either the Wald test ("Wald") or the likelihood ratio test ("LRT"). See difGenLord for further details.

The threshold for detecting DIF items depends on the method and is depending on the significance level set by alpha.

Adjustment for multiple comparisons is possible with the argument p.adjust.method. See the corresponding methods for further information.

The output of the genDichoDif function can be stored in a text file by fixing save.output and output appropriately. See the help file of selectGenDif function (or any other DIF method) for further information.

Value

Either the output of one of the DIF detection methods, or a list of class "genDichoDif" with the following arguments:

`DIF`	a character matrix with one row per item and whose columns refer to the different specified detection methods. See Details.
`alpha`	the significance level `alpha`.
`method`	the value of `method`argument.
`match`	the value of `match` argument.
`type`	the value of `type` argument.
`criterion`	the value of the `criterion` argument.
`model`	the value of `model` option.
`c`	the value of `c` option.
`engine`	The value of the `engine` argument.
`discr`	the value of the `discr` argument.
`irtParam`	the value of `irtParam` option.
`same.scale`	the value of `same.scale` option.
`p.adjust.method`	the value of the `p.adjust.method` argument.
`purification`	the value of `purify` option.
`nrPur`	an integer vector (of length equal to the number of methods) with the number of iterations in the purification process. Returned only if `purify` is TRUE.
`convergence`	a logical vector (of length equal to the number of methods) indicating whether the iterative purification process converged. Returned only if `purify` is TRUE.
`anchor.names`	the value of the `anchor` argument.
`save.output`	the value of the `save.output` argument.
`output`	the value of the `output` argument.

Author(s)

References

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Creating four groups according to gender ("Man" or "Woman") and trait 
 # anger score ("Low" or "High")
 group <- rep("WomanLow", nrow(verbal))
 group[Anger>20 & Gender==0] <- "WomanHigh"
 group[Anger<=20 & Gender==1] <- "ManLow"
 group[Anger>20 & Gender==1] <- "ManHigh"

 # New data set
 Verbal <- cbind(verbal[,1:24], group)

 # Reference group: "WomanLow"
 names <- c("WomanHigh", "ManLow", "ManHigh")

 # Comparing the three available methods
 # with item purification 
 genDichoDif(Verbal, group = 25, focal.names = names, method = c("GMH", "genLogistic",
             "genLord"), purify = TRUE)
   
 # Same analysis, but saving the output into the 'genDicho' file
 genDichoDif(Verbal, group = 25, focal.names = names, method = c("GMH", "genLogistic", 
             "genLord"), purify = TRUE, save.output = TRUE, 
             output = c("genDicho", "default"))

## End(Not run)
## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Creating four groups according to gender ("Man" or "Woman") and trait 
 # anger score ("Low" or "High")
 group <- rep("WomanLow", nrow(verbal))
 group[Anger>20 & Gender==0] <- "WomanHigh"
 group[Anger<=20 & Gender==1] <- "ManLow"
 group[Anger>20 & Gender==1] <- "ManHigh"

 # New data set
 Verbal <- cbind(verbal[,1:24], group)

 # Reference group: "WomanLow"
 names <- c("WomanHigh", "ManLow", "ManHigh")

 # Comparing the three available methods
 # with item purification 
 genDichoDif(Verbal, group = 25, focal.names = names, method = c("GMH", "genLogistic",
             "genLord"), purify = TRUE)
   
 # Same analysis, but saving the output into the 'genDicho' file
 genDichoDif(Verbal, group = 25, focal.names = names, method = c("GMH", "genLogistic", 
             "genLord"), purify = TRUE, save.output = TRUE, 
             output = c("genDicho", "default"))

## End(Not run)

Generalized logistic regression DIF statistic

Description

Calculates the "generalized logistic regression" likelihood-ratio or Wald statistics for DIF detection among multiple groups.

Usage

genLogistik(data, member, match = "score", anchor = 1:ncol(data), 
 	type = "both", criterion = "LRT") 
 genLogistik(data, member, match = "score", anchor = 1:ncol(data), 
 	type = "both", criterion = "LRT")

Arguments

`data`	numeric: the data matrix (one row per subject, one column per item).
`member`	numeric: the vector of group membership with zero and positive integer entries only. See Details.
`match`	specifies the type of matching criterion. Can be either `"score"` (default) to compute the test score, or any continuous or discrete variable with the same length as the number of rows of `data`. See Details.
`anchor`	a vector of integer values specifying which items (all by default) are currently considered as anchor (DIF free) items. See Details.
`type`	a character string specifying which DIF effects must be tested. Possible values are `"both"` (default), `"udif"` and `"nudif"`. See Details.
`criterion`	character: the type of test statistic used to detect DIF items. Possible values are `"LRT"` (default) and `"Wald"`. See Details.

Details

This command computes the generalized logistic regression statistic (Magis, Raiche, Beland and Gerard, 2011) in the specific framework of differential item functioning among $(J+1)$ groups and J is the number of focal groups. It forms the basic command of difGenLogistic and is specifically designed for this call.

The three possible models to be fitted are:

$M_0: logit (\pi_i) = \alpha + \beta X + \gamma_i + \delta_i X$

$M_1: logit (\pi_i) = \alpha + \beta X + \gamma_i$

$M_2: logit (\pi_i) = \alpha + \beta X$

where $\pi_i$ is the probability of answering correctly the item in group i ( $i = 0, ..., J$ ) and $X$ is the matching criterion. Parameters $\alpha$ and $\beta$ are the common intercept and the slope of the logistic curves, while $\gamma_i$ and $\delta_i$ are group-specific parameters. For identification reasons the parameters $\gamma_0$ and $\delta_0$ of the reference group are set to zero. The set of parameters $\{\gamma_i: i = 1, ..., J\}$ of the focal groups ( $g=i$ ) represents the uniform DIF effect across all groups, and the set of parameters $\{\delta_i: i = 1, ..., n\}$ is used to model nonuniform DIF effect across all groups. The models are fitted with the glm function.

Two tests are available: the Wald test and the likelihood ratio test. With the likelihood ratio test, two nested models are fitted and compared by means of Wilks' Lambda (or likelihood ratio) statistic (Wilks, 1938). With the Wald test, the model parameters are statistically tested using an appropriate contrast matrix. Each test is set with the criterion argument, with the values "LRT" and "Wald" respectively.

The argument type determines the type of DIF effect to be tested. The three possible values of type are: type="both" which tests the hypothesis $H_0: \gamma_i = \delta_i=0$ for all i; type="nudif" which tests the hypothesis $H_0: \delta_i = 0$ for all i; and type="udif" which tests the hypothesis $H_0: \gamma_i = 0 | \delta_i = 0$ for all i. In other words, type="both" tests for DIF (without distinction between uniform and nonuniform effects), while type="udif" and type="nudif" test for uniform and nonuniform DIF, respectively. Whatever the tested DIF effects, this is a simultaneous test of the equality of focal group parameters to zero.

The data are passed through the data argument, with one row per subject and one column per item. Missing values are allowed but must be coded as NA values. They are discarded from the fitting of the logistic models (see glm for further details).

The vector of group membership, specified with member argument, must hold only zeros and positive integers. The value zero corresponds to the reference group, and each positive integer value corresponds to one focal group. At least two different positive integers must be supplied.

Option anchor sets the items which are considered as anchor items for computing the logistic regression DIF statistics. Items other than the anchor items and the tested item are discarded. anchor must hold integer values specifying the column numbers of the corresponding anchor items. It is mainly designed to perform item purification.

In addition to the results of the fitted models (model parameters, covariance matrices, test statistics), Nagelkerke's $R^2$ coefficients (Nagelkerke, 1991) are computed for each model and the output returns the differences in these coefficients. Such differences are used as measures of effect size by the difGenLogistic command; see Gomez-Benito, Dolores Hidalgo and Padilla (2009), Jodoin and Gierl (2001) and Zumbo and Thomas (1997).

Value

A list with nine components:

`stat`	the values of the generalized logistic regression DIF statistics (that is, the likelihood ratio test statistics).
`R2M0`	the values of Nagelkerke's R^2 coefficients for the "full" model.
`R2M1`	the values of Nagelkerke's R^2 coefficients for the "simpler" model.
`deltaR2`	the differences between Nagelkerke's $R^2$ coefficients of the tested models. See Details.
`parM0`	a matrix with one row per item and $2+J*2$ columns (where J is the number of focal groups), holding successively the fitted parameters $\hat{\alpha}$ , $\hat{\beta}$ , $\hat{\gamma}_i$ and $\hat{\delta}_i$ ( $i = 1, ..., J$ ) of the "full" model ( $M_0$ if `type="both"` or `type="nudif"`, $M_1$ if `type="udif"`).
`parM1`	the same matrix as `parM0` but with fitted parameters for the "simpler" model ( $M_1$ if `type="nudif"`, $M_2$ if `type="both"` or `type="udif"`).
`covMat`	a 3-dimensional matrix of size p x p x K, where p is the number of estimated parameters and K is the number of items, holding the p x p covariance matrices of the estimated parameters (one matrix for each tested item).
`criterion`	the value of the `criterion` argument.
`match`	a character string, either `"score"` or `"matching variable"` depending on the `match` argument.

