| Title: | IRT Separate Calibration Linking Methods |
|---|---|
| Description: | Item response theory based methods are used to compute linking constants and conduct chain linking of unidimensional or multidimensional tests for multiple groups under a common item design. The unidimensional methods include the Mean/Mean, Mean/Sigma, Haebara, and Stocking-Lord methods for dichotomous (1PL, 2PL and 3PL) and/or polytomous (graded response, partial credit/generalized partial credit, nominal, and multiple-choice model) items. The multidimensional methods include the least squares method and extensions of the Haebara and Stocking-Lord method using single or multiple dilation parameters for multidimensional extensions of all the unidimensional dichotomous and polytomous item response models. The package also includes functions for importing item and/or ability parameters from common IRT software, conducting IRT true score and observed score equating, and plotting item response curves/surfaces, vector plots, information plots, and comparison plots for examining parameter drift. |
| Authors: | Jonathan P. Weeks <[email protected]> |
| Maintainer: | Jonathan P. Weeks <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 1.5-1 |
| Built: | 2024-11-02 06:25:59 UTC |
| Source: | CRAN |
This package uses unidimensional and multidimensional item response theory methods to compute linking constants and conduct chain linking of tests for multiple groups under a nonequivalent groups common item design.
The package consists of three types of functions:
Response Probability Functions: These functions compute response probabilities for specified theta values (for as many dimensions and theta values as allowed by your computer's memory). The package can estimate probabilities for the item response models listed below. For all of these models, I have tried to include examples from the corresponding journal articles as a way to show that the equations are correct.
1PL
2PL
3PL
Graded Response Model* (cumulative or category probabilities)
Partial Credit Model*
Generalized Partial Credit Model*
Nominal Response Model
Multiple-Choice Model
M1PL
M2PL
M3PL
MD Graded Response Model* (cumulative or category probabilities)
MD Partial Credit Model*
MD Generalized Partial Credit Model*
MD Nominal Response Model
MD Multiple-Choice Model (this model has not formally been presented in the literature)
* These models can be specified using a location parameter
Linking Function: There is only one linking function, but it includes a variety of linking methods. The most notable feature of this function (as compared to other software) is that it allows you to input parameters for more than two groups and then chain link all of the tests together in a single run. Below are several options.
Symmetric or Non-Symmetric linking (as originally presented by Haebara)
Specification of weights (can include uniform, quadrature, and normal density weights for default or specified theta points)
Choice of base group
Choice of method to rescale item parameters and/or ability estimates (if included)
When the item parameters correspond to a unidimensional model, linking constants can be estimated using the following methods
Mean/Mean
Mean/Sigma
Haebara
Stocking-Lord
When the item parameters correspond to a multidimensional model, linking constants can be estimated using the following methods (with different options for the first two)
Multidimensional extension of Haebara
Multidimensional extension of Stocking-Lord
Reckase-Martineau (based on the oblique procrustes rotation and least squares method
for estimating the translation vector m.
For the first two approaches there is a "dilation" argument where
all of the elements of the rotation/scaling matrix A and the translation vector m
are estimated via the optimization routine (see Oshima, Davey, & Lee, 2000)
an orthongonal procrustes rotation to resolve the rotational indeterminacy and a single dilation parameter is estimated (see Li & Lissitz, 2000)
an orthongonal procrustes rotation to resolve the rotational indeterminacy and different parameters are estimated to adjust the scale of each dimension (see Min, 2003).
For all of these approaches an optional matrix can be specified to identify the ordering of dimensions. It allows you to specify an overall factor structure that can take into account construct shift or different orderings of the factor loadings.
Utility Functions:
Import item parameters and/or ability estimates from BILOG-MG 3, PARSCALE 4, MULTILOG 7, TESTFACT 4,
ICL, BMIRT, and the eRm and ltm packages.
Separate item parameters from matrices or lists into the slope, difficulty, category, etc. values for use in computing response probabilities, descriptive statistics, etc.
Combine multiple sets of parameters into a single object for use in the linking function (this essentially creates a blueprint of all the items including the IRT model used for each item, the number of response categories, and the mapping of common items between groups)
Plot item characteristic/category curves (for unidimensional items), item response/category surfaces, contour plots, level plots, and three types of vector plots (for multidimensional items). The first three types of multidimensional plots include functionality for creating "conditional" surfaces when there are more than two dimensions).
Summarize the item parameters (unique and/or common) for each item response model separately, and overall
Running the separate calibration is typically a two-step process. The first step is to format the
item parameters and the second step is to run the function plink. In the
simplest scenario, the parameters should be formatted as a single irt.pars
object with multiple groups. Refer to the function as.irt.pars for specific details. Once
in this format, the linking constants can be computed using plink. The
summary function can be used to summarize the common item parameters
(including descriptive statistics) and the linking constants.
To compute response probabilities for a given model, the following functions can be used:
drm, gpcm, grm,
mcm, or nrm. The plot
function can be used to create item/category characteristic curves, item/category response surfaces,
and vector plots.
Jonathan P. Weeks [email protected]
Weeks, J. P. (2010) plink: An R package for linking mixed-format tests using IRT-based methods. Journal of Statistical Software, 35(12), 1–33. URL http://www.jstatsoft.org/v35/i12/
This (unidimensional) dataset includes multiple-choice model item parameter estimates
for two groups. There are 60 items, all of which are common items. The estimates
are from the 1997 and 1998 ACT mathematics test. The item parameters were
obtained from the examples for the software mceq written by Bradley
Hanson.
act.mcmact.mcm
A list of length two. The matrices in the two list elements contain multiple-choice model item parameters for 1997 and 1998 respectively. The first six columns in each matrix are the slope parameters, the next six columns are the category difficulty parameters, and the last five columns are the lower asymptote parameters.
Kim, J.-S. & Hanson, B. A. (2002). Test Equating Under the Multiple-Choice Model. Applied Psychological Measurement, 26(3), 255-270.
The item parameters can be found here http://www.b-a-h.com/software/mcmequate/index.html
This (unidimensional )dataset includes nominal response model item parameter estimates
for two groups. There are 60 items, all of which are common items. The estimates
are from the 1997 and 1998 ACT mathematics test. The item parameters were
obtained from the examples for the software nreq written by Bradley
Hanson.
act.nrmact.nrm
A list of length two. The matrices in the two list elements contain nominal response model item parameters for 1997 and 1998 respectively. The first six columns in each matrix are the slope parameters and the next six columns are the category difficulty parameters.
Kim, J.-S. & Hanson, B. A. (2002). Test Equating Under the Multiple-Choice Model. Applied Psychological Measurement, 26(3), 255-270.
The item parameters can be found here http://www.b-a-h.com/software/mcmequate/index.html
This function attempts to turn the given values into an irt.pars object
that is used primarily with the plink function.
as.irt.pars(x, common, cat, poly.mod, dimensions = 1, location = FALSE, grp.names, ...) ## S4 method for signature 'numeric', 'missing' as.irt.pars(x, common, cat, poly.mod, dimensions, location, grp.names, ...) ## S4 method for signature 'matrix', 'missing' as.irt.pars(x, common, cat, poly.mod, dimensions, location, grp.names, ...) ## S4 method for signature 'data.frame', 'missing' as.irt.pars(x, common, cat, poly.mod, dimensions, location, grp.names, ...) ## S4 method for signature 'list', 'missing' as.irt.pars(x, common, cat, poly.mod, dimensions, location, grp.names, ...) ## S4 method for signature 'sep.pars', 'missing' as.irt.pars(x, common, cat, poly.mod, dimensions, location, grp.names, ...) ## S4 method for signature 'list', 'matrix' as.irt.pars(x, common, cat, poly.mod, dimensions, location, grp.names, ...) ## S4 method for signature 'list', 'list' as.irt.pars(x, common, cat, poly.mod, dimensions, location, grp.names, ...)as.irt.pars(x, common, cat, poly.mod, dimensions = 1, location = FALSE, grp.names, ...) ## S4 method for signature 'numeric', 'missing' as.irt.pars(x, common, cat, poly.mod, dimensions, location, grp.names, ...) ## S4 method for signature 'matrix', 'missing' as.irt.pars(x, common, cat, poly.mod, dimensions, location, grp.names, ...) ## S4 method for signature 'data.frame', 'missing' as.irt.pars(x, common, cat, poly.mod, dimensions, location, grp.names, ...) ## S4 method for signature 'list', 'missing' as.irt.pars(x, common, cat, poly.mod, dimensions, location, grp.names, ...) ## S4 method for signature 'sep.pars', 'missing' as.irt.pars(x, common, cat, poly.mod, dimensions, location, grp.names, ...) ## S4 method for signature 'list', 'matrix' as.irt.pars(x, common, cat, poly.mod, dimensions, location, grp.names, ...) ## S4 method for signature 'list', 'list' as.irt.pars(x, common, cat, poly.mod, dimensions, location, grp.names, ...)
x |
an |
common |
j x 2 matrix or list of matrices identifying the common items between
adjacent groups in |
cat |
vector or list of the number of response categories for each item in |
poly.mod |
a |
dimensions |
numeric vector identifying the number of modeled dimensions in each group. |
location |
logical vector identifying whether the parameters in |
grp.names |
character vector of group names |
... |
further arguments passed to or from other methods |
Returns an object of class irt.pars
This method should only be used for the Rasch model
where x is a vector of difficulty parameters (or parameters related to item difficulty
in the multidimensional case). Under this method the slopes and lower asymptote values for
all items will default to one and zero respectively. This is true for both the unidimensional
and multidimensional case.
x can include item parameters from
multiple models. The general format for structuring x is an additive column
approach. That is, the left-most columns are typically for discrimination/slope parameters,
the next column, if applicable, is for location parameters, the next set of columns
is for difficulty/threshold/step/category parameters, and the final set of columns
is for lower asymptote (guessing) parameters. When multiple models are included, or models
with differing numbers of response categories, not all cells in x may have data.
In these instances, cells with no data should be NA. The resulting matrix will be an
n x k matrix for n items and k equal to the maximum number of columns (across all item
response models). In essence, the combination of multiple models is equivalent to formatting
the item parameters for each response model separately, stacking the matrices on top of one
another and then filling in any missing cells with NAs.
See the method for x = "matrix"
This method can include a list with one, two, or three elements. In general, these elements correspond to discrimination/slope parameters, difficulty/threshold/ step/category parameters, and lower asymptote (guessing) parameters, although this may not be the case depending on the supplied parameters. If a combination of models are used, the number of list elements should correspond with the maximum number of elements across models. For example, if the 3PL model (3 list elements) and nominal response model (2 list elements) are used, the list should include three elements. Within each list element, the parameters should be formatted as vectors or matrices (depending on the specific item response models). For the format of the matrices, see the method for x = "matrix" above and/or the methods corresponding to the specific response models. When location parameters are included, these values should be placed in the first column of the matrix of difficulty/threshold/step/category parameters (in the corresponding list element).
x is an object of class sep.pars.
The arguments cat, poly.mod, and location do not need to be included.
This method is intended for the creation of an
object with only two groups. Each list element should conform to one of the formats
listed above (i.e. a numeric vector, matrix/data.frame, list, or sep.pars object)
or be an irt.pars object. The format of each list element does
not need to be the same. That is, the first list element might be a "sep.pars"
object while the second element is a list of item parameters. If an object of class
irt.pars is included, the object can contain information for a single group
or for multiple groups. If the irt.pars object includes multiple groups,
common should identify the common items between the last group in x[[1]]
and the first group in x[[2]].
For this method common is a j x 2 matrix where the first column identifies the
common items for the first group (x[[1]]). The second column in common
identifies the corresponding set of common items from the next list element. For example,
if item 4 in group one (row 4 in x[[1]]) is the same as item 6 in group two, the
first row of common would be "4,6".
If the objects in x are of class sep.pars or irt.pars, cat,
poly.mod, and location do not need to be included.
This method is intended for the creation of an object
with more than two groups; however, if there are two groups common can be
defined as a list with length one. Each list element should conform to one of the formats
listed above (i.e. a numeric vector, matrix/data.frame, list, or sep.pars object)
or be an irt.pars object. The format of each list element does
not need to be the same. That is, the first list element might be a "sep.pars"
object while the second element is a list of item parameters. If an object of class
irt.pars is included, the object can contain information for a single group
or for multiple groups. The list elements in x should be ordered such that
adjacent elements correspond to adjacent groups. If an "irt.pars" object is
included with multiple groups, the list element following this object should contain
information for a group that is adjacent to the last group in the "irt.pars" object
For this method common is a list of j x 2 matrices where the first column identifies
the identifies the common items for the first group of two adjacent list elements in
The second column in common identifies the corresponding set of common items from the
next list element in x. For example, if x contains only two list elements, a
single set of common items links them together. If item 4 in group one (row 4 in slot pars)
is the same as item 6 in group two, the first row of common would be "4,6".
If all the objects in x are of class sep.pars or irt.pars, the arguments
cat, poly.mod, and location do not need to be included.