Author(s)

References

Nagelkerke, N. J. D. (1991). A note on a general definition of the coefficient of determination. Biometrika, 78, 691-692. doi:10.1093/biomet/78.3.691

Wilks, S. S. (1938). The large-sample distribution of the likelihood ratio for testing composite hypotheses. Annals of Mathematical Statistics, 9, 60-62. doi:10.1214/aoms/1177732360

Zumbo, B. D. and Thomas, D. R. (1997). A measure of effect size for a model-based approach for studying DIF. Prince George, Canada: University of Northern British Columbia, Edgeworth Laboratory for Quantitative Behavioral Science.

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Creating four groups according to gender (0 or 1) and trait anger score
 # ("Low" or "High")
 # Reference group: women with low trait anger score (<=20)
 group <- rep(0,nrow(verbal))
 group[Anger>20 & Gender==0] <- 1
 group[Anger<=20 & Gender==1] <- 2
 group[Anger>20 & Gender==1] <- 3

 # Testing both types of DIF simultaneously
 # With all items
 genLogistik(verbal[,1:24], group)
 genLogistik(verbal[,1:24], group, criterion = "Wald")

 # Removing item 6 from the set of anchor items
 genLogistik(verbal[,1:24], group, anchor = c(1:5, 7:24))
 genLogistik(verbal[,1:24], group, anchor = c(1:5, 7:24), criterion = "Wald")

 # Testing nonuniform DIF effect
 genLogistik(verbal[,1:24], group, type = "nudif")
 genLogistik(verbal[,1:24], group, type = "nudif", criterion="Wald")

 # Testing uniform DIF effect
 genLogistik(verbal[,1:24], group, type = "udif")
 genLogistik(verbal[,1:24], group, type = "udif", criterion="Wald")

 # Using trait anger score as matching criterion
 genLogistik(verbal[,1:24], group, match = verbal[,25])
 
## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Creating four groups according to gender (0 or 1) and trait anger score
 # ("Low" or "High")
 # Reference group: women with low trait anger score (<=20)
 group <- rep(0,nrow(verbal))
 group[Anger>20 & Gender==0] <- 1
 group[Anger<=20 & Gender==1] <- 2
 group[Anger>20 & Gender==1] <- 3

 # Testing both types of DIF simultaneously
 # With all items
 genLogistik(verbal[,1:24], group)
 genLogistik(verbal[,1:24], group, criterion = "Wald")

 # Removing item 6 from the set of anchor items
 genLogistik(verbal[,1:24], group, anchor = c(1:5, 7:24))
 genLogistik(verbal[,1:24], group, anchor = c(1:5, 7:24), criterion = "Wald")

 # Testing nonuniform DIF effect
 genLogistik(verbal[,1:24], group, type = "nudif")
 genLogistik(verbal[,1:24], group, type = "nudif", criterion="Wald")

 # Testing uniform DIF effect
 genLogistik(verbal[,1:24], group, type = "udif")
 genLogistik(verbal[,1:24], group, type = "udif", criterion="Wald")

 # Using trait anger score as matching criterion
 genLogistik(verbal[,1:24], group, match = verbal[,25])
 
## End(Not run)

Generalized Lord's chi-squared DIF statistic

Description

Calculates the generalized Lord's chi-squared statistics for DIF detection among multiple groups.

Usage

genLordChi2(irtParam, nrFocal)
 genLordChi2(irtParam, nrFocal)

Arguments

`irtParam`	numeric: the matrix of item parameter estimates. See Details.
`nrFocal`	numeric: the number of focal groups.

Details

This command computes the generalized Lord's chi-squared statistic (Kim, Cohen and Park, 1995), also called the Qj statistics, in the specific framework of differential item functioning with multiple groups. It forms the basic command of difGenLord and is specifically designed for this call.

In addition, the item parameters of the reference group and the focal groups must be placed on the same scale. This can be done by using itemRescale command, which performs equal means anchoring between two groups of item estimates (Cook and Eignor, 1991).

The number of focal groups has to be specified with argument nrFocal.

Value

A vector with the values of the generalized Lord's chi-squared DIF statistics.

Author(s)

References

Cook, L. L. and Eignor, D. R. (1991). An NCME instructional module on IRT equating methods. Educational Measurement: Issues and Practice, 10, 37-45.

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Creating four groups according to gender ("Man" or "Woman") and
 # trait anger score ("Low" or "High")
 group <- rep("WomanLow",nrow(verbal))
 group[Anger>20 & Gender==0] <- "WomanHigh"
 group[Anger<=20 & Gender==1] <- "ManLow"
 group[Anger>20 & Gender==1] <- "ManHigh"

 # Splitting the data into the four subsets according to "group"
 data0 <- data1 <- data2 <- data3 <- NULL
 for (i in 1:nrow(verbal)){
 if (group[i]=="WomanLow") data0 <- rbind(data0, as.numeric(verbal[i,1:24]))
 if (group[i]=="WomanHigh") data1 <- rbind(data1, as.numeric(verbal[i,1:24]))
 if (group[i]=="ManLow") data2 <- rbind(data2, as.numeric(verbal[i,1:24]))
 if (group[i]=="ManHigh") data3 <- rbind(data3, as.numeric(verbal[i,1:24]))
 }

 # Estimation of the item parameters (1PL model)
 m0.1PL <- itemParEst(data0, model = "1PL")
 m1.1PL <- itemParEst(data1, model = "1PL")
 m2.1PL <- itemParEst(data2, model = "1PL")
 m3.1PL <- itemParEst(data3, model = "1PL")

 # merging the item parameters with rescaling
 irt.scale <- rbind(m0.1PL, itemRescale(m0.1PL, m1.1PL), itemRescale(m0.1PL, m2.1PL), 
                    itemRescale(m0.1PL, m3.1PL))

 # Generalized Lord's chi-squared statistics
 genLordChi2(irt.scale, nrFocal = 3)
 
## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Creating four groups according to gender ("Man" or "Woman") and
 # trait anger score ("Low" or "High")
 group <- rep("WomanLow",nrow(verbal))
 group[Anger>20 & Gender==0] <- "WomanHigh"
 group[Anger<=20 & Gender==1] <- "ManLow"
 group[Anger>20 & Gender==1] <- "ManHigh"

 # Splitting the data into the four subsets according to "group"
 data0 <- data1 <- data2 <- data3 <- NULL
 for (i in 1:nrow(verbal)){
 if (group[i]=="WomanLow") data0 <- rbind(data0, as.numeric(verbal[i,1:24]))
 if (group[i]=="WomanHigh") data1 <- rbind(data1, as.numeric(verbal[i,1:24]))
 if (group[i]=="ManLow") data2 <- rbind(data2, as.numeric(verbal[i,1:24]))
 if (group[i]=="ManHigh") data3 <- rbind(data3, as.numeric(verbal[i,1:24]))
 }

 # Estimation of the item parameters (1PL model)
 m0.1PL <- itemParEst(data0, model = "1PL")
 m1.1PL <- itemParEst(data1, model = "1PL")
 m2.1PL <- itemParEst(data2, model = "1PL")
 m3.1PL <- itemParEst(data3, model = "1PL")

 # merging the item parameters with rescaling
 irt.scale <- rbind(m0.1PL, itemRescale(m0.1PL, m1.1PL), itemRescale(m0.1PL, m2.1PL), 
                    itemRescale(m0.1PL, m3.1PL))

 # Generalized Lord's chi-squared statistics
 genLordChi2(irt.scale, nrFocal = 3)
 
## End(Not run)

Generalized Mantel-Haenszel DIF statistic

Description

Calculates the generalized Mantel-Haenszel statistics for DIF detection among multiple groups.

Usage

genMantelHaenszel(data, member, match = "score", anchor = 1:ncol(data))
genMantelHaenszel(data, member, match = "score", anchor = 1:ncol(data))

Arguments

`data`	numeric: the data matrix (one row per subject, one column per item).
`member`	numeric: the vector of group membership with zero and positive integer entries only. See Details.
`match`	specifies the type of matching criterion. Can be either `"score"` (default) to compute the test score, or any continuous or discrete variable with the same length as the number of rows of `data`. See Details.
`anchor`	a vector of integer values specifying which items (all by default) are currently considered as anchor (DIF free) items. See Details.

Details

This command computes the generalized Mantel-Haenszel statistic (Somes, 1986) in the specific framework of differential item functioning. It forms the basic command of difGMH and is specifically designed for this call.

Option anchor sets the items which are considered as anchor items for computing generalized Mantel-Haenszel statistics. Items other than the anchor items and the tested item are discarded. anchor must hold integer values specifying the column numbers of the corresponding anchor items. It is primarily designed to perform item purification.