Jonathan P. Weeks [email protected]
Weeks, J. P. (2010) plink: An R package for linking mixed-format tests using IRT-based methods. Journal of Statistical Software, 35(12), 1–33. URL http://www.jstatsoft.org/v35/i12/
irt.pars, as.poly.mod, poly.mod,
sep.pars
# Create object for three dichotomous (1PL) items with difficulties # equal to -1, 0, 1 x <- as.irt.pars(c(-1,0,1)) # Create object for three dichotomous (3PL) items and two polytomous # (gpcm) items without a location parameter # (use signature matrix, missing) dichot <- matrix(c(1.2, .8, .9, 2.3, -1.1, -.2, .24, .19, .13),3,3) poly <- matrix(c(.64, -1.8, -.73, .45, NA, .88, .06, 1.4, 1.9, 2.6), 2,5,byrow=TRUE) pars <- rbind(cbind(dichot,matrix(NA,3,2)),poly) cat <- c(2,2,2,4,5) pm <- as.poly.mod(5, c("drm","gpcm"), list(1:3,4:5)) x <- as.irt.pars(pars, cat=cat, poly.mod=pm) summary(x) # Create object for three dichotomous (3PL) items and two polytomous # (gpcm) items without a location parameter # (use signature list, missing) a <- c(1.2, .8, .9, .64, .88) b <- matrix(c( 2.3, rep(NA,3), -1.1, rep(NA,3), -.2, rep(NA,3), -1.8, -.73, .45, NA, .06, 1.4, 1.9, 2.6),5,4,byrow=TRUE) c <- c(1.4, 1.9, 2.6, NA, NA) pars <- list(a,b,c) cat <- c(2,2,2,4,5) pm <- as.poly.mod(5, c("drm","gpcm"), list(1:3,4:5)) x <- as.irt.pars(pars, cat=cat, poly.mod=pm) summary(x) # Create object for three dichotomous (3PL) items, four polytomous items, # two gpcm items and two nrm items. Include a location parameter for the # gpcm items (use signature list, missing) a <- matrix(c( 1.2, rep(NA,4), .8, rep(NA,4), .9, rep(NA,4), .64, rep(NA,4), .88, rep(NA,4), .905, .522, -.469, -.959, NA, .828, .375, -.357, -.079, -.817),7,5,byrow=TRUE) b <- matrix(c( 2.3, rep(NA,4), -1.1, rep(NA,4), -.2, rep(NA,4), -.69, -1.11, -.04, 1.14, NA, 1.49, -1.43, -.09, .41, 1.11, .126, -.206, -.257, .336, NA, .565, .865, -1.186, -1.199, .993),7,5,byrow=TRUE) c <- c(.14, .19, .26, rep(NA,4)) pars <- list(a,b,c) cat <- c(2,2,2,4,5,4,5) pm <- as.poly.mod(7, c("drm","gpcm","nrm"), list(1:3,4:5,6:7)) x <- as.irt.pars(pars, cat=cat, poly.mod=pm, location=TRUE) summary(x, TRUE) # Create object with two groups (all dichotomous items) pm <- as.poly.mod(36) x <- as.irt.pars(KB04$pars, KB04$common, cat=list(rep(2,36),rep(2,36)), list(pm,pm), grp.names=c("form.x","form.y")) summary(x, TRUE) # Create object with six groups (all dichotomous items) pars <- TK07$pars common <- TK07$common cat <- list(rep(2,26),rep(2,34),rep(2,37),rep(2,40),rep(2,41),rep(2,43)) pm1 <- as.poly.mod(26) pm2 <- as.poly.mod(34) pm3 <- as.poly.mod(37) pm4 <- as.poly.mod(40) pm5 <- as.poly.mod(41) pm6 <- as.poly.mod(43) pm <- list(pm1, pm2, pm3, pm4, pm5, pm6) x <- as.irt.pars(pars, common, cat, pm, grp.names=paste("grade",3:8,sep="")) # Create an object with two groups using mixed-format items and # a mixed placement of common items. This example uses the dgn dataset. pm1=as.poly.mod(55,c("drm","gpcm","nrm"),dgn$items$group1) pm2=as.poly.mod(55,c("drm","gpcm","nrm"),dgn$items$group2) x=as.irt.pars(dgn$pars,dgn$common,dgn$cat,list(pm1,pm2)) summary(x, TRUE)# Create object for three dichotomous (1PL) items with difficulties # equal to -1, 0, 1 x <- as.irt.pars(c(-1,0,1)) # Create object for three dichotomous (3PL) items and two polytomous # (gpcm) items without a location parameter # (use signature matrix, missing) dichot <- matrix(c(1.2, .8, .9, 2.3, -1.1, -.2, .24, .19, .13),3,3) poly <- matrix(c(.64, -1.8, -.73, .45, NA, .88, .06, 1.4, 1.9, 2.6), 2,5,byrow=TRUE) pars <- rbind(cbind(dichot,matrix(NA,3,2)),poly) cat <- c(2,2,2,4,5) pm <- as.poly.mod(5, c("drm","gpcm"), list(1:3,4:5)) x <- as.irt.pars(pars, cat=cat, poly.mod=pm) summary(x) # Create object for three dichotomous (3PL) items and two polytomous # (gpcm) items without a location parameter # (use signature list, missing) a <- c(1.2, .8, .9, .64, .88) b <- matrix(c( 2.3, rep(NA,3), -1.1, rep(NA,3), -.2, rep(NA,3), -1.8, -.73, .45, NA, .06, 1.4, 1.9, 2.6),5,4,byrow=TRUE) c <- c(1.4, 1.9, 2.6, NA, NA) pars <- list(a,b,c) cat <- c(2,2,2,4,5) pm <- as.poly.mod(5, c("drm","gpcm"), list(1:3,4:5)) x <- as.irt.pars(pars, cat=cat, poly.mod=pm) summary(x) # Create object for three dichotomous (3PL) items, four polytomous items, # two gpcm items and two nrm items. Include a location parameter for the # gpcm items (use signature list, missing) a <- matrix(c( 1.2, rep(NA,4), .8, rep(NA,4), .9, rep(NA,4), .64, rep(NA,4), .88, rep(NA,4), .905, .522, -.469, -.959, NA, .828, .375, -.357, -.079, -.817),7,5,byrow=TRUE) b <- matrix(c( 2.3, rep(NA,4), -1.1, rep(NA,4), -.2, rep(NA,4), -.69, -1.11, -.04, 1.14, NA, 1.49, -1.43, -.09, .41, 1.11, .126, -.206, -.257, .336, NA, .565, .865, -1.186, -1.199, .993),7,5,byrow=TRUE) c <- c(.14, .19, .26, rep(NA,4)) pars <- list(a,b,c) cat <- c(2,2,2,4,5,4,5) pm <- as.poly.mod(7, c("drm","gpcm","nrm"), list(1:3,4:5,6:7)) x <- as.irt.pars(pars, cat=cat, poly.mod=pm, location=TRUE) summary(x, TRUE) # Create object with two groups (all dichotomous items) pm <- as.poly.mod(36) x <- as.irt.pars(KB04$pars, KB04$common, cat=list(rep(2,36),rep(2,36)), list(pm,pm), grp.names=c("form.x","form.y")) summary(x, TRUE) # Create object with six groups (all dichotomous items) pars <- TK07$pars common <- TK07$common cat <- list(rep(2,26),rep(2,34),rep(2,37),rep(2,40),rep(2,41),rep(2,43)) pm1 <- as.poly.mod(26) pm2 <- as.poly.mod(34) pm3 <- as.poly.mod(37) pm4 <- as.poly.mod(40) pm5 <- as.poly.mod(41) pm6 <- as.poly.mod(43) pm <- list(pm1, pm2, pm3, pm4, pm5, pm6) x <- as.irt.pars(pars, common, cat, pm, grp.names=paste("grade",3:8,sep="")) # Create an object with two groups using mixed-format items and # a mixed placement of common items. This example uses the dgn dataset. pm1=as.poly.mod(55,c("drm","gpcm","nrm"),dgn$items$group1) pm2=as.poly.mod(55,c("drm","gpcm","nrm"),dgn$items$group2) x=as.irt.pars(dgn$pars,dgn$common,dgn$cat,list(pm1,pm2)) summary(x, TRUE)
This function attempts to turn the given values into a poly.mod object that
associates each item with a specific unidimensional or multidimensional item response model.
as.poly.mod(n, model = "drm", items = NULL)as.poly.mod(n, model = "drm", items = NULL)
n |
total number of items |
model |
character vector identifying the IRT models used to estimate
the item parameters. The only acceptable models are
|
items |
list identifying the item numbers from a set of parameters
that correspond to the given model in |
When creating a poly.mod object, there is no difference in the specification for
unidimensional versus multidimensional item response models. If all the items are dichotomous,
it is only necessary to specify a value for n. If all the items correspond to a
single model (other than drm), only n and model need to be specified.
The IRT models associated with the codes:
drm:dichotomous response models (includes the 1PL, 2PL, 3PL, M1PL, M2PL, and M3PL)
gpcm:partial credit model, generalized partial credit model, multidimensional partial credit model, and multidimensional generalized partial credit model
grm:graded response model and multidimensional graded response model
mcm:multiple-choice model and multidimensional multiple-choice model
nrm:nominal response model and multidimensional nominal response model
Returns an object of class poly.mod
Jonathan P. Weeks [email protected]
Weeks, J. P. (2010) plink: An R package for linking mixed-format tests using IRT-based methods. Journal of Statistical Software, 35(12), 1–33. URL http://www.jstatsoft.org/v35/i12/
# Ten dichotomous items as.poly.mod(10) # The first ten items in the set of associated (not present here) item # parameters are dichotomous and the last five were estimated using the # generalized partial credit model as.poly.mod(15, c("drm", "gpcm"), list(1:10,11:15) ) # Ten multidimensional graded response model items # Note: This same specification would be used for a unidimensional # graded response model as.poly.mod(10, "grm")# Ten dichotomous items as.poly.mod(10) # The first ten items in the set of associated (not present here) item # parameters are dichotomous and the last five were estimated using the # generalized partial credit model as.poly.mod(15, c("drm", "gpcm"), list(1:10,11:15) ) # Ten multidimensional graded response model items # Note: This same specification would be used for a unidimensional # graded response model as.poly.mod(10, "grm")
This function facilitates the creation of either a set of quadrature weights or
weights based on a set of theta values for use in the function
plink
as.weight(n, theta, weight, quadrature = FALSE, normal.wt = FALSE, dimensions = 1, ...)as.weight(n, theta, weight, quadrature = FALSE, normal.wt = FALSE, dimensions = 1, ...)
n |
numeric value or vector identifying the number of theta values
to use for each dimension. If only one value is supplied but |
theta |
vector or list of theta values. If |
weight |
vector or list of weights. If |
quadrature |
if |
normal.wt |
if |
dimensions |
number of dimensions for which the weights should be created |
... |
further arguments passed to other methods |
When weighting expected response probabilities at different theta values using
characteristic curve linking methods, there are a variety of approaches one can take.
These range from uniform weights to normal weights, to quadrature weights corresponding
to some a priori specified distribution. The purpose of this function is to facilitate
the creation of these weights for use in plink .
For all approaches, when more than one dimension is specified, the weights for each combined set of theta values will be a multiplicative weight. For example, if there are two dimensions and the specified weights corresponding to two specific theta values on each dimension respectively are 0.8 and 1.2, the final weight for this pair of theta values will be 0.96.
Five general approaches can be used to create uniform weights.
If no arguments are supplied, a set of weights (all equal to one) will be
returned for a single dimension, for 40 equal interval theta values ranging from
-4 to 4. If dimensions is greater than one, seven equal interval theta values
ranging from -4 to 4 will be specified for each dimension. For instance, for two
dimensions, there will be weights for 7 x 7 (49) points.
If only a value for n is supplied, uniform weights (all equal to one)
will be created for n points ranging from -4 to 4 for each dimension specified.
If values are only specified for theta, uniform weights (all equal to one)
will be created for each of these values for each dimension specified.
If values are only specified for weight where the values are all equal. In
this case, equal interval theta values will be selected from -4 to 4 to correspond to
the number of specified weights.
If values are specified for theta and uniform values are specified for
weight.
Four general approaches can be used to create non-uniform weights.
If values are only specified for weight where the values are not equal. In
this case, equal interval theta values will be selected from -4 to 4 to correspond to
the number of specified weights.
If values are specified for theta and varying values are specified for
weight.
If quadrature is equal to TRUE, and no other arguments are specified
n will default to 40. if dimensions is greater than one, n will default
to seven for each dimension. In either case n quadrature points and weights will be
selected from a standard normal distribution. To change the mean and/or standard deviation of
this distribution, values for mu and sigma respectively should be specified. See
gauss.quad.prob for more information. Different means
and/or SDs can be supplied for each dimension. If values are specified for theta or
weight, the quadrature approach will not be used.
If quadrature equals TRUE other distributions can be specified for
n points. See gauss.quad.prob for more
information.
If normal.wt equals TRUE, normal densities will be created for the
specified theta values (if supplied) or equal interval values. The default distribution
will be standard normal, but different means and/or standard deviations can be specified by
passing arguments for mean and sd respectively. Different means and/or SDs can be
supplied for each dimension. If no values are included for theta, equal interval theta
values will be created for the range of three SDs above and below the mean. If values are
specified for weight, the normal.wt argument will be ignored.
See Kolen & Brennan (2004) for more information on calibration weights.
Returns a list of length two. The first list element is an n x m matrix of theta values for m dimensions. The second list element is a vector of weights corresponding to the theta values in the first list element.
Jonathan P. Weeks [email protected]
Kolen, M. J., & Brennan, R. L. (2004) Test Equating, Scaling, and Linking. New York: Springer
###### Unidimensional Examples ###### # Create a set of 40 (default) theta values and uniform weights wt <- as.weight() # Create a set of 40 quadrature points and weights using a standard # normal distribution wt <- as.weight(quadrature=TRUE) # Create a set of 30 quadrature points and weights using a normal # distribution with a mean of 0.5 wt <- as.weight(n=30, quadrature=TRUE, mu=0.5) # Create weights for a set of random normal theta values wt <- as.weight(theta=rnorm(100)) # Create an object with equal interval theta values and standard # normal density weights wt <- as.weight(theta=seq(-4,4,0.05), normal.wt=TRUE) # Create an object with equal interval theta values and normal # density weights with a mean of 0.5 and SD equal to .92 wt <- as.weight(theta=seq(-4,4,0.05), normal.wt=TRUE, mean=0.5, sd=0.92) ###### Multidimensional Examples ###### # Create a set of 49 theta values and uniform weights # (based on seven points for each dimension) wt <- as.weight(dimensions=2) # Create a set of 100 quadrature points and weights using a normal # distribution with a means of 0 and 0.5 for the two dimensions respectively wt <- as.weight(n=10, quadrature=TRUE, mu=c(0,0.5), dimensions=2) # Create an object with equal interval theta values and standard # normal density weights for three dimensions wt <- as.weight(theta=seq(-3,3), normal.wt=TRUE, dimensions=3) # Create an object with two sets of equal interval theta values for # two dimensions wt <- as.weight(theta=list(seq(-4,4),seq(-3,3)), dimensions=2) # Create an object with two sets of random normal theta values and # standard normal density weights for two dimensions wt <- as.weight(theta=list(rnorm(10),rnorm(10)), normal.wt=TRUE, dimensions=2)###### Unidimensional Examples ###### # Create a set of 40 (default) theta values and uniform weights wt <- as.weight() # Create a set of 40 quadrature points and weights using a standard # normal distribution wt <- as.weight(quadrature=TRUE) # Create a set of 30 quadrature points and weights using a normal # distribution with a mean of 0.5 wt <- as.weight(n=30, quadrature=TRUE, mu=0.5) # Create weights for a set of random normal theta values wt <- as.weight(theta=rnorm(100)) # Create an object with equal interval theta values and standard # normal density weights wt <- as.weight(theta=seq(-4,4,0.05), normal.wt=TRUE) # Create an object with equal interval theta values and normal # density weights with a mean of 0.5 and SD equal to .92 wt <- as.weight(theta=seq(-4,4,0.05), normal.wt=TRUE, mean=0.5, sd=0.92) ###### Multidimensional Examples ###### # Create a set of 49 theta values and uniform weights # (based on seven points for each dimension) wt <- as.weight(dimensions=2) # Create a set of 100 quadrature points and weights using a normal # distribution with a means of 0 and 0.5 for the two dimensions respectively wt <- as.weight(n=10, quadrature=TRUE, mu=c(0,0.5), dimensions=2) # Create an object with equal interval theta values and standard # normal density weights for three dimensions wt <- as.weight(theta=seq(-3,3), normal.wt=TRUE, dimensions=3) # Create an object with two sets of equal interval theta values for # two dimensions wt <- as.weight(theta=list(seq(-4,4),seq(-3,3)), dimensions=2) # Create an object with two sets of random normal theta values and # standard normal density weights for two dimensions wt <- as.weight(theta=list(rnorm(10),rnorm(10)), normal.wt=TRUE, dimensions=2)
This function combines objects of class irt.pars and/or
sep.pars into a single irt.pars object
combine.pars(x, common, grp.names)combine.pars(x, common, grp.names)
x |
an ordered list containing two or more |
common |
an n x 2 matrix or list of matrices identifying the common items between
adjacent pairs of objects in |
grp.names |
character vector of names for all the groups in the returned object |
Although many of the methods in this package allow for lists containing irt.pars
and sep.pars objects, it may be helpful to combine the item parameters for multiple
groups into a single object. x can include a combination of irt.pars and
sep.pars objects. The irt.pars objects can contain information for a
single group or multiple groups. The list elements in x should be ordered such that
adjacent elements correspond to adjacent groups. If an irt.pars object is included
with multiple groups, the list element following this object should contain information for a
group that is adjacent to the last group in the irt.pars object.
If x contains only two elements, common should be a matrix. If x contains
more than two elements, common should be a list. In any of the common matrices
the first column identifies the common items for the first group of two adjacent list elements
in x. The second column in common identifies the corresponding set of common
items from the next list element in x. For example, if x contains only two list
elements, a single set of common items links them together. If item 4 in group one (row 4 in
slot pars) is the same as item 6 in group two, the first row of common would be "4,6".
If an irt.pars object is included with multiple groups, common should correspond
to the set of common items between the last group in the irt.pars object and the
group in the adjacent element in x.
Returns an object of class irt.pars
Jonathan P. Weeks [email protected]
Weeks, J. P. (2010) plink: An R package for linking mixed-format tests using IRT-based methods. Journal of Statistical Software, 35(12), 1–33. URL http://www.jstatsoft.org/v35/i12/
irt.pars, as.irt.pars, sep.pars,
sep.pars
This (unidimensional) dataset includes item parameters for two groups estimated under the three parameter logistic model, the generalized partial credit model, and the nominal response model. There are 15 common items (7 drm, 3 gpcm, 5 nrm). This dataset is for illustrative purposes to show how items from different response models can be mixed together and common items can be in different positions across groups.
dgndgn
A list of length four. The first element is a list of length two with item parameter estimates for groups 1 and 2 respectively. The gpcm items do not include a location parameter. The second list element is a list identifying the number of response categories for the two groups. The third element, specifies which items correspond to the different item response models for each group respectively. The last element is a matrix of common items between the two groups.
The item parameters are loosely based on parameters listed in the following articles
Kim, J.-S. (2006). Using the Distractor Categories of Multiple-Choice Items to Improve IRT Linking. Journal of Educational Measurement, 43(3), 193-213.
Kim, S. & Lee, W.-C. (2006). An Extension of Four IRT Linking Methods for Mixed-Format Tests. Journal of Educational Measurement, 43(1), 53-76.
This function computes the probability of correct responses (and optionally, incorrect responses) for one or more items for a given set of theta values using the 1PL, 2PL, 3PL, M1PL, M2PL or M3PL model, depending on the included item parameters and the specified number of dimensions.
drm(x, theta, dimensions = 1, D = 1, incorrect = FALSE, print.mod = FALSE, items, information = FALSE, angle, ...) ## S4 method for signature 'numeric' drm(x, theta, dimensions, D, incorrect, print.mod, items, information, angle, ...) ## S4 method for signature 'matrix' drm(x, theta, dimensions, D, incorrect, print.mod, items, information, angle, ...) ## S4 method for signature 'data.frame' drm(x, theta, dimensions, D, incorrect, print.mod, items, information, angle, ...) ## S4 method for signature 'list' drm(x, theta, dimensions, D, incorrect, print.mod, items, information, angle, ...) ## S4 method for signature 'irt.pars' drm(x, theta, dimensions, D, incorrect, print.mod, items, information, angle, ...) ## S4 method for signature 'sep.pars' drm(x, theta, dimensions, D, incorrect, print.mod, items, information, angle, ...)drm(x, theta, dimensions = 1, D = 1, incorrect = FALSE, print.mod = FALSE, items, information = FALSE, angle, ...) ## S4 method for signature 'numeric' drm(x, theta, dimensions, D, incorrect, print.mod, items, information, angle, ...) ## S4 method for signature 'matrix' drm(x, theta, dimensions, D, incorrect, print.mod, items, information, angle, ...) ## S4 method for signature 'data.frame' drm(x, theta, dimensions, D, incorrect, print.mod, items, information, angle, ...) ## S4 method for signature 'list' drm(x, theta, dimensions, D, incorrect, print.mod, items, information, angle, ...) ## S4 method for signature 'irt.pars' drm(x, theta, dimensions, D, incorrect, print.mod, items, information, angle, ...) ## S4 method for signature 'sep.pars' drm(x, theta, dimensions, D, incorrect, print.mod, items, information, angle, ...)
x |
an |
theta |
vector, matrix, or list of theta values for which probabilities will be computed.
If |
dimensions |
number of modeled dimensions |
D |
scaling constant. The default value assumes that the parameters are already in the desired metric. If the parameters are in the logistic metric, they can be transformed to a normal metric by setting D = 1.7 |
incorrect |
if |
print.mod |
if |
items |
numeric vector identifying the items for which probabilities should be computed |
information |
logical value. If |
angle |
vector or matrix of angles between the dimension 1 axis and the corresponding axes for each
of the other dimensions for which information will be computed. When there are more than two dimensions
and |
... |
further arguments passed to or from other methods |
theta can be specified as a vector, matrix, or list. For the unidimensional case, theta
should be a vector. If a matrix or list of values is supplied, they will be converted to a single vector
of theta values. For the multidimensional case, if a vector of values is supplied it will be assumed
that this same set of values should be used for each dimension. Probabilities will be computed for each
combination of theta values. Similarly, if a list is supplied, probabilities will be computed for each
combination of theta values. In instances where probabilities are desired for specific combinations of
theta values, a j x m matrix should be specified for j ability points and m dimensions where the columns
are ordered from dimension 1 to m.
Returns an object of class irt.prob
This method should only be used for the Rasch model where x is a vector of
difficulty parameters (or parameters related to item difficulty in the multidimensional case). Under
this method the slopes and lower asymptote values for all items will default to one and zero respectively.
This is true for both the unidimensional and multidimensional case.
This method allows one to specify an n x k matrix for n items, for m dimensions, and k equal to m, m+1, or m+2. The first m columns generally correspond to the discrimination/slope parameters, and the last two columns correspond to the difficulty, and lower asymptote parameters, although this may not be the case depending on the supplied parameters.
For the 1PL and M1PL model with discriminations equal to 1 (Rasch Model),
an n x 1 matrix of item difficulties can be supplied. An n x (m+1) matrix can also be used with
all the values in the first m columns equal to 1 and difficulty parameters in the last
column. For discrimination values other than 1, x should include at least m+1
columns where the first m columns contain the item discriminations (identical for all items)
and the last column is for the item difficulties. The lower asymptote defaults to zero for
all items; however, m+2 columns can be specified where the values in the last column all
equal zero.
x should include at least m+1 columns where the first m columns contain
the item discriminations and the last column is for the item difficulties. The lower asymptote
defaults to zero for all items; however, m+2 columns can be specified where the values in the
last column all equal zero.
x should include m+2 columns where the first m columns contain the item
discriminations, the m+1 column is for item difficulties, and the last column is for lower
asymptote values.
See the method for x = "matrix"
This method can include a list with 1, 2, or 3 elements. In general, these elements correspond to discrimination, difficulty, and lower asymptote parameters, although this may not be the case depending on the supplied parameters. For the unidimensional case, each list element can be a vector of length n or an n x 1 matrix for n items. For the multidimensional case, the list element for the discrimination parameters should be an n x m matrix for m dimensions. The other list elements can be vectors of length n or an n x 1 matrix for n items.
For the 1PL and M1PL model with discriminations equal to 1 (Rasch Model),
one element with item difficulties can be supplied. Alternatively, two elements can be used
with the first list element containing a matrix/vector of ones and difficulty parameters in
the second list element. For discrimination values other than 1, x should contain
at least two list elements where the first contains the item discriminations (identical
for all items) and the second is for item difficulties. The lower asymptote defaults to
zero for all items; however, a third element with a vector/matrix of zeros can be included.
x should contain at least two list elements where the first element
contains the item discriminations and the second element includes the item difficulties.
The lower asymptote defaults to zero for all items; however, a third element with a
vector/matrix of zeros can be included.
x should include three list elements where the first element contains the
item discriminations, the second element includes the item difficulties, and the third element
contains the lower asymptote values.
This method can be used to compute probabilities for the dichotomous
items in an object of class "irt.pars". If x contains polytomous items, a
warning will be displayed stating that probabilities will be computed for the
dichotomous items only. If x contains parameters for multiple groups, a list of
"irt.prob" objects will be returned. The argument dimensions does not need
to be included for this method.
This method can be used to compute probabilities for the dichotomous
items in an object of class sep.pars. If x contains polytomous items, a
warning will be displayed stating that probabilities will be computed for the
dichotomous items only. The argument dimensions does not need to be included for
this method.
The indices 0 and 1 are used in the column labels for incorrect and correct responses
respectively (e.g. the label d1.0 indicates the incorrect response column for item 1, d1.1
indicates the correct response column for the same item). If incorrect = FALSE,
all column labels will end with .1
Jonathan P. Weeks [email protected]
Adams, R. J., Wilson, M., & Wang, W. (1997). The multidimensional random coefficients multinomial logit model. Applied Psychological Measurement, 21(1), 1-23.
Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability In F. M. Lord & M. R. Novick (Eds.) Statistical Theories of Mental Test Scores. Reading, MA: Addison-Wesley.
Rasch, G. (1960). Probabilistic Models for Some Intelligence and Attainment Tests Copenhagen, Denmark: Danish Institute for Educational Research.
Reckase, M. D. (1985). The difficulty of items that measure more than one ability. Applied Psychological Measurement, 9(4), 401-412.
Reckase, M. D. (1997). A linear logistic multidimensional model for dichotomous item response data. In W. J. van der Linden & R. K. Hambleton (Eds.) Handbook of Modern Item Response Theory (pp. 271-286). New York: Springer-Verlag.