Value

A vector with the values of the generalized Mantel-Haenszel DIF statistics.

Author(s)

References

Somes, G. W. (1986). The generalized Mantel-Haenszel statistic. The American Statistician, 40, 106-108. doi:10.2307/2684866

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Creating four groups according to gender (0 or 1) and trait anger
 # score ("Low" or "High")
 # Reference group: women with low trait anger score (<=20)
 group <- rep(0, nrow(verbal))
 group[Anger>20 & Gender==0] <- 1
 group[Anger<=20 & Gender==1] <- 2
 group[Anger>20 & Gender==1] <- 3

 # Without continuity correction
 genMantelHaenszel(verbal[,1:24], group)

 # Removing item 6 from the set of anchor items
 genMantelHaenszel(verbal[,1:24], group, anchor = c(1:5, 7:24))
 
## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Creating four groups according to gender (0 or 1) and trait anger
 # score ("Low" or "High")
 # Reference group: women with low trait anger score (<=20)
 group <- rep(0, nrow(verbal))
 group[Anger>20 & Gender==0] <- 1
 group[Anger<=20 & Gender==1] <- 2
 group[Anger>20 & Gender==1] <- 3

 # Without continuity correction
 genMantelHaenszel(verbal[,1:24], group)

 # Removing item 6 from the set of anchor items
 genMantelHaenszel(verbal[,1:24], group, anchor = c(1:5, 7:24))
 
## End(Not run)

Item parameter estimation for DIF detection using Rasch (1PL) model

Description

Fits the Rasch (1PL) model and returns related item parameter estimates.

Usage

itemPar1PL(data, engine = "ltm", discr = 1)
 itemPar1PL(data, engine = "ltm", discr = 1)

Arguments

`data`	numeric: the data matrix.
`engine`	character: the engine for estimating the 1PL model, either `"ltm"` (default) or `"lme4"`.
`discr`	either `NULL` or a real positive value for the common discrimination parameter (default is 1). Not used if `engine` is `"lme4"`. See Details.

Details

itemPar1PL permits to get item parameter estimates from the Rasch or 1PL model. The output is ordered such that it can be directly used with the general itemParEst command, as well as the methods of Lord (difLord) and Raju (difRaju) and Generalized Lord's (difGenLord) to detect differential item functioning.

The data is a matrix whose rows correspond to the subjects and columns to the items.

Missing values are allowed but must be coded as NA values. They are discarded for item parameter estimation.

The estimation engine is set by the engine argument. By default (engine="ltm"), the Rasch model is fitted using marginal maximum likelihood, by means of the function rasch from the ltm package (Rizopoulos, 2006). The other option, engine="lme4", permits to fit the Rasch model as a generalized linear mixed model, by means of the glmer function of the lme4 package (Bates and Maechler, 2009).

With the "ltm" engine, the common discrimination parameter is set equal to 1 by default. It is possible to fix another value through the argumentdiscr. Alternatively, this common discrimination parameter can be estimated (though not returned) by fixing discr to NULL. See the functionalities of rasch command for further details.

Value

A matrix with one row per item and two columns, the first one with item parameter estimates and the second one with the related standard errors.

Author(s)

References

Bates, D. and Maechler, M. (2009). lme4: Linear mixed-effects models using S4 classes. R package version 0.999375-31. http://CRAN.R-project.org/package=lme4

Rizopoulos, D. (2006). ltm: An R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17, 1–25. doi:10.18637/jss.v017.i05

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)
 
 # Getting item parameter estimates ('ltm' engine)
 itemPar1PL(verbal[, 1:24])

 # Estimating the common discrimination parameter instead
 itemPar1PL(verbal[, 1:24], discr = NULL)

 # Getting item parameter estimates ('lme4' engine) 
 itemPar1PL(verbal[, 1:24], engine = "lme4")
 
## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)
 
 # Getting item parameter estimates ('ltm' engine)
 itemPar1PL(verbal[, 1:24])

 # Estimating the common discrimination parameter instead
 itemPar1PL(verbal[, 1:24], discr = NULL)

 # Getting item parameter estimates ('lme4' engine) 
 itemPar1PL(verbal[, 1:24], engine = "lme4")
 
## End(Not run)

Item parameter estimation for DIF detection using 2PL model

Description

Fits the 2PL model and returns related item parameter estimates, standard errors and covariances between item parameters.

Usage

itemPar2PL(data)
 itemPar2PL(data)

Arguments

data

numeric: the data matrix.

Details

itemPar2PL permits to get item parameter estimates from the 2PL model. The output is ordered such that it can be directly used with the general itemParEst command, as well as the methods of Lord (difLord) and Raju (difRaju) and Generalized Lord's (difGenLord) to detect differential item functioning.

The data is a matrix whose rows correspond to the subjects and columns to the items.

Missing values are allowed but must be coded as NA values. They are discarded for item parameter estimation.

The 2PL model is fitted using marginal maximum likelihood by means of the functions from the ltm package (Rizopoulos, 2006).

Value

A matrix with one row per item and five columns: the estimates of item discrimination a and difficulty b parameters, the related standard errors se(a) and se(b), and the covariances cov(a,b), in this order.

Note

The 2PL model is fitted under the linear parametrization in ltm, the covariance matrix is extracted with the vcov() function, and final standard errors and covariances are derived by the Delta method. See Rizopoulos (2006) for further details, and the Note.pdf document in the difR package for mathematical details.

Author(s)

References

Rizopoulos, D. (2006). ltm: An R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17, 1–25. doi:10.18637/jss.v017.i05

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)

 # Getting item parameter estimates
 itemPar2PL(verbal[,1:24])
 
## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)

 # Getting item parameter estimates
 itemPar2PL(verbal[,1:24])
 
## End(Not run)

Item parameter estimation for DIF detection using 3PL model

Description

Fits the 3PL model and returns related item parameter estimates.

Usage

itemPar3PL(data)
itemPar3PL(data)

Arguments

data

numeric: the data matrix.

Details

itemPar3PL permits to get item parameter estimates from the 3PL model. The output is ordered such that it can be directly used with the general itemParEst command, as well as the methods of Lord (difLord) and Raju (difRaju) and Generalized Lord's (difGenLord) to detect differential item functioning.

The output consists of nine columns which are displayed in the following order. The first three columns hold the estimates of item discrimination a, difficulty b and pseudo-guessing c parameters. In the next three columns one can find the related standard errors se(a), se(b) and se(c). Eventually, the last three columns contain the covariances between item parameters, respectively cov(a,b), cov(a,c) and cov(b,c).

The data is a matrix whose rows correspond to the subjects and columns to the items.

Missing values are allowed but must be coded as NA values. They are discarded for item parameter estimation.

The 3PL model is fitted using marginal maximum likelihood by means of the functions from the ltm package (Rizopoulos, 2006).

Value

A matrix with one row per item and nine columns. See Details.

Note

The 3PL model is fitted under the linear parametrization in tpm, the covariance matrix is extracted with the vcov() function, and final standard errors and covariances are derived by the Delta method. See Rizopoulos (2006) for further details, and the Note.pdf document in the difR package for mathematical details.

Author(s)

References

Rizopoulos, D. (2006). ltm: An R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17, 1–25. doi:10.18637/jss.v017.i05

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)

 # Getting item parameter estimates
 itemPar3PL(verbal[,1:24])
 
## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)

 # Getting item parameter estimates
 itemPar3PL(verbal[,1:24])
 
## End(Not run)

Item parameter estimation for DIF detection using constrained 3PL model

Description

Fits the 3PL model with constrained pseudo-guessing values and returns related item parameter estimates.

Usage

itemPar3PLconst(data, c=rep(0,ncol(data)))
itemPar3PLconst(data, c=rep(0,ncol(data)))

Arguments

`data`	numeric: the data matrix.
`c`	numeric value or vector of constrained pseudo-guessing parameters. See Details.

Details

itemPar3PLconst permits to get item parameter estimates from the 3PL model for which the pseudo-guessing parameters are constrained to some fixed values. The output is ordered such that it can be directly used with the general itemParEst command, as well as the methods of Lord (difLord) and Raju (difRaju) and Generalized Lord's (difGenLord) to detect differential item functioning.

The output is similar to that of itemPar2PL method to fit the 2PL model; an additional column is included and holds the fixed pseudo-guessing parameter values.

The data is a matrix whose rows correspond to the subjects and columns to the items.

Missing values are allowed but must be coded as NA values. They are discarded for item parameter estimation.

The argument c can be either a single numeric value or a numeric vector of the same length of the number of items. In the former case, the pseudo-guessing parameters are considered to be all identical to the given c value; otherwise c is directly used to constraint these parameters.

The constrained 3PL model is fitted using marginal maximum likelihood by means of the functions from the ltm package (Rizopoulos, 2006).

Value

A matrix with one row per item and six columns: the item discrimination a and difficulty estimates b, the corresponding standard errors se(a) and se(b), the covariances cov(a,b) and the constrained pseudo-guessing values c.

Note

The constrained 3PL model is fitted under the linear parametrization in tpm, the covariance matrix is extracted with the vcov() function, and final standard errors and covariances are derived by the Delta method. See Rizopoulos (2006) for further details, and the Note.pdf document in the difR package for mathematical details.

Author(s)

References

Rizopoulos, D. (2006). ltm: An R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17, 1–25. doi:10.18637/jss.v017.i05

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)

 # Constraining all pseudo-guessing parameters to be equal to 0.05
 itemPar3PLconst(verbal[,1:24], c = 0.05)

 # Constraining pseudo-guessing values to  0.1 for the first 10 items,
 # and to 0.05 for the remaining ones
 itemPar3PLconst(verbal[,1:24], c = c(rep(0.1, 10), rep(0.05, 14)))
 
## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)

 # Constraining all pseudo-guessing parameters to be equal to 0.05
 itemPar3PLconst(verbal[,1:24], c = 0.05)

 # Constraining pseudo-guessing values to  0.1 for the first 10 items,
 # and to 0.05 for the remaining ones
 itemPar3PLconst(verbal[,1:24], c = c(rep(0.1, 10), rep(0.05, 14)))
 
## End(Not run)

Item parameter estimation for DIF detection

Description

Fits a specified logistic IRT model and returns related item parameter estimates.

Usage

itemParEst(data, model, c = NULL, engine = "ltm", discr = 1)
 itemParEst(data, model, c = NULL, engine = "ltm", discr = 1)

Arguments

`data`	numeric: the data matrix.
`model`	character: the IRT model to be fitted (either `"1PL"`, `"2PL"` or `"3PL"`).
`c`	optional numeric value or vector giving the values of the constrained pseudo-guessing parameters. See Details.
`engine`	character: the engine for estimating the 1PL model, either `"ltm"` (default) or `"lme4"`.
`discr`	either `NULL` or a real positive value for the common discrimination parameter (default is 1). Used only if `model` is `"1PL"` and `engine` is `"ltm"`. See Details.

Details

itemParEst permits to get item parameter estimates of some pre-specified logistic IRT model, together with estimates of the standard errors and the covariances between item parameters, if any. The output is ordered such that it can be directly used with the methods of Lord (difLord) and Raju (difRaju) and Generalized Lord's (difGenLord) to detect differential item functioning.

The data is a matrix whose rows correspond to the subjects and columns to the items.

Missing values are allowed but must be coded as NA values. They are discarded for item parameter estimation.

If the model is not the 1PL model, or if engine is equal to "ltm", the selected IRT model is fitted using marginal maximum likelihood by means of the functions from the ltm package (Rizopoulos, 2006). Otherwise, the 1PL model is fitted as a generalized linear mixed model, by means of the glmer function of the lme4 package (Bates and Maechler, 2009). With the "ltm" engine, the common discrimination parameter can be either fixed to a constant value using the discr argument, or it can be estimated (though not returned) by specifying discr to NULL. The default value of the common discrimination is 1.

The 3PL model can be fitted either unconstrained or by fixing the pseudo-guessing values. In the latter case the argument c holds either a numeric vector of same length of the number of items, with one value per item pseudo-guessing parameter, or a single value which is duplicated for all the items. If c is different from NULL then the 3PL model is always fitted (whatever the value of model).

Each row of the output matrix corresponds to one item of the data set; the number of columns depends on the fitted model. At most, nine columns are produced, with the unconstrained 3PL model. The order of the columns is the following: first, the estimates of item discrimination a, difficulty b and pseudo-guessing c; second, the corresponding standard errors se(a), se(b) and se(c); finally, the covariances between the item parameters, cov(a,b), cov(a,c) and cov(b,c).

If the 2PL model is fitted, only five columns are displayed: a, b, se(a), se(b) and cov(a,b). In case of the 1PL model, only b and se(b) are returned. If the constrained 3PL is considered, the output matrix holds six columns, the first five being identical to those from the 2PL model, and the last one holds the fixed pseudo-guessing parameters.

Value

A matrix with one row per item and at most nine columns, with item parameter estimates, standard errors and covariances, if any. See Details.

Note

Whenever making use of the ltm package to fit the IRT models, the linear parametrization is used, the covariance matrix is extracted with the vcov() function, and final standard errors and covariances are derived by the Delta method. See Rizopoulos (2006) for further details, and the Note.pdf document in the difR package for mathematical details.

Author(s)

References

Bates, D. and Maechler, M. (2009). lme4: Linear mixed-effects models using S4 classes. R package version 0.999375-31. http://CRAN.R-project.org/package=lme4

Rizopoulos, D. (2006). ltm: An R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17, 1–25. doi:10.18637/jss.v017.i05

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)

 # Estimation of the item parameters (1PL model, "ltm" engine)
 items.1PL <- itemParEst(verbal[,1:24], model = "1PL")

 # Estimation of the item parameters (1PL model, "ltm" engine,
 # estimated common discrimination parameter)
 items.1PL <- itemParEst(verbal[,1:24], model = "1PL", discr = NULL)

 # Estimation of the item parameters (1PL model, "lme4" engine)
 items.1PL <- itemParEst(verbal[,1:24], model = "1PL", engine = "lme4")

 # Estimation of the item parameters (2PL model)
 items.2PL <- itemParEst(verbal[,1:24], model = "2PL")

 # Estimation of the item parameters (3PL model)
 # items.3PL <- itemParEst(verbal[,1:24], model = "3PL")

 # Constraining all pseudo-guessing values to be equal to 0.05
 items.3PLc <- itemParEst(verbal[,1:24], model = "3PL", c = 0.05)

## End(Not run)
## Not run: 

 # Loading of the verbal data
 data(verbal)

 # Estimation of the item parameters (1PL model, "ltm" engine)
 items.1PL <- itemParEst(verbal[,1:24], model = "1PL")

 # Estimation of the item parameters (1PL model, "ltm" engine,
 # estimated common discrimination parameter)
 items.1PL <- itemParEst(verbal[,1:24], model = "1PL", discr = NULL)

 # Estimation of the item parameters (1PL model, "lme4" engine)
 items.1PL <- itemParEst(verbal[,1:24], model = "1PL", engine = "lme4")

 # Estimation of the item parameters (2PL model)
 items.2PL <- itemParEst(verbal[,1:24], model = "2PL")

 # Estimation of the item parameters (3PL model)
 # items.3PL <- itemParEst(verbal[,1:24], model = "3PL")

 # Constraining all pseudo-guessing values to be equal to 0.05
 items.3PLc <- itemParEst(verbal[,1:24], model = "3PL", c = 0.05)

## End(Not run)

Rescaling item parameters by equal means anchoring

Description

Rescale the item parameters from one data set to the scale of the parameters from another data set, using equal means anchoring.

Usage

itemRescale(mR, mF, items = 1:nrow(mR))
 itemRescale(mR, mF, items = 1:nrow(mR))

Arguments

`mR`	numeric: a matrix of item parameter estimates (one row per item) which constitutes the reference scale. See Details.
`mF`	numeric: a matrix of item parameter estimates (one row per item) which have to be rescaled. See Details.
`items`	a numeric vector of integer values specifying which items are used for equal means anchoring. See Details.

Details

The matrices mR and mF must have the same format as the output of the command itemParEst and one the possible models (1PL, 2PL, 3PL or constrained 3PL). The number of columns therefore equals two, five, nine or six, respectively.

Rescaling is performed by equal means anchoring (Cook and Eignor, 1991). The items involved in the anchoring process are specified by means of their row number in either mR or mF, and are passed through the items argument.

itemRescale primarily serves as a routine for item purification in Lord (difLord) and Raju (difRaju) Generalized Lord's (difGenLord) methods of DIF identification (Candell and Drasgow, 1988).

Value

A matrix of the same format as mF with the rescaled item parameters.

Author(s)

References

Cook, L. L. and Eignor, D. R. (1991). An NCME instructional module on IRT equating methods. Educational Measurement: Issues and Practice, 10, 37-45.