Weeks, J. P. (2010) plink: An R package for linking mixed-format tests using IRT-based methods. Journal of Statistical Software, 35(12), 1–33. URL http://www.jstatsoft.org/v35/i12/
mixed: compute probabilities for mixed-format itemsplot: plot item characteristic/category curvesirt.prob, irt.pars, sep.pars: classes
###### Unidimensional Examples ###### ## 1PL # A single item with a difficulty at 1 logit x <- drm(1) plot(x) # Three items with a difficulties at -2.2, -1.7, and 0.54 logits # respectively. Assume a constant discrimination of 1 for each item b <- c(-2.2,-.7,1.54) x <- drm(b) # Plot the expected probabilites plot(x) # Five items with a constant discrimination equal to 1.2 # Use the scaling constant for the normal metric a <- rep(1.2,5) b <- c(-2,-1,0,1,2) pars <- list(a,b) x <- drm(pars,D=1.7) # Extract the expected probabilities get.prob(x) # Plot the expected probabilities plot(x,combine=5,item.names="Items 1-5") ## 2PL # Compute probabilities for five items at a theta value of 1.5 logits # Print the model a <- c(1,.8,.6,1.2,.9) b <- c(-1.7,2.2,.4,-1,-.4) pars <- cbind(a,b) drm(pars,theta=1.5,print.mod=TRUE) # Include a vector of zeros for the lower asymptote # Compute probabilities for incorrect responses a <- c(.63,1.14,.89,1.01,1.51,.58) b <- c(-1.9,.08,1.6,-1.4,.5,-2.3) c <- rep(0,6) pars <- cbind(a,b,c) x <- drm(pars,incorrect=TRUE) plot(x) ## 3PL a <- c(1,.8,.4,1.5,.9) b <- c(-2,-1,0,1,2) c <- c(.2,.25,.18,.2,.22) pars <- list(a,b,c) x <- drm(pars) plot(x) # Use theta values from -3 to 3 with an increment of 0.2 a <- c(.71,.96,.36,1.05,1.76,.64) b <- c(-.16,1.18,2.03,1.09,.82,-1.56) c <- c(.22,.17,.24,.08,.20,.13) theta <- seq(-3,3,.2) pars <- cbind(a,b,c) x <- drm(pars,theta) plot(x,combine=6,item.names="Items 1-6",auto.key=list(space="right")) ###### Multidimensional Examples ###### ## M1PL # A single item with a parameter related to item difficulty at 1 logit x <- drm(1, dimensions=2) plot(x) # Three items with a difficulties at -2.2, -1.7, and 0.54 logits # respectively. Assume a constant discrimination of 1 for each item d <- c(-2.2,-.7,1.54) x <- drm(d, dimensions=2) plot(x, drape=TRUE) ## M2PL # Items 27-30 from Reckase (1985) a <- matrix(c(1.66,1.72,.69,.19,.88,1.12,.68,1.21),4,2,byrow=TRUE) d <- c(-.38,-.68,-.91,-1.08) pars <- list(a,d) x <- drm(pars,dimensions=2) plot(x, drape=TRUE, item.names=paste("Item",27:30)) # Create contourplots for these items plot(x,type="contourplot",cuts=10) ## M3PL # Single item from Reckase (1997, p. 274) pars <- t(c(.8,1.4,-2,.2)) x <- drm(pars, dimensions=2) plot(x, default.scales=list(arrows=FALSE),drape=TRUE) # Compute the probabilities of an incorrect response # for the Reckase (1997) item x <- drm(pars, dimensions=2, incorrect=TRUE) plot(x, screen=list(z=-40,x=-60), auto.key=list(space="right")) # Four items from the included example for BMIRT (Yao, 2003) # modeled using four dimensions pars <- matrix(c(0.5038,2.1910,1.1317,0.2493,0.5240,0.1763, 0.2252,1.1999,0.5997,0.2087, 2.1730,0.4576, 0.2167,0.2487,1.4009,0.3865,-1.5270,0.3507, 2.3428,1.1530,0.3577,0.4240,-1.4971,0.3641),4,6,byrow=TRUE) colnames(pars) <- c("a1","a2","a3","a4","d","c") x <- drm(pars, dimensions=4, print.mod=TRUE) ## Not run: # Plot the item response surfaces for item 4 plot(x,items=4,item.names="Item 4",drape=TRUE) ## End(Not run)###### Unidimensional Examples ###### ## 1PL # A single item with a difficulty at 1 logit x <- drm(1) plot(x) # Three items with a difficulties at -2.2, -1.7, and 0.54 logits # respectively. Assume a constant discrimination of 1 for each item b <- c(-2.2,-.7,1.54) x <- drm(b) # Plot the expected probabilites plot(x) # Five items with a constant discrimination equal to 1.2 # Use the scaling constant for the normal metric a <- rep(1.2,5) b <- c(-2,-1,0,1,2) pars <- list(a,b) x <- drm(pars,D=1.7) # Extract the expected probabilities get.prob(x) # Plot the expected probabilities plot(x,combine=5,item.names="Items 1-5") ## 2PL # Compute probabilities for five items at a theta value of 1.5 logits # Print the model a <- c(1,.8,.6,1.2,.9) b <- c(-1.7,2.2,.4,-1,-.4) pars <- cbind(a,b) drm(pars,theta=1.5,print.mod=TRUE) # Include a vector of zeros for the lower asymptote # Compute probabilities for incorrect responses a <- c(.63,1.14,.89,1.01,1.51,.58) b <- c(-1.9,.08,1.6,-1.4,.5,-2.3) c <- rep(0,6) pars <- cbind(a,b,c) x <- drm(pars,incorrect=TRUE) plot(x) ## 3PL a <- c(1,.8,.4,1.5,.9) b <- c(-2,-1,0,1,2) c <- c(.2,.25,.18,.2,.22) pars <- list(a,b,c) x <- drm(pars) plot(x) # Use theta values from -3 to 3 with an increment of 0.2 a <- c(.71,.96,.36,1.05,1.76,.64) b <- c(-.16,1.18,2.03,1.09,.82,-1.56) c <- c(.22,.17,.24,.08,.20,.13) theta <- seq(-3,3,.2) pars <- cbind(a,b,c) x <- drm(pars,theta) plot(x,combine=6,item.names="Items 1-6",auto.key=list(space="right")) ###### Multidimensional Examples ###### ## M1PL # A single item with a parameter related to item difficulty at 1 logit x <- drm(1, dimensions=2) plot(x) # Three items with a difficulties at -2.2, -1.7, and 0.54 logits # respectively. Assume a constant discrimination of 1 for each item d <- c(-2.2,-.7,1.54) x <- drm(d, dimensions=2) plot(x, drape=TRUE) ## M2PL # Items 27-30 from Reckase (1985) a <- matrix(c(1.66,1.72,.69,.19,.88,1.12,.68,1.21),4,2,byrow=TRUE) d <- c(-.38,-.68,-.91,-1.08) pars <- list(a,d) x <- drm(pars,dimensions=2) plot(x, drape=TRUE, item.names=paste("Item",27:30)) # Create contourplots for these items plot(x,type="contourplot",cuts=10) ## M3PL # Single item from Reckase (1997, p. 274) pars <- t(c(.8,1.4,-2,.2)) x <- drm(pars, dimensions=2) plot(x, default.scales=list(arrows=FALSE),drape=TRUE) # Compute the probabilities of an incorrect response # for the Reckase (1997) item x <- drm(pars, dimensions=2, incorrect=TRUE) plot(x, screen=list(z=-40,x=-60), auto.key=list(space="right")) # Four items from the included example for BMIRT (Yao, 2003) # modeled using four dimensions pars <- matrix(c(0.5038,2.1910,1.1317,0.2493,0.5240,0.1763, 0.2252,1.1999,0.5997,0.2087, 2.1730,0.4576, 0.2167,0.2487,1.4009,0.3865,-1.5270,0.3507, 2.3428,1.1530,0.3577,0.4240,-1.4971,0.3641),4,6,byrow=TRUE) colnames(pars) <- c("a1","a2","a3","a4","d","c") x <- drm(pars, dimensions=4, print.mod=TRUE) ## Not run: # Plot the item response surfaces for item 4 plot(x,items=4,item.names="Item 4",drape=TRUE) ## End(Not run)
This function conducts IRT true score and observed score equating for unidimensional
single-format or mixed-format item parameters for two or more groups. This function
supports all item response models available in plink with the exception of
the multiple-choice model.
equate(x, method=c("TSE", "OSE"), true.scores, ts.low, base.grp=1, score=1, startval, weights1, weights2, syn.weights, exclude, max.tse.iter, ...) ## S4 method for signature 'list' equate(x, method, true.scores, ts.low, base.grp, score, startval, weights1, weights2, syn.weights, exclude, max.tse.iter, ...) ## S4 method for signature 'irt.pars', 'ANY' equate(x, method, true.scores, ts.low, base.grp, score, startval, weights1, weights2, syn.weights, exclude, max.tse.iter, ...)equate(x, method=c("TSE", "OSE"), true.scores, ts.low, base.grp=1, score=1, startval, weights1, weights2, syn.weights, exclude, max.tse.iter, ...) ## S4 method for signature 'list' equate(x, method, true.scores, ts.low, base.grp, score, startval, weights1, weights2, syn.weights, exclude, max.tse.iter, ...) ## S4 method for signature 'irt.pars', 'ANY' equate(x, method, true.scores, ts.low, base.grp, score, startval, weights1, weights2, syn.weights, exclude, max.tse.iter, ...)
x |
an object of class |
method |
character vector identifying the equating method(s) to use.
Values can include |
true.scores |
numeric vector of true score values to be equated |
ts.low |
logical value. If TRUE, interpolate values for the equated true scores in the range of observed scores from one to the value below the lowest estimated true score (a rounded sum of guessing parameters) |
base.grp |
integer identifying the group for the base scale |
score |
if |
startval |
integer starting value for the first value of |
weights1 |
list containing information about the theta values and weights to be used in the observed score equating for population 1. See below for more details. |
weights2 |
list containing information about the theta values and weights to be used in the observed score equating for population 2. See below for more details. |
syn.weights |
vector of length two or a list containing vectors of length two
with synthetic population weights to be used for each pair of tests for populations
1 and 2 respectively. If missing, weights of 0.5 will be used for both populations
for all groups. If |
exclude |
character vector or list identifying common items that should be excluded when estimating the linking constants. See below for more details. |
max.tse.iter |
maximum number of iterations to identify the theta value associated with each true score. The default is 50. |
... |
further arguments passed to or from other methods. See below for details. |
weights1 can be a list or a list of lists. The purpose of this object is to specify
the theta values for population 1 to integrate over in the observed score equating as well as
any weights associated with the theta values. The function as.weight
can be used to facilitate the creation of this object. If weights1 is missing, the
default is to use equally spaced theta values ranging from -4 to 4 with an increment of 0.05
and normal density weights for all groups.
To better understand the elements of weights1, let us assume for a moment that x
has parameters for only two groups. In this instance, weights1 would be a single list
with length two. The first element should be a vector of theta values corresponding to points
on the base scale. The second list element should be a vector of weights corresponding the
theta values. If x contains more than two groups, a single weights1 object can
be supplied, and the same set of thetas and weights will be used for all adjacent groups.
However, a separate list of theta values and weights for each adjacent group in x can be
supplied.
The specification of weights2 is the same as that for weights1, although the
theta values and weights for this object correspond to theta values for population 2.
This argument is only used when the synthetic weight associated with population 2 is greater
than zero. If weights2 is missing, the same theta values and weights used for
weights1 will be used for weights2.
For both equating methods, response probabilities are computed using the functions drm,
grm, gpcm, and nrm for the associated models respectively. Various
arguments from these functions can be passed to equate. Specifically, the argument
incorrect can be passed to drm and catprob can be passed to grm.
In the functions drm, grm, and gpcm there is an argument D
for the value of a scaling constant. In plink, a single argument D can be passed
that will be applied to all applicable models, or arguments D.drm, D.grm, and
D.gpcm can be specified for each model respectively. If an argument is specified for D
and, say D.drm, the values for D.grm and D.gpcm (if applicable) will be
set equal to D. If only D.drm is specified, the values for D.grm and
D.gpcm (if applicable) will be set to 1.
There are instances where certain items should not be included in the computation of total scores
(e.g., when the common items correspond to an external anchor test or when using field test items)
The exclude argument can be used to remove these items prior to conducting the equationg.
exclude can be specified as a character vector or a list. In the former case, a single value
"all.common" can be used to remove all common items or a vector of model names (i.e., "drm", "grm",
"gpcm", "nrm", "mcm") can be supplied, indicating that any item on any test associated with the given
model(s) would be excluded. If the argument is specified as a list, exclude should have as
any elements as groups in x. Each list element can include model names and/or item numbers
corresponding to the items on each test that should be excluded. If no items need to be excluded for
a given group, the list element should be equal NA. For example, say we have two groups and we would
like to exclude the GRM items and item 23 from the first group, we would specify exclude as
exclude <- list(c("grm",23),NA).
Returns a matrix of equated true scores and/or a list of equated observed scores with associated marginal distributions or a list combining these two objects. The output for the observed-score equating also includes EAP scores and SDs for each of the observed scores (Thissen & Orlando, 2001).
Jonathan P. Weeks [email protected]
Kolen, M. J. (1981). Comparison of traditional and item response theory methods for equating tests. Journal of Educational Measurement, 18(1), 1-11.
Kolen, M. J. & Brennan, R. L. (2004) Test Equating, Scaling, and Linking (2nd ed.). New York: Springer
Thissen, D. & Orlando, M. (2001) Item response theory for items scored in two categories. In D. Thissen & H. Wainer (Eds.) Test Scoring (p. 23 - 72). Hillsdale, NJ: Lawrence Erlbaum Associates.