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Splitting the data set into reference and focal groups
 nF <- sum(Gender)
 nR <- nrow(verbal)-nF
 data.ref <- verbal[,1:24][order(Gender),][1:nR,]
 data.focal <- verbal[,1:24][order(Gender),][(nR+1):(nR+nF),]

 # Estimating item parameters in each data set with 1PL model
 mR <- itemPar1PL(data.ref)
 mF <- itemPar1PL(data.focal)

 # Rescaling focal group item parameters, using all items for anchoring
 itemRescale(mR, mF)

 # Rescaling focal group item parameters, using the first 10 items for anchoring
 itemRescale(mR, mF, items = 1:10)

 # Estimating item parameters in each data set with 2PL model
 mR <- itemPar2PL(data.ref)
 mF <- itemPar2PL(data.focal)

 # Rescaling focal group item parameters, using all items for anchoring
 itemRescale(mR, mF)
 
## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Splitting the data set into reference and focal groups
 nF <- sum(Gender)
 nR <- nrow(verbal)-nF
 data.ref <- verbal[,1:24][order(Gender),][1:nR,]
 data.focal <- verbal[,1:24][order(Gender),][(nR+1):(nR+nF),]

 # Estimating item parameters in each data set with 1PL model
 mR <- itemPar1PL(data.ref)
 mF <- itemPar1PL(data.focal)

 # Rescaling focal group item parameters, using all items for anchoring
 itemRescale(mR, mF)

 # Rescaling focal group item parameters, using the first 10 items for anchoring
 itemRescale(mR, mF, items = 1:10)

 # Estimating item parameters in each data set with 2PL model
 mR <- itemPar2PL(data.ref)
 mF <- itemPar2PL(data.focal)

 # Rescaling focal group item parameters, using all items for anchoring
 itemRescale(mR, mF)
 
## End(Not run)

Logistic regression DIF statistic

Description

Calculates the "logistic regression" likelihood-ratio statistics and effect sizes for DIF detection.

Usage

Logistik(data, member, member.type = "group", match = "score",
	 anchor = 1:ncol(data), type = "both", criterion = "LRT", all.cov = FALSE)
 Logistik(data, member, member.type = "group", match = "score",
	 anchor = 1:ncol(data), type = "both", criterion = "LRT", all.cov = FALSE)

Arguments

`data`	numeric: the data matrix (one row per subject, one column per item).
`member`	numeric or factor: the vector of group membership. Can either take two distinct values (zero for the reference group and one for the focal group) or be a continuous vector. See Details.
`member.type`	character: either `"group"` (default) to specify that group membership is made of two groups, or `"cont"` to indicate that group membership is based on a continuous criterion. See Details.
`match`	specifies the type of matching criterion. Can be either `"score"` (default) to compute the test score, or any continuous or discrete variable with the same length as the number of rows of `data`. See Details.
`anchor`	a vector of integer values specifying which items (all by default) are currently considered as anchor (DIF free) items. Ignored if `match` is not `"score"`. See Details.
`type`	a character string specifying which DIF effects must be tested. Possible values are `"both"` (default), `"udif"` and `"nudif"`. See Details.
`criterion`	a character string specifying which DIF statistic is computed. Possible values are `"LRT"` (default) or `"Wald"`. See Details.
`all.cov`	logical: should all covariance matrices of model parameter estimates be returned (as lists) for both nested models and all items? (default is `FALSE`.

Details

This command computes the logistic regression statistic (Swaminathan and Rogers, 1990) in the specific framework of differential item functioning. It forms the basic command of difLogistic and is specifically designed for this call.

If the member.type argument is set to "group", the member argument must be a vector with two distinct (numeric or factor) values, say 0 and 1 (for the reference and focal groups respectively). Those values are internally transformed onto factors to denote group membership. The three possible models to be fitted are then:

$M_0: logit (\pi_g) = \alpha + \beta X + \gamma_g + \delta_g X$

$M_1: logit (\pi_g) = \alpha + \beta X + \gamma_g$

$M_2: logit (\pi_g) = \alpha + \beta X$

where $\pi_g$ is the probability of answering correctly the item in group g and $X$ is the matching variable. Parameters $\alpha$ and $\beta$ are the intercept and the slope of the logistic curves (common to all groups), while $\gamma_g$ and $\delta_g$ are group-specific parameters. For identification reasons the parameters $\gamma_0$ and $\delta_0$ for reference group ( $g=0$ ) are set to zero. The parameter $\gamma_1$ of the focal group ( $g=1$ ) represents the uniform DIF effect, and the parameter $\delta_1$ is used to model nonuniform DIF effect. The models are fitted with the glm function.

If member.type is set to "cont", then "group membership" is replaced by a continuous or discrete variable, given by the member argument, and the models above are written as

$M_0: logit (\pi_g) = \alpha + \beta X + \gamma Y+ \delta X Y$

$M_1: logit (\pi_g) = \alpha + \beta X + \gamma Y$

$M_2: logit (\pi_g) = \alpha + \beta X$

where Y is the group variable. Parameters $\gamma$ and $\delta$ act now as the $\gamma_1$ and $\delta_1$ DIF parameters.

Two types of DIF statistics can be computed: the likelihood ratio test statistics, obtained by comparing the fit of two nested models, and the Wald statistics, obtained with an appropriate contrast matrix for testing the model parameters (Johnson and Wichern, 1998). These are specified by the argument criterion, with respective values "LRT" and "Wald". By default, the LRT statistics are computed.

If criterion is "LRT", the argument type determines the models to be compared by means of the LRT statistics. The three possible values of type are: type="both" (default) which tests the hypothesis $H_0: \gamma_1 = \delta_1=0$ (or $H_0: \gamma = \delta=0$ ) by comparing models $M_0$ and $M_2$ ; type="nudif" which tests the hypothesis $H_0: \delta_1 = 0$ (or $H_0: \delta = 0$ ) by comparing models $M_0$ and $M_1$ ; and type="udif" which tests the hypothesis $H_0: \gamma_1 = 0$ (or $H_0: \gamma = 0$ ) by comparing models $M_1$ and $M_2$ (assuming that $\delta_1 = 0$ or $\delta = 0$ ). In other words, type="both" tests for DIF (without distinction between uniform and nonuniform effects), while type="udif" and type="nudif" test for uniform and nonuniform DIF, respectively.

If criterion is "Wald", the argument type determines the logistic model to be considered and the appropriate contrast matrix. If type=="both", the considered model is model $M_0$ and the contrast matrix has two rows, (0,0,1,0) and (0,0,0,1). If type=="nudif", the considered model is also model $M_0$ but the contrast matrix has only one row, (0,0,0,1). Eventually, if type=="udif", the considered model is model $M_1$ and the contrast matrix has one row, (0,0,1).

The vector of group membership, specified with member argument, must hold only zeros and ones, a value of zero corresponding to the reference group and a value of one to the focal group.

Option anchor sets the items which are considered as anchor items for computing the test scores and related logistic regression DIF statistics. Items other than the anchor items and the tested item are discarded. anchor must hold integer values specifying the column numbers of the corresponding anchor items. It is mainly designed to perform item purification. Note that this option is discarded when match is not "score".

The output contains: the selected DIF statistics (either the LRT or the Wald statistic) computed for each item, two matrices with the parameter estimates of both models (for each item) and two matrices of related standard error values. In addition, Nagelkerke's $R^2$ coefficients (Nagelkerke, 1991) are computed for each model and the output returns both, the vectors of $R^2$ coefficients for each model and the differences in these coefficients. Such differences are used as measures of effect size by the difLogistic command; see Gomez-Benito, Dolores Hidalgo and Padilla (2009), Jodoin and Gierl (2001) and Zumbo and Thomas (1997). The criterion and member.type arguments are also returned, as well as a character argument named match that specifies the type of matching criterion that was used.

Value

A list with nine components:

`stat`	the values of the logistic regression DIF statistics.
`R2M0`	the values of Nagelkerke's R^2 coefficients for the "full" model.
`R2M1`	the values of Nagelkerke's R^2 coefficients for the "simpler" model.
`deltaR2`	the differences between Nagelkerke's $R^2$ coefficients of the tested models. See Details.
`parM0`	a matrix with one row per item and four columns, holding successively the fitted parameters $\hat{\alpha}$ , $\hat{\beta}$ , $\hat{\gamma}_1$ and $\hat{\delta}_1$ of the "full" model ( $M_0$ if `type="both"` or `type="nudif"`, $M_1$ if `type="udif"`).
`parM1`	the same matrix as `parM0` but with fitted parameters for the "simpler" model ( $M_1$ if `type="nudif"`, $M_2$ if `type="both"` or `type="udif"`).
`seM0`	a matrix with the standard error values of the parameter estimates in matrix `parM0`.
`seM1`	a matrix with the standard error values of the parameter estimates in matrix `parM1`.
`cov.M0`	either `NULL` (if `all.cov` argument is `FALSE`) or a list of covariance matrices of parameter estimates of the "full" model ( $M_0$ ) for each item (if `all.cov` argument is `TRUE`).
`cov.M1`	either `NULL` (if `all.cov` argument is `FALSE`) or a list of covariance matrices of parameter estimates of the "reduced" model ( $M_1$ ) for each item (if `all.cov` argument is `TRUE`).
`criterion`	the value of the `criterion` argument.
`member.type`	the value of the `member.type` argument.
`match`	a character string, either `"score"` or `"matching variable"` depending on the `match` argument.

Author(s)

References

Johnson, R. A. and Wichern, D. W. (1998). Applied multivariate statistical analysis (fourth edition). Upper Saddle River, NJ: Prentice-Hall.