Weeks, J. P. (2010) plink: An R package for linking mixed-format tests using IRT-based methods. Journal of Statistical Software, 35(12), 1–33. URL http://www.jstatsoft.org/v35/i12/
# IRT true score and observed score examples from # Kolen & Brennan (2004, ch. 6) pm <- as.poly.mod(36) x <- as.irt.pars(KB04$pars, KB04$common, cat=list(rep(2,36),rep(2,36)), poly.mod=list(pm,pm)) out <- plink(x, rescale="MS", base.grp=2, D=1.7, exclude=list(27,NA)) # Create the quadrature points and weights wt <- as.weight( theta=c(-5.2086,-4.163,-3.1175,-2.072,-1.0269,0.0184, 1.0635,2.109,3.1546,4.2001), weight=c(0.000101,0.00276,0.03021,0.142,0.3149,0.3158, 0.1542,0.03596,0.003925,0.000186)) # Conduct the equating equate(out,weights1=wt, synth.weights=c(1,0),D=1.7) # Conduct true score equating for specific true scores equate(out, true.scores=7:15, ts.low=FALSE, D=1.7) # Exclude all common items (assume they correspond to an external anchor) equate(out, D=1.7, exclude="all.common") # Observed score equating for mixed-format tests pm1 <- as.poly.mod(55,c("drm","gpcm","nrm"),dgn$items$group1) pm2 <- as.poly.mod(55,c("drm","gpcm","nrm"),dgn$items$group2) x <- as.irt.pars(dgn$pars,dgn$common,dgn$cat,list(pm1,pm2)) out <- plink(x, rescale="HB") OSE <- equate(out, method="OSE", score=2) # Display the equated scores OSE[[1]] # Multiple group equating pars <- TK07$pars common <- TK07$common cat <- list(rep(2,26),rep(2,34),rep(2,37),rep(2,40),rep(2,41),rep(2,43)) pm1 <- as.poly.mod(26) pm2 <- as.poly.mod(34) pm3 <- as.poly.mod(37) pm4 <- as.poly.mod(40) pm5 <- as.poly.mod(41) pm6 <- as.poly.mod(43) pm <- list(pm1, pm2, pm3, pm4, pm5, pm6) x <- as.irt.pars(pars, common, cat, pm, grp.names=paste("grade",3:8,sep="")) out <- plink(x, rescale="SL") # True score equating equate(out, method="TSE") # True score equating with the base group changed to 3 equate(out, method="TSE", base.grp=3) # Observed score equating (These data are for non-equivalent groups, but # this example is included to illustrate the multigroup capabilities) OSE <- equate(out, method="OSE", base.grp=3) # Display the equated scores for each group OSE[[1]]# IRT true score and observed score examples from # Kolen & Brennan (2004, ch. 6) pm <- as.poly.mod(36) x <- as.irt.pars(KB04$pars, KB04$common, cat=list(rep(2,36),rep(2,36)), poly.mod=list(pm,pm)) out <- plink(x, rescale="MS", base.grp=2, D=1.7, exclude=list(27,NA)) # Create the quadrature points and weights wt <- as.weight( theta=c(-5.2086,-4.163,-3.1175,-2.072,-1.0269,0.0184, 1.0635,2.109,3.1546,4.2001), weight=c(0.000101,0.00276,0.03021,0.142,0.3149,0.3158, 0.1542,0.03596,0.003925,0.000186)) # Conduct the equating equate(out,weights1=wt, synth.weights=c(1,0),D=1.7) # Conduct true score equating for specific true scores equate(out, true.scores=7:15, ts.low=FALSE, D=1.7) # Exclude all common items (assume they correspond to an external anchor) equate(out, D=1.7, exclude="all.common") # Observed score equating for mixed-format tests pm1 <- as.poly.mod(55,c("drm","gpcm","nrm"),dgn$items$group1) pm2 <- as.poly.mod(55,c("drm","gpcm","nrm"),dgn$items$group2) x <- as.irt.pars(dgn$pars,dgn$common,dgn$cat,list(pm1,pm2)) out <- plink(x, rescale="HB") OSE <- equate(out, method="OSE", score=2) # Display the equated scores OSE[[1]] # Multiple group equating pars <- TK07$pars common <- TK07$common cat <- list(rep(2,26),rep(2,34),rep(2,37),rep(2,40),rep(2,41),rep(2,43)) pm1 <- as.poly.mod(26) pm2 <- as.poly.mod(34) pm3 <- as.poly.mod(37) pm4 <- as.poly.mod(40) pm5 <- as.poly.mod(41) pm6 <- as.poly.mod(43) pm <- list(pm1, pm2, pm3, pm4, pm5, pm6) x <- as.irt.pars(pars, common, cat, pm, grp.names=paste("grade",3:8,sep="")) out <- plink(x, rescale="SL") # True score equating equate(out, method="TSE") # True score equating with the base group changed to 3 equate(out, method="TSE", base.grp=3) # Observed score equating (These data are for non-equivalent groups, but # this example is included to illustrate the multigroup capabilities) OSE <- equate(out, method="OSE", base.grp=3) # Display the equated scores for each group OSE[[1]]
This function computes the probability of responding in a specific category for one or more items for a given set of theta values using the partial credit model, generalized partial credit model, or multidimensional extension of these models, depending on the included item parameters and the specified number of dimensions.
gpcm(x, cat, theta, dimensions = 1, D = 1, location = FALSE, print.mod = FALSE, items, information = FALSE, angle, ...) ## S4 method for signature 'matrix', 'numeric' gpcm(x, cat, theta, dimensions, D, location, print.mod, items, information, angle, ...) ## S4 method for signature 'data.frame', 'numeric' gpcm(x, cat, theta, dimensions, D, location, print.mod, items, information, angle, ...) ## S4 method for signature 'list', 'numeric' gpcm(x, cat, theta, dimensions, D, location, print.mod, items, information, angle, ...) ## S4 method for signature 'irt.pars', 'ANY' gpcm(x, cat, theta, dimensions, D, location, print.mod, items, information, angle, ...) ## S4 method for signature 'sep.pars', 'ANY' gpcm(x, cat, theta, dimensions, D, location, print.mod, items, information, angle, ...)gpcm(x, cat, theta, dimensions = 1, D = 1, location = FALSE, print.mod = FALSE, items, information = FALSE, angle, ...) ## S4 method for signature 'matrix', 'numeric' gpcm(x, cat, theta, dimensions, D, location, print.mod, items, information, angle, ...) ## S4 method for signature 'data.frame', 'numeric' gpcm(x, cat, theta, dimensions, D, location, print.mod, items, information, angle, ...) ## S4 method for signature 'list', 'numeric' gpcm(x, cat, theta, dimensions, D, location, print.mod, items, information, angle, ...) ## S4 method for signature 'irt.pars', 'ANY' gpcm(x, cat, theta, dimensions, D, location, print.mod, items, information, angle, ...) ## S4 method for signature 'sep.pars', 'ANY' gpcm(x, cat, theta, dimensions, D, location, print.mod, items, information, angle, ...)
x |
an |
cat |
vector identifying the number of response categories (not the number of step parameters) for each item. |
theta |
vector, matrix, or list of theta values for which probabilities will be computed.
If |
dimensions |
number of modeled dimensions |
D |
scaling constant. The default value assumes that the parameters are already in the desired metric. If the parameters are in the logistic metric, they can be transformed to a normal metric by setting D = 1.7 |
location |
if |
print.mod |
if |
items |
numeric vector identifying the items for which probabilities should be computed |
information |
logical value. If |
angle |
vector or matrix of angles between the dimension 1 axis and the corresponding axes for each
of the other dimensions for which information will be computed. When there are more than two dimensions
and |
... |
further arguments passed to or from other methods |
theta can be specified as a vector, matrix, or list. For the unidimensional case, theta
should be a vector. If a matrix or list of values is supplied, they will be converted to a single vector
of theta values. For the multidimensional case, if a vector of values is supplied it will be assumed
that this same set of values should be used for each dimension. Probabilities will be computed for each
combination of theta values. Similarly, if a list is supplied, probabilities will be computed for each
combination of theta values. In instances where probabilities are desired for specific combinations of
theta values, a j x m matrix should be specified for j ability points and m dimensions where the columns
are ordered from dimension 1 to m.
Returns an object of class irt.prob
In the following description, references to the partial credit model and generalized partial credit model should be thought of as encompassing both the unidimensional and multidimensional models.
This method allows one to specify an n x k matrix for n items. The number of columns can vary depending on the model (partial credit or generalized partial credit model), number of dimensions, and whether a location parameter is included. Generally, the first m columns, for m dimensions, are for item slopes and the remaining columns are for step parameters.
The partial credit model is typically specified with all slopes equal to 1. For this model it is unnecessary (although optional) to include ones in the first m columns. For slope values other than one (equal for all items) or for the generalized partial credit model, slope parameters should be included in the first m columns.
Step parameters can be characterized in
two ways: as the actual steps or deviations from an overall item difficulty
(location). In the deviation scenario the location argument should be
equal to TRUE. If column(s) are included for the slope parameters,
the location parameters should be in the m+1 column; otherwise, they should be
in the first column. The columns for the step/step deviation parameters will always
follow the slope and/or location columns (or they may potentially
start in the first column for the partial credit model with no location parameter).
The number of step/step deviation parameters can vary for each item. In these
instances, all cells with missing values should be filled with NAs. For
example, for a unidimensional generalized partial credit model with no location
parameter, if one item has five categories (four step parameters) and another item
has three categories (two step parameters), there should be five columns. The first
column includes the slope parameters and columns 2-5 include the step parameters.
The values in the last two columns for the item with three categories should be NA.
See the method for x = "matrix"
This method can include a list with one or two elements. Generally, the first element is for item slopes and the second is for step/step deviation parameters.
For the partial credit model with all slopes equal to 1 it is unnecessary (although optional) to include a list element for the item slopes. If no slope values are included, the first element would contain the step/deviation step parameters. For slopes other than 1 (equal for all items) or for the generalized partial credit model, slope values should be included in the first list element. For the unidimensional case, these values should be a vector of length n or an n x 1 matrix for n items. For the multidimensional case, an n x m matrix of values for m dimensions should be supplied
The step/step deviation parameters should be
formatted as an n x k matrix for n items. If the steps are deviations from a location
parameter, the argument location should equal TRUE and the location
parameters should be in the first column. The number of step/step deviation parameters
can vary for each item. In these instances, all cells with missing values should be
filled with NAs (See the example in the method for x = "matrix").
This method can be used to compute probabilities
for the gpcm items in an object of class "irt.pars". If x contains
dichotomous items or items associated with another polytomous model, a warning
will be displayed stating that probabilities will be computed for the gpcm items only.
If x contains parameters for multiple groups, a list of "irt.prob" objects
will be returned. The argument dimensions does not need to be included for this method.
This method can be used to compute probabilities
for the gpcm items in an object of class sep.pars. If x contains
dichotomous items or items associated with another polytomous model, a warning will be
displayed stating that probabilities will be computed for the gpcm items only. The argument
dimensions does not need to be included for this method.
The determination of the model (partial credit or generalized partial credit) is based on
the number of non-NA columns for each item in x and the corresponding values in
cat.
Jonathan P. Weeks [email protected]
Adams, R. J., Wilson, M., & Wang, W. (1997). The multidimensional random coefficients multinomial logit model. Applied Psychological Measurement, 21(1), 1-23.
Embretson, S. E., & Reise, S. P. (2000). Item Response Theory for Psychologists. Mahwah, New Jersey: Lawrence Erlbaum Associates.
Masters, G. N. (1982). A rasch model for partial credit scoring. Psychometrika, 47(2), 149-174.
Masters, G. N. & Wright, B. D. (1996) The partial credit model. In W. J. van der Linden & Hambleton, R. K. (Eds.) Handbook of Modern Item Response Theory (pp. 101-121). New York: Springer-Verlag.
Muraki, E. (1992) A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159-176.
Muraki, E. (1996) A generalized partial credit model. In W. J. van der Linden & Hambleton, R. K. (Eds.) Handbook of Modern Item Response Theory (pp. 153-164). New York: Springer-Verlag.
Weeks, J. P. (2010) plink: An R package for linking mixed-format tests using IRT-based methods. Journal of Statistical Software, 35(12), 1–33. URL http://www.jstatsoft.org/v35/i12/
Yao, L. (2003). BMIRT: Bayesian multivariate item response theory [Computer Program]. Monterey, CA: CTB/McGraw-Hill.
Yao, L., & Schwarz, R. D. (2006). A multidimensional partial credit model with associated item and test statistics: An application to mixed-format tests. Applied Psychological Measurement, 30(6), 469-492.
mixed: compute probabilities for mixed-format itemsplot: plot item characteristic/category curvesirt.prob, irt.pars, sep.pars: classes
###### Unidimensional Examples ###### ## Partial Credit Model ## Item parameters from Embretson & Reise (2000, p. 108) item 5 b <- t(c(-2.519,-.063,.17,2.055)) x <- gpcm(b,5) plot(x) ## Generalized Partial Credit Model ## Item parameters from Embretson & Reise (2000, p. 112) items 5-7 a <- c(.683,1.073,.583) b <- matrix(c(-3.513,-.041,.182,NA,-.873,.358,-.226, 1.547,-4.493,-.004,NA,NA),3,4,byrow=TRUE) pars <- cbind(a,b) # Does not include a location parameter rownames(pars) <- paste("Item",5:7,sep="") colnames(pars) <- c("a",paste("b",1:4,sep="")) cat <- c(4,5,3) x <- gpcm(pars,cat,seq(-3,3,.05)) plot(x) ## Item parameters from Muraki (1996, p. 154) a <- c(1,.5) b <- matrix(c(.25,-1.75,1.75,.75,-1.25,1.25),2,3,byrow=TRUE) pars <- cbind(a,b) # Include a location parameter rownames(pars) <- paste("Item",1:2,sep="") colnames(pars) <- c("a","b",paste("d",1:2,sep="")) cat <- c(3,3) x <- gpcm(pars,cat,location=TRUE,print.mod=TRUE, D=1.7) # Plot category curves for two items matplot(x@prob$theta,x@prob[,2:4],xlab="Theta",ylab="Probability", ylim=c(0,1),lty=1,type="l",col="black") par(new=TRUE) matplot(x@prob$theta,x@prob[,5:7],xlab="Theta",ylab="Probability", ylim=c(0,1),lty=3,type="l",col="black") ###### Multidimensional Examples ###### ## Multidimensional Partial Credit Model pars <- matrix(c(2.4207,0.245,-1.1041,NA, 2.173,-0.4576,NA,NA, 2.1103,-0.8227,.4504,NA, 3.2023,1.0251,-.7837,-1.3062),4,4,byrow=TRUE) cat <- c(4,3,4,5) x <- gpcm(pars,cat,dimensions=2,print.mod=TRUE) # plot combined item category surfaces # The screen argument adjusts the orientation of the axes plot(x,screen=list(z=-60,x=-70)) ## Multidimensional Generalized Partial Credit Model a <- matrix(c(.873, .226, .516, .380, .613, .286 ),3,2,byrow=TRUE) b <- matrix(c(2.255, 1.334, -.503, -2.051, -3.082, 1.917, 1.074, -.497, -1.521, -2.589, 1.624, .994, -.656, -1.978, NA),3,5,byrow=TRUE) pars <- cbind(a,b) cat <- c(6,6,5) x <- gpcm(pars,cat,dimensions=2,print.mod=TRUE) # plot combined item category surfaces plot(x,screen=list(z=-40,x=-60), auto.key=list(space="right")) # plot separated item category surfaces for item two plot(x,items=2,separate=TRUE,drape=TRUE,panels=1) # Compute response probabilities for a single three-category item with # three dimensions. Plot the response surfaces for the first two # dimensions conditional on each theta value on the third dimension pars <- matrix(c(1.1999,0.5997,0.8087,2.1730,-1.4576),1,5) x <- gpcm(pars,3,dimensions=3,theta=-4:4) plot(x, screen=list(z=-30,x=-60))###### Unidimensional Examples ###### ## Partial Credit Model ## Item parameters from Embretson & Reise (2000, p. 108) item 5 b <- t(c(-2.519,-.063,.17,2.055)) x <- gpcm(b,5) plot(x) ## Generalized Partial Credit Model ## Item parameters from Embretson & Reise (2000, p. 112) items 5-7 a <- c(.683,1.073,.583) b <- matrix(c(-3.513,-.041,.182,NA,-.873,.358,-.226, 1.547,-4.493,-.004,NA,NA),3,4,byrow=TRUE) pars <- cbind(a,b) # Does not include a location parameter rownames(pars) <- paste("Item",5:7,sep="") colnames(pars) <- c("a",paste("b",1:4,sep="")) cat <- c(4,5,3) x <- gpcm(pars,cat,seq(-3,3,.05)) plot(x) ## Item parameters from Muraki (1996, p. 154) a <- c(1,.5) b <- matrix(c(.25,-1.75,1.75,.75,-1.25,1.25),2,3,byrow=TRUE) pars <- cbind(a,b) # Include a location parameter rownames(pars) <- paste("Item",1:2,sep="") colnames(pars) <- c("a","b",paste("d",1:2,sep="")) cat <- c(3,3) x <- gpcm(pars,cat,location=TRUE,print.mod=TRUE, D=1.7) # Plot category curves for two items matplot(x@prob$theta,x@prob[,2:4],xlab="Theta",ylab="Probability", ylim=c(0,1),lty=1,type="l",col="black") par(new=TRUE) matplot(x@prob$theta,x@prob[,5:7],xlab="Theta",ylab="Probability", ylim=c(0,1),lty=3,type="l",col="black") ###### Multidimensional Examples ###### ## Multidimensional Partial Credit Model pars <- matrix(c(2.4207,0.245,-1.1041,NA, 2.173,-0.4576,NA,NA, 2.1103,-0.8227,.4504,NA, 3.2023,1.0251,-.7837,-1.3062),4,4,byrow=TRUE) cat <- c(4,3,4,5) x <- gpcm(pars,cat,dimensions=2,print.mod=TRUE) # plot combined item category surfaces # The screen argument adjusts the orientation of the axes plot(x,screen=list(z=-60,x=-70)) ## Multidimensional Generalized Partial Credit Model a <- matrix(c(.873, .226, .516, .380, .613, .286 ),3,2,byrow=TRUE) b <- matrix(c(2.255, 1.334, -.503, -2.051, -3.082, 1.917, 1.074, -.497, -1.521, -2.589, 1.624, .994, -.656, -1.978, NA),3,5,byrow=TRUE) pars <- cbind(a,b) cat <- c(6,6,5) x <- gpcm(pars,cat,dimensions=2,print.mod=TRUE) # plot combined item category surfaces plot(x,screen=list(z=-40,x=-60), auto.key=list(space="right")) # plot separated item category surfaces for item two plot(x,items=2,separate=TRUE,drape=TRUE,panels=1) # Compute response probabilities for a single three-category item with # three dimensions. Plot the response surfaces for the first two # dimensions conditional on each theta value on the third dimension pars <- matrix(c(1.1999,0.5997,0.8087,2.1730,-1.4576),1,5) x <- gpcm(pars,3,dimensions=3,theta=-4:4) plot(x, screen=list(z=-30,x=-60))
This function computes the cumulative probability of responding within or above a certain category or the probability of responding in a specific category for one or more items for a given set of theta values using the graded response model or multidimensional graded response model.
grm(x, cat, theta, dimensions = 1, catprob = FALSE, D = 1, location = FALSE, items, information = FALSE, angle, ...) ## S4 method for signature 'matrix', 'numeric' grm(x, cat, theta, dimensions, catprob, D, location, items, information, angle, ...) ## S4 method for signature 'data.frame', 'numeric' grm(x, cat, theta, dimensions, catprob, D, location, items, information, angle, ...) ## S4 method for signature 'list', 'numeric' grm(x, cat, theta, dimensions, catprob, D, location, items, information, angle, ...) ## S4 method for signature 'irt.pars', 'ANY' grm(x, cat, theta, dimensions, catprob, D, location, items, information, angle, ...) ## S4 method for signature 'sep.pars', 'ANY' grm(x, cat, theta, dimensions, catprob, D, location, items, information, angle, ...)grm(x, cat, theta, dimensions = 1, catprob = FALSE, D = 1, location = FALSE, items, information = FALSE, angle, ...) ## S4 method for signature 'matrix', 'numeric' grm(x, cat, theta, dimensions, catprob, D, location, items, information, angle, ...) ## S4 method for signature 'data.frame', 'numeric' grm(x, cat, theta, dimensions, catprob, D, location, items, information, angle, ...) ## S4 method for signature 'list', 'numeric' grm(x, cat, theta, dimensions, catprob, D, location, items, information, angle, ...) ## S4 method for signature 'irt.pars', 'ANY' grm(x, cat, theta, dimensions, catprob, D, location, items, information, angle, ...) ## S4 method for signature 'sep.pars', 'ANY' grm(x, cat, theta, dimensions, catprob, D, location, items, information, angle, ...)
x |
ans |
cat |
vector identifying the number of response categories (not the number of threshold parameters) for each item. |
theta |
vector, matrix, or list of theta values for which probabilities will be computed.
If |
dimensions |
number of modeled dimensions |
catprob |
if |
D |
scaling constant. The default value assumes that the parameters are already in the desired metric. If the parameters are in the logistic metric, they can be transformed to a normal metric by setting D = 1.7 |
location |
if |
items |
numeric vector identifying the items for which probabilities should be computed |
information |
logical value. If |
angle |
vector or matrix of angles between the dimension 1 axis and the corresponding axes for each
of the other dimensions for which information will be computed. When there are more than two dimensions
and |
... |
further arguments passed to or from other methods |
theta can be specified as a vector, matrix, or list. For the unidimensional case, theta
should be a vector. If a matrix or list of values is supplied, they will be converted to a single vector
of theta values. For the multidimensional case, if a vector of values is supplied it will be assumed
that this same set of values should be used for each dimension. Probabilities will be computed for each
combination of theta values. Similarly, if a list is supplied, probabilities will be computed for each
combination of theta values. In instances where probabilities are desired for specific combinations of
theta values, a j x m matrix should be specified for j ability points and m dimensions where the columns
are ordered from dimension 1 to m.
Returns an object of class irt.prob
This method allows one to specify an n x k matrix for n items. The number of columns can vary depending on the number of dimensions and whether a location parameter is included. The first m columns, for m dimensions, are for item slopes and the remaining columns are for the threshold/threshold deviation parameters.
Threshold parameters can be characterized in two ways: the actual thresholds or
deviations from an overall item difficulty (location). In the deviation scenario the
location argument should be TRUE and the location parameters should be
in the m+1 column. The columns for the threshold/threshold deviation parameters
will always follow the slope column(s) and, if applicable, the location column. The number
of threshold/threshold deviation parameters can vary for each item. In these instances,
all cells with missing values should be filled with NAs. For example, for a
unidimensional model with no location parameter, if one item has five categories (four
threshold parameters) and another item has three categories (two threshold parameters),
there should be five columns. The first column includes the slope parameters and columns
2-5 include the threshold parameters. The values in the last two columns for the item with
three categories should be NA.
See the method for x = "matrix"
This method is for a list with two elements. The first
list element is for item slopes and the second for the threshold/threshold deviation
parameters. For the unidimensional case, the slope values should be a vector of length n
or an n x 1 matrix for n items. For the multidimensional case, the slopes should be
specified in an n x m matrix. For both the unidimensional and multidimensional cases,
the threshold/threshold deviation parameters should be formatted as an n x k matrix. If
the thresholds are deviations from a location parameter, the argument location
should be TRUE and the location parameters should be in the first column. The
number of threshold/threshold deviation parameters can vary for each item. In these
instances, all cells with missing values should be filled with NAs (See the
example in the method for x = "matrix").
This method can be used to compute probabilities
for the grm items in an object of class "irt.pars". If x contains
dichotomous items or items associated with another polytomous model, a warning will be
displayed stating that probabilities will be computed for the grm items only. If x
contains parameters for multiple groups, a list of "irt.prob" objects will be
returned. The argument dimensions does not need to be included for this method.
This method can be used to compute probabilities
for the grm items in an object of class sep.pars. If x contains
dichotomous items or items associated with another polytomous model, a warning will be
displayed stating that probabilities will be computed for the grm items only. The
argument dimensions does not need to be included for this method.
Jonathan P. Weeks [email protected]
Embretson, S. E., & Reise, S. P. (2000) Item Response Theory for Psychologists. Mahwah, New Jersey: Lawrence Erlbaum Associates.
Kolen, M. J. & Brennan, R. L. (2004) Test Equating, Scaling, and Linking (2nd ed.). New York: Springer.
Muraki, E., & Carlson, J. E. (1995). Full-information factor analysis for polytomous item responses. Applied Psychological Measurement, 19(1), 73-90.
Samejima, F. (1969) Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, No. 17.
Samejima, F. (1996) The graded response model. In W. J. van der Linden & Hambleton, R. K. (Eds.) Handbook of Modern Item Response Theory (pp. 85-100). New York: Springer-Verlag.
Weeks, J. P. (2010) plink: An R package for linking mixed-format tests using IRT-based methods. Journal of Statistical Software, 35(12), 1–33. URL http://www.jstatsoft.org/v35/i12/
mixed: compute probabilities for mixed-format itemsplot: plot item characteristic/category curvesirt.prob, irt.pars, sep.pars: classes
###### Unidimensional Examples ###### ## Cumulative probabilities ## Item parameters from Embretson & Reise (2000, p. 101) items 9-11 # Includes a location parameter a <- c(2.09,1.18,1.69) b <- matrix(c(-1.07,-1.03,.39,.86,1.78,1.85,-.87,-.39,.31,NA, -1.4,-.42,NA,NA,-1.74),3,5) pars <- cbind(a,b) rownames(pars) <- paste("Item",9:11,sep=" ") colnames(pars) <- c("a","b",paste("c",1:4,sep="")) cat <- c(3,4,5) x <- grm(pars,cat,location=TRUE) plot(x) ## Item parameters from Kolen & Brennan (2004, p. 210) # Use theta values from -3 to 3 with an increment of 0.5 pars <- t(c(1.2,-.5,.6,1.1,1.3)) x <- grm(pars,5,theta=seq(-3,3,.05)) plot(x,item.lab=FALSE) ## Category probabilities ## Single item pars <- t(c(1.2,-.5,.6,1.1,1.3)) x <- grm(pars,5,seq(-3,3,.05),catprob=TRUE) plot(x,item.lab=FALSE) # Items without location parameter a <- c(2.09,1.18,1.69) b <- matrix(c(-1.93,-2.81,-1.46,-.2,-.64,.08,NA,.37,.81,NA,NA,2.13),3,4) pars <- cbind(a,b) rownames(pars) <- paste("Item",9:11,sep=" ") colnames(pars) <- c("a",paste("b",1:4,sep="")) cat <- c(3,4,5) x <- grm(pars,cat,catprob=TRUE) plot(x) ## Create sep.pars object then compute category probabilities a <- c(2.09,1.18,1.69) b <- matrix(c(-1.93,-2.81,-1.46,-.2,-.64,.08,NA,.37,.81,NA,NA,2.13),3,4) pars <- cbind(a,b) cat <- c(3,4,5) pm <- as.poly.mod(3,"grm") out <- sep.pars(pars,cat,pm) x <- grm(out,catprob=TRUE) plot(x) ###### Multidimensional Examples ###### ## Cumulative probabilities a <- matrix(c(.873, .226, .516, .380, .613, .286 ),3,2,byrow=TRUE) d <- matrix(c(2.255, 1.334, -.503, -2.051, -3.082, 1.917, 1.074, -.497, -1.521, -2.589, 1.624, .994, -.656, -1.978, NA),3,5,byrow=TRUE) pars <- cbind(a,d) cat <- c(6,6,5) x <- grm(pars,cat,dimensions=2) plot(x) # Plot separated response surfaces plot(x, separate=TRUE, drape=TRUE) ## Category Probabilities ## Use {pars} an {cat} from the example above x <- grm(pars,cat,dimensions=2, catprob=TRUE) # plot combined item category surfaces # The screen argument adjusts the orientation of the axes plot(x,screen=list(z=-40,x=-60), auto.key=list(space="right"))###### Unidimensional Examples ###### ## Cumulative probabilities ## Item parameters from Embretson & Reise (2000, p. 101) items 9-11 # Includes a location parameter a <- c(2.09,1.18,1.69) b <- matrix(c(-1.07,-1.03,.39,.86,1.78,1.85,-.87,-.39,.31,NA, -1.4,-.42,NA,NA,-1.74),3,5) pars <- cbind(a,b) rownames(pars) <- paste("Item",9:11,sep=" ") colnames(pars) <- c("a","b",paste("c",1:4,sep="")) cat <- c(3,4,5) x <- grm(pars,cat,location=TRUE) plot(x) ## Item parameters from Kolen & Brennan (2004, p. 210) # Use theta values from -3 to 3 with an increment of 0.5 pars <- t(c(1.2,-.5,.6,1.1,1.3)) x <- grm(pars,5,theta=seq(-3,3,.05)) plot(x,item.lab=FALSE) ## Category probabilities ## Single item pars <- t(c(1.2,-.5,.6,1.1,1.3)) x <- grm(pars,5,seq(-3,3,.05),catprob=TRUE) plot(x,item.lab=FALSE) # Items without location parameter a <- c(2.09,1.18,1.69) b <- matrix(c(-1.93,-2.81,-1.46,-.2,-.64,.08,NA,.37,.81,NA,NA,2.13),3,4) pars <- cbind(a,b) rownames(pars) <- paste("Item",9:11,sep=" ") colnames(pars) <- c("a",paste("b",1:4,sep="")) cat <- c(3,4,5) x <- grm(pars,cat,catprob=TRUE) plot(x) ## Create sep.pars object then compute category probabilities a <- c(2.09,1.18,1.69) b <- matrix(c(-1.93,-2.81,-1.46,-.2,-.64,.08,NA,.37,.81,NA,NA,2.13),3,4) pars <- cbind(a,b) cat <- c(3,4,5) pm <- as.poly.mod(3,"grm") out <- sep.pars(pars,cat,pm) x <- grm(out,catprob=TRUE) plot(x) ###### Multidimensional Examples ###### ## Cumulative probabilities a <- matrix(c(.873, .226, .516, .380, .613, .286 ),3,2,byrow=TRUE) d <- matrix(c(2.255, 1.334, -.503, -2.051, -3.082, 1.917, 1.074, -.497, -1.521, -2.589, 1.624, .994, -.656, -1.978, NA),3,5,byrow=TRUE) pars <- cbind(a,d) cat <- c(6,6,5) x <- grm(pars,cat,dimensions=2) plot(x) # Plot separated response surfaces plot(x, separate=TRUE, drape=TRUE) ## Category Probabilities ## Use {pars} an {cat} from the example above x <- grm(pars,cat,dimensions=2, catprob=TRUE) # plot combined item category surfaces # The screen argument adjusts the orientation of the axes plot(x,screen=list(z=-40,x=-60), auto.key=list(space="right"))
The formal S4 class for irt.pars. This class contains the item parameters and characteristics of the item parameters for one or more groups. When parameters are included for two or more groups, the common items between the different groups are also included.