Nagelkerke, N. J. D. (1991). A note on a general definition of the coefficient of determination. Biometrika, 78, 691-692. doi:10.1093/biomet/78.3.691

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)

 # Testing both types of DIF simultaneously
 # With all items, test score as matching criterion
 Logistik(verbal[,1:24], verbal[,26])

 # Returning all covariance matrices of model parameters
 Logistik(verbal[,1:24], verbal[,26], all.cov = TRUE)

 # Testing both types of DIF simultaneously
 # With all items and Wald test
 Logistik(verbal[,1:24], verbal[,26], criterion = "Wald")

 # Removing item 6 from the set of anchor items
 Logistik(verbal[,1:24], verbal[,26], anchor = c(1:5, 7:24))

 # Testing for nonuniform DIF
 Logistik(verbal[,1:24], verbal[,26], type = "nudif")

 # Testing for uniform DIF
 Logistik(verbal[,1:24], verbal[,26], type = "udif")

 # Using the "anger" trait variable as matching criterion
 Logistik(verbal[,1:24],verbal[,26], match = verbal[,25])

 # Using the "anger" trait variable as group membership
 Logistik(verbal[,1:24],verbal[,25], member.type = "cont")
 
## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)

 # Testing both types of DIF simultaneously
 # With all items, test score as matching criterion
 Logistik(verbal[,1:24], verbal[,26])

 # Returning all covariance matrices of model parameters
 Logistik(verbal[,1:24], verbal[,26], all.cov = TRUE)

 # Testing both types of DIF simultaneously
 # With all items and Wald test
 Logistik(verbal[,1:24], verbal[,26], criterion = "Wald")

 # Removing item 6 from the set of anchor items
 Logistik(verbal[,1:24], verbal[,26], anchor = c(1:5, 7:24))

 # Testing for nonuniform DIF
 Logistik(verbal[,1:24], verbal[,26], type = "nudif")

 # Testing for uniform DIF
 Logistik(verbal[,1:24], verbal[,26], type = "udif")

 # Using the "anger" trait variable as matching criterion
 Logistik(verbal[,1:24],verbal[,26], match = verbal[,25])

 # Using the "anger" trait variable as group membership
 Logistik(verbal[,1:24],verbal[,25], member.type = "cont")
 
## End(Not run)

Lord's chi-square DIF statistic

Description

Calculates the Lord's chi-square statistics for DIF detection.

Usage

LordChi2(mR, mF)
 LordChi2(mR, mF)

Arguments

`mR`	numeric: the matrix of item parameter estimates (one row per item) for the reference group. See Details.
`mF`	numeric: the matrix of item parameter estimates (one row per item) for the focal group. See Details.

Details

This command computes the Lord's chi-square statistic (Lord, 1980) in the specific framework of differential item functioning. It forms the basic command of difLord and is specifically designed for this call.

The matrices mR and mF must have the same format as the output of the command itemParEst with one the possible models (1PL, 2PL, 3PL or constrained 3PL). The number of columns therefore equals two, five, nine or six, respectively. Moreover, item parameters of the focal must be on the same scale of that of the reference group. If not, make use of e.g. equal means anchoring (Cook and Eignor, 1991) and itemRescale to transform them adequately.

Value

A vector with the values of the Lord's chi-square DIF statistics.

Note

WARNING: the previous versions of LordChi2 were holding an error: under the 3PL model, the covariance matrices $Sig_1$ and $Sig_2$ were wrongly computed as the variance of the pseudo-guessing parameters were replaced by the parameter estimates. This has been fixed from version 4.0 of difR. Many thanks to J. Patrick Meyer (Curry School of Education, University of Virginia) for having discovered this mistake.

Author(s)

References

Cook, L. L. and Eignor, D. R. (1991). An NCME instructional module on IRT equating methods. Educational Measurement: Issues and Practice, 10, 37-45.

Lord, F. (1980). Applications of item response theory to practical testing problems. Hillsdale, NJ: Lawrence Erlbaum Associates.

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Splitting the data into reference and focal groups
 nF <- sum(Gender)
 nR <- nrow(verbal)-nF
 data.ref <- verbal[, 1:24][order(Gender),][1:nR,]
 data.focal <- verbal[, 1:24][order(Gender),][(nR+1):(nR+nF),]

 # Pre-estimation of the item parameters (1PL model)
 mR <- itemParEst(data.ref, model = "1PL")
 mF <- itemParEst(data.focal, model = "1PL")
 mF <- itemRescale(mR, mF)
 LordChi2(mR, mF)

 # Pre-estimation of the item parameters (2PL model)
 mR <- itemParEst(data.ref, model = "2PL")
 mF <- itemParEst(data.focal, model = "2PL")
 mF <- itemRescale(mR, mF)
 LordChi2(mR, mF)

 # Pre-estimation of the item parameters (constrained 3PL model)
 mR <- itemParEst(data.ref, model = "3PL", c = 0.05)
 mF <- itemParEst(data.focal, model = "3PL", c = 0.05)
 mF <- itemRescale(mR, mF)
 LordChi2(mR, mF)
 
## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Splitting the data into reference and focal groups
 nF <- sum(Gender)
 nR <- nrow(verbal)-nF
 data.ref <- verbal[, 1:24][order(Gender),][1:nR,]
 data.focal <- verbal[, 1:24][order(Gender),][(nR+1):(nR+nF),]

 # Pre-estimation of the item parameters (1PL model)
 mR <- itemParEst(data.ref, model = "1PL")
 mF <- itemParEst(data.focal, model = "1PL")
 mF <- itemRescale(mR, mF)
 LordChi2(mR, mF)

 # Pre-estimation of the item parameters (2PL model)
 mR <- itemParEst(data.ref, model = "2PL")
 mF <- itemParEst(data.focal, model = "2PL")
 mF <- itemRescale(mR, mF)
 LordChi2(mR, mF)

 # Pre-estimation of the item parameters (constrained 3PL model)
 mR <- itemParEst(data.ref, model = "3PL", c = 0.05)
 mF <- itemParEst(data.focal, model = "3PL", c = 0.05)
 mF <- itemRescale(mR, mF)
 LordChi2(mR, mF)
 
## End(Not run)

Likelihood-Ratio Test DIF statistic

Description

Calulates Likelihoo-Ratio Test (LRT) statistics for DIF detection.

Usage

LRT(data, member)
LRT(data, member)

Arguments

`data`	numeric: the data matrix (one row per subject, one column per item).
`member`	numeric: the vector of group membership with zero and one entries only. See Details.

Details

This command computes the likelihood-ratio test statistic (Thissen, Steinberg and Wainer, 1988) in the specific framework of differential item functioning. It forms the basic command of difLRT and is specifically designed for this call.

The data are passed through the data argument, with one row per subject and one column per item. Missing values are allowed but must be coded as NA values.

The vector of group membership, specified with member argument, must hold only zeros and ones, a value of zero corresponding to the reference group and a value of one to the focal group.

The LRT DIF statistic is computed for each item separately, using all other items as anchor items.

Value

A vector with the values of the LRT DIF statistics.

Note

Because of the fitting of the modified Rasch model with glmer the process can be very time consuming (see the Details section of difLRT).

Author(s)

References

Bates, D. and Maechler, M. (2009). lme4: Linear mixed-effects models using S4 classes. R package version 0.999375-31. http://CRAN.R-project.org/package=lme4

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Excluding the "Anger" variable
 verbal <- verbal[colnames(verbal)!="Anger"]

 # Keeping the first 5 items and the first 50 subjects
 # (this is an artificial simplification to reduce the computational time)
 # Sixth column holds the group membership
 verbal <- verbal[1:50, c(1:5, 25)]

 # Likelihood-ratio statistics
 LRT(verbal[,1:5], verbal[,6])
 
## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Excluding the "Anger" variable
 verbal <- verbal[colnames(verbal)!="Anger"]

 # Keeping the first 5 items and the first 50 subjects
 # (this is an artificial simplification to reduce the computational time)
 # Sixth column holds the group membership
 verbal <- verbal[1:50, c(1:5, 25)]

 # Likelihood-ratio statistics
 LRT(verbal[,1:5], verbal[,6])
 
## End(Not run)

Mantel-Haenszel DIF statistic

Description

Calculates Mantel-Haenszel statistics for DIF detection.

Usage

mantelHaenszel(data, member, match = "score", correct = TRUE, exact = FALSE,
  anchor = 1:ncol(data))
 mantelHaenszel(data, member, match = "score", correct = TRUE, exact = FALSE,
  anchor = 1:ncol(data))

Arguments

`data`	numeric: the data matrix (one row per subject, one column per item).
`member`	numeric: the vector of group membership with zero and one entries only. See Details.
`match`	specifies the type of matching criterion. Can be either `"score"` (default) to compute the test score, or any continuous or discrete variable with the same length as the number of rows of `data`. See Details.
`correct`	logical: should the continuity correction be used? (default is `TRUE`).
`exact`	logical: should an exact test be computed? (default is `FALSE`).
`anchor`	a vector of integer values specifying which items (all by default) are currently considered as anchor (DIF free) items. See Details.