Objects of class irt.pars contain all the information necessary to produce
expected response probabilities, linking constants (when data for two or more groups are
included), and to plot item response curves/surfaces, vector plots, and parameter
comparison plots.
Objects can be created by calls of the form new("irt.pars", ...), but this is
not encouraged. Use the function as.irt.pars instead.
pars:matrix of item parameters for a single group or a list of matrices containing item parameters for two or more groups
cat:vector with length equal to the number of items, identifying the number of response categories for each item for a single group or a list of response category vectors for two or more groups
poly.mod:a poly.mod object for one group or a list of
poly.mod objects identifying the items associated with each IRT model (see class
poly.mod for more information)
common:a j x 2 matrix for j common items or a list of matrices identifying the common items between pairs of item parameters
location:logical vector identifying whether a given set of item parameters (i.e., for a given group) includes a location parameter
groups:the number of groups (i.e. sets of item parameters)
dimensions:numeric vector identifying the number of modeled dimensions
Class poly.mod, directly.
Class list.poly, by class poly.mod, distance 2.
Jonathan P. Weeks [email protected]
The formal S4 class for irt.prob. This class contains the expected probabilities of responding in a given category for a set of items and theta values under the specified IRT models. The class also includes characteristics of the items.
Objects can be created by calls of the form new("irt.prob", ...), but this is
not encouraged. Use one of the functions drm, gpcm, grm, mcm,
nrm, or mixed instead.
prob:data.frame of item probabilities with n rows and j+m columns for n theta values and j items (the first m column contains theta values for m dimensions)
info:data.frame of item information
p.cat:vector identifying the number of categories for each item for which probabilities were computed
mod.lab:character vector of labels for the model(s).
dimensions:numeric value identifying the number of modeled dimensions
D:numeric vector identifying scaling constants for drm, grm,
and gpcm
pars:list of the item parameters used to compute the probabilities
model:character vector identifying all the models used to compute the
probabilities in prob. The only acceptable models are drm, gpcm,
grm, mcm, and nrm (see class poly.mod for more
information).
items:list with the same length as model, where each element
identifies the items associated with the model(s) specified in model.
Class poly.mod, directly.
Class list.poly, by class poly.mod, distance 2.
Jonathan P. Weeks [email protected]
These models provide information on both unidimensional and multidimensional formulationsdrm: for dichotomous response models (1PL, 2PL, and 3PL)gpcm: for the partial credit/generalized partial credit modelgrm: for the graded response modelmcm: for the multiple-choice modelnrm: for the nominal response modelmixed: for mixed-format itemsirt.pars
These functions test whether a given object is of a specified class
is.irt.pars(x) is.sep.pars(x) is.poly.mod(x) is.irt.prob(x) is.link(x)is.irt.pars(x) is.sep.pars(x) is.poly.mod(x) is.irt.prob(x) is.link(x)
x |
an |
Returns a logical value
Jonathan P. Weeks [email protected]
irt.pars, sep.pars, poly.mod,
irt.prob, link
This (unidimensional) dataset includes three parameter logistic model (3PL) item parameter estimates for two groups. The Form X items are from a "new" form and the Form Y items are from an "old" form. The data are listed in Table 6.5 in Test Equating, Scaling, and Linking (2nd ed.). There are 12 common items between the two forms.
KB04KB04
A list of length two. The first element pars is a list of length two.
Each of the list elements in pars is a 36 x 3 matrix of item parameters.
Element one is for Form X, and element two is for Form Y.
The second list element in KB04 is a matrix identifying the common items
between the two sets of parameters in pars. That is, the first column in
common identifies the rows in pars$form.x that are common items.
The second column identifies the corresponding set of common items in pars$form.y.
In the book, the authors transform the Form X parameters to the Form Y scale.
Kolen, M. J., & Brennan, R. L. (2004) Test Equating, Scaling, and Linking. (2nd ed.), p. 192 New York: Springer
The formal S4 class for link. This class compiles the linking constants, item parameter descriptive statistics, and iteration/objective/convergence information for the moment methods and characteristic curve methods from a separate calibration linking procedure. It is also able to store rescaled item and ability parameters.
Objects can be created by calls of the form new("link", ...), but this is
not encouraged. Use the function plink instead.
constants:list of linking methods where the list elements include the linking constants. For unidimensional models, each element is a vector of length two containing a slope and intercept to adjust the SD and mean respectively. For multidimensional models, each element is a list with a matrix (or set of matrices) to resolve rotational indeterminacy and scale, and a vector of translation constants to adjust the mean of each dimension.
descriptives:data.frame or list containing the item parameter descriptive statistics
iterations:vector of the number of optimization iterations for the Haebara and Stocking-Lord methods
objective:vector of criterion values for HBcrit and SLcrit at the point of convergence for the Haebara and Stocking-Lord methods
convergence:character vector identifying the type of convergence reached under the Haebara or Stocking-Lord optimizations.
base.grp:numeric value indicating the base group for the calibration
n:vector identifying the total number of common items, the number of dichotomous common items, and the number of polytomous common items
grp.names:character vector of group names
mod.lab:character vector identifying the dichotomous and/or polytomous models used to model the item responses
dilation:character value identifying the dilation approach used when estimating multidimensional linking constants
Jonathan P. Weeks [email protected]
These functions extract information from an object of class irt.pars,
typically created using plink, or an object of
class irt.prob containing expected probabilities.
link.con(x, method = "ALL") link.pars(x) link.ability(x) get.prob(x)link.con(x, method = "ALL") link.pars(x) link.ability(x) get.prob(x)
x |
an |
method |
character vector identifying the linking methods for which
constants should be returned. The only acceptable values are
|
link.con extracts the linking constants,
link.pars extracts the rescaled item parameters (if present), and
link.ability extracts the rescaled ability estimates (if present)
get.prob extracts expected probabilities for an irt.prob object
For link.con, the following values can be included for method.
For multidimensional constants, although matrices A, K, and T are
included in the object of class "link" for the Haebara and Stocking-Lord
methods, only the matrix A will be returned for these methods.
Returns the constants for all linking methods
MM:Mean/Mean
MS:Mean/Sigma
HB:Haebara
SL:Stocking-Lord
RM:Reckase-Martineau (for multidimensional constants only)
Returns a matrix or list of linking constants
Jonathan P. Weeks [email protected]
These classes are combinations of classes
combines class list and class poly.mod
combines class list and class data.frame
combines class list and class matrix
combines class list and class numeric
combines class numeric and class NULL
combines class list and class NULL
combines class irt.pars and class NULL
combines class data.frame and class NULL
A virtual Class: No objects may be created from it.
Jonathan P. Weeks [email protected]
This function computes the probability of responding in a specific category for one or more items for a given set of theta values using the multiple-choice model or multidimensional multiple-choice model.
mcm(x, cat, theta, dimensions = 1, items, information = FALSE, angle, ...) ## S4 method for signature 'matrix', 'numeric' mcm(x, cat, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'data.frame', 'numeric' mcm(x, cat, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'list', 'numeric' mcm(x, cat, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'irt.pars', 'ANY' mcm(x, cat, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'sep.pars', 'ANY' mcm(x, cat, theta, dimensions, items, information, angle, ...)mcm(x, cat, theta, dimensions = 1, items, information = FALSE, angle, ...) ## S4 method for signature 'matrix', 'numeric' mcm(x, cat, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'data.frame', 'numeric' mcm(x, cat, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'list', 'numeric' mcm(x, cat, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'irt.pars', 'ANY' mcm(x, cat, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'sep.pars', 'ANY' mcm(x, cat, theta, dimensions, items, information, angle, ...)
x |
an |
cat |
vector identifying the number of response categories plus one for each item (the additional category is for 'do not know') |
theta |
vector, matrix, or list of theta values for which probabilities will be computed.
If |
dimensions |
number of modeled dimensions |
items |
numeric vector identifying the items for which probabilities should be computed |
information |
logical value. If |
angle |
vector or matrix of angles between the dimension 1 axis and the corresponding axes for each
of the other dimensions for which information will be computed. When there are more than two dimensions
and |
... |
further arguments passed to or from other methods |
theta can be specified as a vector, matrix, or list. For the unidimensional case, theta
should be a vector. If a matrix or list of values is supplied, they will be converted to a single vector
of theta values. For the multidimensional case, if a vector of values is supplied it will be assumed
that this same set of values should be used for each dimension. Probabilities will be computed for each
combination of theta values. Similarly, if a list is supplied, probabilities will be computed for each
combination of theta values. In instances where probabilities are desired for specific combinations of
theta values, a j x m matrix should be specified for j ability points and m dimensions where the columns
are ordered from dimension 1 to m.
Returns an object of class irt.prob
This method allows one to specify an n x h matrix
for n items and h equal to [m x 2k + (k-1)] where m is the number of modeled dimensions
and k is equal to the maximum number of response categories (including the 'do not know'
category) across items. The first (m x k) columns are for category slope parameters,
the next block of (m x k) columns are for category difficulty parameters, and the
remaining columns are for the lower asymptote (guessing) parameters. For any items
with fewer categories than the maximum, the remaining cells in each block of (m x k)
columns or the last k columns should be NA.
Say we have one item with four actual response
categories and one item with five response categories. There will be 17 columns.
The first six columns are for the category slope parameters. The first column should
contain the parameters for the 'do not know' category. Column six for the four category
item should be NA. The next six columns (7-12) are for category difficulty
parameters. The first column of this subset of columns (column 7) should contain the
category difficulties for the 'do not know' category. Similar to the block of columns
containing the slopes, the last column in this subset of columns (column 12) for the
four category item should be NA. The remaining five columns are for the lower
asymptote (guessing) parameters. The last column for the four category item would
be NA.
In the multidimensional case, the columns for
the slope and difficulty parameters should be grouped first by dimension and then by
category. Using the same example for the two items with two dimensions there will be
29 columns. The first five columns for the four category item would include the slope
parameters associated with the first dimension for the 'do not know' category and each
of the four actual categories respectively. Columns 11-12 would be NA. Columns
13-17 would include the category difficulties associated with the first dimension (again
the parameters for the 'do not know' category should be in the first column of this
block of columns) and columns 23-24 would be NA. The remaining five columns are
for the lower asymptote (guessing) parameters. The last column for the four category
item would be NA.
See the method for x = "matrix"
This method is for a list with three elements. The
first element is an n x (m x k) matrix of category slope values for n items, m
dimensions, and k equal to the maximum number of response categories across items
(including the 'do not know' category). The second list element is an n x (m x k)
matrix of category difficulty parameters and the last element is an n x (k-1) matrix
of lower asymptote (guessing) parameters. For any list element, for items with fewer
categories than the maximum, the remaining cells in the rows should be NA (see
the examples for method x = "matrix" for specification details).
This method can be used to compute probabilities
for the mcm items in an object of class "irt.pars". If x contains
dichotomous items or items associated with another polytomous model, a warning will be
displayed stating that probabilities will be computed for the mcm items only. If x
contains parameters for multiple groups, a list of "irt.prob" objects will be
returned. The argument dimensions does not need to be included for this method.
This method can be used to compute probabilities
for the mcm items in an object of class sep.pars. If x contains
dichotomous items or items associated with another polytomous model, a warning will be
displayed stating that probabilities will be computed for the mcm items only. The argument
dimensions does not need to be included for this method.
No multidimensional extension of the multiple-choice model has officially been presented in the literature; however, this model is consistent with how the 3PL and nominal response model were extended to the multidimensional context. Additionally, item information is not currently available for the multiple-choice model even though the arguments are included in the function. Information will be available in a later release.
Jonathan P. Weeks [email protected]
Bolt, D. M. & Johnson, T. J. (in press) Applications of a MIRT model to self-report measures: Addressing score bias and DIF due to individual differences in response style. Applied Psychological Measurement.
Thissen, D., & Steinberg, L. (1984). A response model for multiple choice items. Psychometrika, 49(4), 501-519.