Details

This command basically computes the Mantel-Haenszel (1959) statistic in the specific framework of differential item functioning. It forms the basic command of difMH and is specifically designed for this call.

The data are passed through the data argument, with one row per subject and one column per item.

Missing values are allowed for item responses (not for group membership) but must be coded as NA values. They are discarded from sum-score computation.

The vector of group membership, specified with member argument, must hold only zeros and ones, a value of zero corresponding to the reference group and a value of one to the focal group.

By default, the continuity correction factor -0.5 is used (Holland and Thayer, 1988). One can nevertheless remove it by specifying correct=FALSE.

Option anchor sets the items which are considered as anchor items for computing Mantel-Haenszel statistics. Items other than the anchor items and the tested item are discarded. anchor must hold integer values specifying the column numbers of the corresponding anchor items. It is primarily designed to perform item purification.

In addition to the Mantel-Haenszel statistics to identify DIF items, mantelHaenszel computes the estimates of the common odds ratio $\alpha_{MH}$ which are used for measuring the effect size of the items (Holland and Thayer, 1985, 1988). They are returned in the resAlpha argument of the output list. Moreover, the logarithm of $\alpha_{MH}$ , say $\lambda_{MH}$ , is asymptotically distributed and its variance is computed and returned into the varLambda argument. Note that this variance is the one proposed by Philips and Holland (1987), since it seems the most accurate expression for the variance of $\lambda_{MH}$ (Penfield and Camilli, 2007).

Value

A list with several arguments:

`resMH`	the vector of the Mantel-Haenszel DIF statistics (either asymptotic or exact).
`resAlpha`	the vector of the (asymptotic) Mantel-Haenszel estimates of the common odds ratios. Returned only if `exact` is `FALSE`.
`varLambda`	the (asymptotic) variance of the $\lambda_{MH}$ statistic. Returned only if `exact` is `FALSE`.
`Pval`	the exact P-values of the MH test. Returned only if `exact` is `TRUE`.
`match`	a character string, either `"score"` or `"matching variable"` depending on the `match` argument.

Author(s)

References

Agresti, A. (1990). Categorical data analysis. New York: Wiley.

Agresti, A. (1992). A survey of exact inference for contingency tables. Statistical Science, 7, 131-177. doi:10.1214/ss/1177011454

Holland, P. W. and Thayer, D. T. (1985). An alternative definition of the ETS delta scale of item difficulty. Research Report RR-85-43. Princeton, NJ: Educational Testing Service.

Mantel, N. and Haenszel, W. (1959). Statistical aspects of the analysis of data from retrospective studies of disease. Journal of the National Cancer Institute, 22, 719-748.

Philips, A., and Holland, P. W. (1987). Estimators of the Mantel-Haenszel log odds-ratio estimate. Biometrics, 43, 425-431. doi:10.2307/2531824

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)

 # With and without continuity correction
 mantelHaenszel(verbal[,1:24], verbal[,26])
 mantelHaenszel(verbal[,1:24], verbal[,26], correct = FALSE)
 
 # Exact test
 mantelHaenszel(verbal[,1:24], verbal[,26], exact = TRUE)

 # Removing item 6 from the set of anchor items
 mantelHaenszel(verbal[,1:24], verbal[,26], anchor = c(1:5,7:24))
 
## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)

 # With and without continuity correction
 mantelHaenszel(verbal[,1:24], verbal[,26])
 mantelHaenszel(verbal[,1:24], verbal[,26], correct = FALSE)
 
 # Exact test
 mantelHaenszel(verbal[,1:24], verbal[,26], exact = TRUE)

 # Removing item 6 from the set of anchor items
 mantelHaenszel(verbal[,1:24], verbal[,26], anchor = c(1:5,7:24))
 
## End(Not run)

Raju's area DIF statistic

Description

Calculates the Raju's statistics for DIF detection.

Usage

RajuZ(mR, mF, signed = FALSE)
 RajuZ(mR, mF, signed = FALSE)

Arguments

`mR`	numeric: the matrix of item parameter estimates (one row per item) for the reference group. See Details.
`mF`	numeric: the matrix of item parameter estimates (one row per item) for the focal group. See Details.
`signed`	logical: should the signed area be computed, or the unsigned (i.e. in absolute value) ara? Default is `FALSE`, i.e. the unsigned area. See Details.

Details

This command computes the Raju's area statistic (Raju, 1988, 1990) in the specific framework of differential item functioning. It forms the basic command of difRaju and is specifically designed for this call.

The matrices mR and mF must have the same format as the output of the command itemParEst and one the possible models (1PL, 2PL or constrained 3PL). The number of columns therefore equals two, five or six, respectively. Note that the unconstrained 3PL model cannot be used in this method: all pseudo-guessing parameters must be equal in both groups of subjects. Moreover, item parameters of the focal must be on the same scale of that of the reference group. If not, make use of e.g. equal means anchoring (Cook and Eignor, 1991) and itemRescale to transform them adequately.

By default, the unsigned area, given by Equation (57) in Raju (1990), is computed. It makes use of Equations (14), (15), (23) and (46) for the numerator, and Equations (17), (33) to (39), and (52) for the denominator of the Z statistic. However, the signed area, given by Equation (56) in Raju (1990), can be used instead. In this case, Equations (14), (21) and (44) are used for the numerator, and Equations (17), (25) and (48) for the denominator. The choice of the type of area is fixed by the logical signed argument, with default value FALSE.

Value

A list with two components:

`res`	a matrix with one row per item and three columns, holding respectively Raju's area between the two item characteristic curves, its standard error and the Raju DIF statistic (the latter being the ratio of the first two columns).
`signed`	the value of the `signed` argument.

Author(s)

References

Cook, L. L. and Eignor, D. R. (1991). An NCME instructional module on IRT equating methods. Educational Measurement: Issues and Practice, 10, 37-45.

Raju, N.S. (1988). The area between two item characteristic curves. Psychometrika, 53, 495-502. doi:10.1007/BF02294403

Examples

## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Splitting the data into reference and focal groups
 nF <- sum(Gender)
 nR <- nrow(verbal)-nF
 data.ref <- verbal[,1:24][order(Gender),][1:nR,]
 data.focal <- verbal[,1:24][order(Gender),][(nR+1):(nR+nF),]

 # Pre-estimation of the item parameters (1PL model)
 mR <- itemParEst(data.ref,model = "1PL")
 mF <- itemParEst(data.focal,model = "1PL")
 mF <- itemRescale(mR, mF)

 # Signed and unsigned Raju statistics
 RajuZ(mR, mF)
 RajuZ(mR, mF, signed = TRUE)

 # Pre-estimation of the item parameters (2PL model)
 mR <- itemParEst(data.ref, model = "2PL")
 mF <- itemParEst(data.focal, model = "2PL")
 mF <- itemRescale(mR, mF)

 # Signed and unsigned Raju statistics
 RajuZ(mR, mF)
 RajuZ(mR, mF, signed = TRUE)
 
 # Pre-estimation of the item parameters (constrained 3PL model)
 mR <- itemParEst(data.ref, model = "3PL", c = 0.05)
 mF <- itemParEst(data.focal, model = "3PL", c =0 .05)
 mF <- itemRescale(mR, mF)
 
 # Signed and unsigned Raju statistics
 RajuZ(mR, mF)
 RajuZ(mR, mF, signed = TRUE)
 
## End(Not run)
 ## Not run: 

 # Loading of the verbal data
 data(verbal)
 attach(verbal)

 # Splitting the data into reference and focal groups
 nF <- sum(Gender)
 nR <- nrow(verbal)-nF
 data.ref <- verbal[,1:24][order(Gender),][1:nR,]
 data.focal <- verbal[,1:24][order(Gender),][(nR+1):(nR+nF),]

 # Pre-estimation of the item parameters (1PL model)
 mR <- itemParEst(data.ref,model = "1PL")
 mF <- itemParEst(data.focal,model = "1PL")
 mF <- itemRescale(mR, mF)

 # Signed and unsigned Raju statistics
 RajuZ(mR, mF)
 RajuZ(mR, mF, signed = TRUE)

 # Pre-estimation of the item parameters (2PL model)
 mR <- itemParEst(data.ref, model = "2PL")
 mF <- itemParEst(data.focal, model = "2PL")
 mF <- itemRescale(mR, mF)

 # Signed and unsigned Raju statistics
 RajuZ(mR, mF)
 RajuZ(mR, mF, signed = TRUE)
 
 # Pre-estimation of the item parameters (constrained 3PL model)
 mR <- itemParEst(data.ref, model = "3PL", c = 0.05)
 mF <- itemParEst(data.focal, model = "3PL", c =0 .05)
 mF <- itemRescale(mR, mF)
 
 # Signed and unsigned Raju statistics
 RajuZ(mR, mF)
 RajuZ(mR, mF, signed = TRUE)
 
## End(Not run)

Selection of one of the DIF detection methods

Description

This function performs DIF detection for one pre-specified method.

Usage

selectDif(Data, group, focal.name, method, anchor = NULL, props = NULL, 
 	thrTID = 1.5, alpha = 0.05, MHstat = "MHChisq", correct = TRUE, 
 	exact = FALSE, stdWeight = "focal", thrSTD = 0.1, BDstat = "BD", 
 	member.type = "group", match = "score", type = "both", criterion = "LRT", 
 	model = "2PL", c = NULL, engine = "ltm", discr = 1, irtParam = NULL, 
 	same.scale = TRUE, signed = FALSE, purify = FALSE, purType = "IPP1", 
 	nrIter = 10, extreme = "constraint", const.range = c(0.001, 0.999), 
 	nrAdd = 1, p.adjust.method = NULL, save.output = FALSE, 
 	output = c("out", "default"))
 selectDif(Data, group, focal.name, method, anchor = NULL, props = NULL, 
 	thrTID = 1.5, alpha = 0.05, MHstat = "MHChisq", correct = TRUE, 
 	exact = FALSE, stdWeight = "focal", thrSTD = 0.1, BDstat = "BD", 
 	member.type = "group", match = "score", type = "both", criterion = "LRT", 
 	model = "2PL", c = NULL, engine = "ltm", discr = 1, irtParam = NULL, 
 	same.scale = TRUE, signed = FALSE, purify = FALSE, purType = "IPP1", 
 	nrIter = 10, extreme = "constraint", const.range = c(0.001, 0.999), 
 	nrAdd = 1, p.adjust.method = NULL, save.output = FALSE, 
 	output = c("out", "default"))

Arguments

`Data`	numeric: either the data matrix only, or the data matrix plus the vector of group membership. See Details.
`group`	numeric or character: either the vector of group membership or the column indicator (within data) of group membership. See Details.
`focal.name`	numeric or character indicating the level of `group` which corresponds to the focal group.
`method`	character: the name of the selected method. Possible values are `"TID"`, `"MH"`, `"Std"`, `"Logistic"`, `"BD"`, `"SIBTEST"`, `"Lord"`, `"Raju"` and `"LRT"`. See Details.
`anchor`	either `NULL` (default) or a vector of item names (or identifiers) to specify the anchor items. See Details.
`props`	either `NULL` (default) or a two-column matrix with proportions of success in the reference group and the focal group. See Details .
`thrTID`	numeric: the threshold for detecting DIF items with TID method (default is 1.5).
`alpha`	numeric: significance level (default is 0.05).
`MHstat`	character: specifies the DIF statistic to be used for DIF identification. Possible values are `"MHChisq"` (default) and `"logOR"`. See Details .
`correct`	logical: should the continuity correction be used? (default is TRUE).
`exact`	logical: should an exact test be computed? (default is `FALSE`).
`stdWeight`	character: the type of weights used for the standardized P-DIF statistic. Possible values are `"focal"` (default), `"reference"` and `"total"`. See Details.
`thrSTD`	numeric: the threshold (cut-score) for standardized P-DIF statistic (default is 0.10).
`BDstat`	character specifying the DIF statistic to be used. Possible values are `"BD"` (default) and `"trend"`. See Details.
`member.type`	character: either `"group"` (default) to specify that group membership is made of two groups, or `"cont"` to indicate that group membership is based on a continuous criterion. See Details.
`match`	specifies the type of matching criterion. Can be either `"score"` (default) to compute the test score, or any continuous or discrete variable with the same length as the number of rows of `Data`. See Details.
`type`	a character string specifying which DIF effects must be tested. Possible values are `"both"` (default), `"udif"` and `"nudif"`. See Details.
`criterion`	a character string specifying which DIF statistic is computed. Possible values are `"LRT"` (default) or `"Wald"`. See Details.
`model`	character: the IRT model to be fitted (either `"1PL"`, `"2PL"` or `"3PL"`). Default is `"2PL"`.
`c`	optional numeric value or vector giving the values of the constrained pseudo-guessing parameters. See Details.
`engine`	character: the engine for estimating the 1PL model, either `"ltm"` (default) or `"lme4"`.
`discr`	either `NULL` or a real positive value for the common discrimination parameter (default is 1). Used onlky if `model` is `"1PL"` and `engine` is `"ltm"`. See Details.
`irtParam`	matrix with 2J rows (where J is the number of items) and at most 9 columns containing item parameters estimates. See Details.
`same.scale`	logical: are the item parameters of the `irtParam` matrix on the same scale? (default is "TRUE"). See Details.
`signed`	logical: should the Raju's statistics be computed using the signed (`TRUE`) or unsigned (`FALSE`, default) area? See Details.
`purify`	logical: should the method be used iteratively to purify the set of anchor items? (default is FALSE).
`purType`	character: the type of purification process to be run. Possible values are `"IPP1"` (default), `"IPP2"` and `"IPP3"`. Ignored if `purify` is `FALSE` or if `method` is not `"TID"`.
`nrIter`	numeric: the maximal number of iterations in the item purification process (default is 10).
`extreme`	character: the method used to modify the extreme proportions. Possible values are `"constraint"` (default) or `"add"`. Ignored if `method` is not `"TID"`.
`const.range`	numeric: a vector of two constraining proportions. Default values are 0.001 and 0.999. Ignored if `method` is not `"TID"` or if `extreme` is `"add"`.
`nrAdd`	integer: the number of successes and the number of failures to add to the data in order to adjust the proportions. Default value is 1. Ignored if `method` is not `"TID"` or if `extreme` is `"constraint"`.
`p.adjust.method`	either `NULL` (default) or the acronym of the method for p-value adjustment for multiple comparisons. See Details.
`save.output`	logical: should the output be saved into a text file? (Default is `FALSE`).
`output`	character: a vector of two components. The first component is the name of the output file, the second component is either the file path or `"default"` (default value). See Details.

Details

This is a generic function which calls one of the DIF detection methods and displays its output. It is mainly used as a routine for dichoDif command.

The possible methods are:

"TID" for Transformed Item Difficulties (TID) method (Angoff and Ford, 1973),
"MH" for mantel-Haenszel (Holland and Thayer, 1988),
"Std" for standardization (Dorans and Kulick, 1986),
"BD" for Breslow-Day method (Penfield, 2003),
"Logistic" for logistic regression (Swaminathan and Rogers, 1990),
"SIBTEST" for SIBTEST (Shealy and Stout) and Crossing-SIBTEST (Chalmers, 2018; Li and Stout, 1996) methods,
"Lord" for Lord's chi-square test (Lord, 1980),
"Raju" for Raju's area method (Raju, 1990), and
"LRT" for likelihood-ratio test method (Thissen, Steinberg and Wainer, 1988).

The vector of group membership must hold only two different values, either as numeric or character. The focal group is defined by the argument focal.name.

For "MH", "Std", "Logistic" and "BD" methods, the matching criterion can be either the test score or any other continuous or discrete variable to be passed in the selected DIF function. This is specified by the match argument. By default, it takes the value "score" and the test score (i.e. raw score) is computed. The second option is to assign to match a vector of continuous or discrete numeric values, which acts as the matching criterion. Note that for consistency this vector should not belong to the Data matrix.

For Breslow-Day method, two test statistics are available: the usual Breslow-Day statistic for testing homogeneous association (Aguerri, Galibert, Attorresi and Maranon, 2009) and the trend test statistic for assessing some monotonic trend in the odds ratios (Penfield, 2003). The DIF statistic is supplied by the BDstat argument, with values "BD" (default) for the usual statistic and "trend" for the trend test statistic.

For logistic regression, the argument type permits to test either both uniform and nonuniform effects simultaneously (type="both"), only uniform DIF effect (type="udif") or only nonuniform DIF effect (type="nudif"). The criterion argument specifies the DIF statistic to be computed, either the likelihood ratio test statistic (with criterion="LRT") or the Wald test (with criterion="Wald"). Moreover, the group membership can be either a vector of two distinct values, one for the reference group and one for the focal group, or a continuous or discrete variable that acts as the "group" membership variable. In the former case, the member.type argument is set to "group" and the focal.name defines which value in the group variable stands for the focal group. In the latter case, member.type is set to "cont", focal.name is ignored and each value of the group represents one "group" of data (that is, the DIF effects are investigated among participants relying on different values of some discrete or continuous trait). See Logistik for further details.

For Raju's method, the type of area (signed or unsigned) is fixed by the logical signed argument, with default value FALSE (i.e. unsigned areas). See RajuZ for further details.

Item purification can be requested by specifying purify option to TRUE. Recall that item purification is slightly different for IRT and for non-IRT based methods. See the corresponding methods for further information.

Adjustment for multiple comparisons is possible with the argument p.adjust.method. See the corresponding methods for further information.

The output of the selected method can be stored in a text file by fixing save.output and output appropriately. See the help file of the corresponding method for further information.

Value

The output of the selected DIF detection method.