Thissen, D., & Steinberg, L. (1996) A response model for multiple choice items. In W.J. van der Linden & Hambleton, R. K. (Eds.) Handbook of Modern Item Response Theory. New York: Springer-Verlag
Weeks, J. P. (2010) plink: An R package for linking mixed-format tests using IRT-based methods. Journal of Statistical Software, 35(12), 1–33. URL http://www.jstatsoft.org/v35/i12/
mixed: compute probabilities for mixed-format itemsplot: plot item characteristic/category curvesirt.prob, irt.pars, sep.pars: classes
###### Unidimensional Examples ###### ## Item parameters from Thissen & Steinberg (1984, p. 510) ## Items R,S,T,U for the whole test a <- matrix(c(-1.7, -1, 1.1, .3, 1.9, -2.1, -.6, 1.2, 2.3, -.8, -1.3, -.9, -.2, 1.9, .5, -1.9, -.5, 0, -.6, 1.9),4,5,byrow=TRUE) c <- matrix(c(.3, -2.3, 2.4, -2.5, 2.1, 2.1, .05, -3, -.6, 1, -.9, -2.5, -.1, 1.8, 1.6, -.1, -2, .5, .8, .8),4,5,byrow=TRUE) d <- matrix(c(.25, .25, .25, .25, .2, .2, .4, .2, .2, .2, .4, .2, .25, .25, .25, .25), 4,4,byrow=TRUE) pars <- cbind(a,c,d) x <- mcm(pars, rep(5,4)) plot(x,item.names=paste("Item",c("R","S","T","U")), auto.key=list(space="right")) ## Item parameters from Thissen & Steinberg (1984, p. 511) ## Items W,X,Y,Z for the pars <- vector("list",3) pars[[1]] <- matrix(c(-2.3, -.2, 2, .9, -.3, -.8, .6, -.5, 1.1, -.4, -.5, -.2, 2, -1.2, 0, -1.5, -.7, -.2, .1, 2.3),4,5,byrow=TRUE) pars[[2]] <- matrix(c(.5, .7, -.5, -1.9, 1.1, 1.6, -2.8, 1.5, 0, -.3, -.3, .7, -1, .7, 0, .4, .4, -.5, .5, -.8),4,5,byrow=TRUE) pars[[3]] <- matrix(c(.2, .4, .2, .2, .2, .2, .4, .2, .2, .4, .2, .2, .2, .2, .2, .4), 4,4,byrow=TRUE) x <- mcm(pars, rep(5,4)) plot(x,item.names=paste("Item",c("W","X","Y","Z")), auto.key=list(space="right")) ###### Multidimensional Example ###### ## Discrimination and category parameters from Bolt & Johnson (in press) pars <- matrix(c(-1.28, -1.029, -0.537, 0.015, 0.519, 0.969, 1.343, 1.473, -0.585, -0.561, -0.445, -0.741, -0.584, 1.444, 0.29, 0.01, 0.04, 0.34, 0, -0.04, -0.63, 0.01, 0.09, 0.09, 0.28, 0.22, 0.31),1,27) x <- mcm(pars, cat=7, dimensions=2) # Plot separated surfaces plot(x,separate=TRUE,drape=TRUE)###### Unidimensional Examples ###### ## Item parameters from Thissen & Steinberg (1984, p. 510) ## Items R,S,T,U for the whole test a <- matrix(c(-1.7, -1, 1.1, .3, 1.9, -2.1, -.6, 1.2, 2.3, -.8, -1.3, -.9, -.2, 1.9, .5, -1.9, -.5, 0, -.6, 1.9),4,5,byrow=TRUE) c <- matrix(c(.3, -2.3, 2.4, -2.5, 2.1, 2.1, .05, -3, -.6, 1, -.9, -2.5, -.1, 1.8, 1.6, -.1, -2, .5, .8, .8),4,5,byrow=TRUE) d <- matrix(c(.25, .25, .25, .25, .2, .2, .4, .2, .2, .2, .4, .2, .25, .25, .25, .25), 4,4,byrow=TRUE) pars <- cbind(a,c,d) x <- mcm(pars, rep(5,4)) plot(x,item.names=paste("Item",c("R","S","T","U")), auto.key=list(space="right")) ## Item parameters from Thissen & Steinberg (1984, p. 511) ## Items W,X,Y,Z for the pars <- vector("list",3) pars[[1]] <- matrix(c(-2.3, -.2, 2, .9, -.3, -.8, .6, -.5, 1.1, -.4, -.5, -.2, 2, -1.2, 0, -1.5, -.7, -.2, .1, 2.3),4,5,byrow=TRUE) pars[[2]] <- matrix(c(.5, .7, -.5, -1.9, 1.1, 1.6, -2.8, 1.5, 0, -.3, -.3, .7, -1, .7, 0, .4, .4, -.5, .5, -.8),4,5,byrow=TRUE) pars[[3]] <- matrix(c(.2, .4, .2, .2, .2, .2, .4, .2, .2, .4, .2, .2, .2, .2, .2, .4), 4,4,byrow=TRUE) x <- mcm(pars, rep(5,4)) plot(x,item.names=paste("Item",c("W","X","Y","Z")), auto.key=list(space="right")) ###### Multidimensional Example ###### ## Discrimination and category parameters from Bolt & Johnson (in press) pars <- matrix(c(-1.28, -1.029, -0.537, 0.015, 0.519, 0.969, 1.343, 1.473, -0.585, -0.561, -0.445, -0.741, -0.584, 1.444, 0.29, 0.01, 0.04, 0.34, 0, -0.04, -0.63, 0.01, 0.09, 0.09, 0.28, 0.22, 0.31),1,27) x <- mcm(pars, cat=7, dimensions=2) # Plot separated surfaces plot(x,separate=TRUE,drape=TRUE)
This dataset includes multidimensional 3PL and multidimensional generalized partial credit model item parameters for two groups modeled with four dimensions. The item parameters were estimated based on item responses provided with BMIRT (Yao, 2008).
md.mixedmd.mixed
A list of length two. There are two matrices in the first list element that correspond to form.x and form.y tests respectively. The second list element identifies the number of response categories for each item. All of the items are common items. There is no location parameter for the polytomous items.
Yao, L. (2008). BMIRT: Bayesian multivariate item response theory [Computer Program]. Monterey, CA: CTB/McGraw-Hill.
This function computes the probability of responding in a specific category for one or more items for a given set of theta values when the items are from a mixed-format test.
mixed(x, cat, poly.mod, theta, dimensions = 1, items, information = FALSE, angle, ...) ## S4 method for signature 'numeric', 'numeric' mixed(x, cat, poly.mod, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'matrix', 'numeric' mixed(x, cat, poly.mod, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'data.frame', 'numeric' mixed(x, cat, poly.mod, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'list', 'numeric' mixed(x, cat, poly.mod, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'irt.pars', 'ANY' mixed(x, cat, poly.mod, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'sep.pars', 'ANY' mixed(x, cat, poly.mod, theta, dimensions, items, information, angle, ...)mixed(x, cat, poly.mod, theta, dimensions = 1, items, information = FALSE, angle, ...) ## S4 method for signature 'numeric', 'numeric' mixed(x, cat, poly.mod, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'matrix', 'numeric' mixed(x, cat, poly.mod, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'data.frame', 'numeric' mixed(x, cat, poly.mod, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'list', 'numeric' mixed(x, cat, poly.mod, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'irt.pars', 'ANY' mixed(x, cat, poly.mod, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'sep.pars', 'ANY' mixed(x, cat, poly.mod, theta, dimensions, items, information, angle, ...)
x |
an |
cat |
vector identifying the number of response categories for each item. If
multiple-choice model items are included, |
poly.mod |
object of class |
theta |
vector, matrix, or list of theta values for which probabilities will be computed.
If |
dimensions |
number of modeled dimensions |
items |
numeric vector identifying the items for which probabilities should be computed |
information |
logical value. If |
angle |
vector or matrix of angles between the dimension 1 axis and the corresponding axes for each
of the other dimensions for which information will be computed. When there are more than two dimensions
and |
... |
further arguments passed to or from other methods. See details below. |
The item parameters supplied to this method can be associated with a single IRT model or
multiple models. When the parameters are tied to only one model, the format of x
(for either unidimensional or multidimensional models) should follow the conventions in
drm for dichotomous response models (i.e. 1PL, 2PL, 3PL), gpcm
for the partial credit model and generalized partial credit model, grm for
the graded response model, mcm for the multiple-choice model, and
nrm for the nominal response model. When the parameters are associated with
two or more models, the parameters should be combined. See as.irt.pars or
for more details on how the parameters from different models can be combined. Additional
arguments for the above models can be passed to this method as well.
theta can be specified as a vector, matrix, or list. For the unidimensional case, theta
should be a vector. If a matrix or list of values is supplied, they will be converted to a single vector
of theta values. For the multidimensional case, if a vector of values is supplied it will be assumed
that this same set of values should be used for each dimension. Probabilities will be computed for each
combination of theta values. Similarly, if a list is supplied, probabilities will be computed for each
combination of theta values. In instances where probabilities are desired for specific combinations of
theta values, a j x m matrix should be specified for j ability points and m dimensions where the columns
are ordered from dimension 1 to m.
The mixed function essentially compiles response probabilities computed using the functions
drm, grm, gpcm, nrm, and mcm for the associated models
respectively. All of the arguments specified in any one of these functions can be passed to
mixed as an additional argument. For example, the argument incorrect can be passed
to drm and catprob can be passed to grm. In the functions drm, grm,
and gpcm there is an argument D for the value of a scaling constant. In mixed,
a single argument D can be passed that will be applied to all applicable models, or arguments
D.drm, D.grm, and D.gpcm can be specified for each model respectively. If an
argument is specified for D and, say D.drm, the values for D.grm and D.gpcm
(if applicable) will be set equal to D. If only D.drm is specified, the values for
D.grm and D.gpcm (if applicable) will be set to 1.
Returns an object of class irt.prob
Jonathan P. Weeks [email protected]
plot: plot item characteristic/category curvesirt.prob, irt.pars, sep.pars: classes
###### Unidimensional Examples ###### # Compute probabilities for three dichotomous (3PL) items and two # polytomous (gpcm) items without a location parameter dichot <- matrix(c(1.2, .8, .9, 2.3, -1.1, -.2, .24, .19, .13),3,3) poly <- matrix(c(.64, -1.8, -.73, .45, NA, .88, .06, 1.4, 1.9, 2.6), 2,5,byrow=TRUE) pars <- rbind(cbind(dichot,matrix(NA,3,2)),poly) cat <- c(2,2,2,4,5) pm <- as.poly.mod(5, c("drm","gpcm"), list(1:3,4:5)) x <- mixed(pars, cat, pm) plot(x) # Specify a different scaling constant for the GPCM items in the # above example x <- mixed(pars, cat, pm, D.gpcm=1.7) plot(x) # Compute probabilities for three dichotomous (3PL) items, four # polytomous items, two gpcm items and two nrm items. Include a # location parameter for the gpcm items a <- matrix(c( 1.2, rep(NA,4), .8, rep(NA,4), .9, rep(NA,4), .64, rep(NA,4), .88, rep(NA,4), .905, .522, -.469, -.959, NA, .828, .375, -.357, -.079, -.817),7,5,byrow=TRUE) b <- matrix(c( 2.3, rep(NA,4), -1.1, rep(NA,4), -.2, rep(NA,4), -.69, -1.11, -.04, 1.14, NA, 1.49, -1.43, -.09, .41, 1.11, .126, -.206, -.257, .336, NA, .565, .865, -1.186, -1.199, .993),7,5,byrow=TRUE) c <- c(.14, .19, .26, rep(NA,4)) pars <- list(a,b,c) cat <- c(2,2,2,4,5,4,5) pm <- as.poly.mod(7, c("drm","gpcm","nrm"), list(1:3,4:5,6:7)) x <- mixed(pars, cat, pm, location=TRUE) plot(x) ###### Multidimensional Example ###### # Compute response probabilities for four dichotomous items # modeled using the M2PL and three polytomous items modeled # using the multidimensional graded response model. For the # later items, cumulative probabilities are computed. a <- matrix(c(1.66,1.72,.69,.19,.88,1.12,.68,1.21, .873, .226, .516, .380, .613, .286 ),7,2,byrow=TRUE) d <- matrix(c(-.38,NA,NA,NA,NA, -.68,NA,NA,NA,NA, -.91,NA,NA,NA,NA, -1.08,NA,NA,NA,NA, 2.255, 1.334, -.503, -2.051, -3.082, 1.917, 1.074, -.497, -1.521, -2.589, 1.624, .994, -.656, -1.978, NA),7,5,byrow=TRUE) cat <- c(2,2,2,2,6,6,5) pars <- cbind(a,d) pm <- as.poly.mod(7,c("drm","grm"),list(1:4,5:7)) x <- mixed(pars, cat, pm, dimensions=2, catprob=TRUE) plot(x)###### Unidimensional Examples ###### # Compute probabilities for three dichotomous (3PL) items and two # polytomous (gpcm) items without a location parameter dichot <- matrix(c(1.2, .8, .9, 2.3, -1.1, -.2, .24, .19, .13),3,3) poly <- matrix(c(.64, -1.8, -.73, .45, NA, .88, .06, 1.4, 1.9, 2.6), 2,5,byrow=TRUE) pars <- rbind(cbind(dichot,matrix(NA,3,2)),poly) cat <- c(2,2,2,4,5) pm <- as.poly.mod(5, c("drm","gpcm"), list(1:3,4:5)) x <- mixed(pars, cat, pm) plot(x) # Specify a different scaling constant for the GPCM items in the # above example x <- mixed(pars, cat, pm, D.gpcm=1.7) plot(x) # Compute probabilities for three dichotomous (3PL) items, four # polytomous items, two gpcm items and two nrm items. Include a # location parameter for the gpcm items a <- matrix(c( 1.2, rep(NA,4), .8, rep(NA,4), .9, rep(NA,4), .64, rep(NA,4), .88, rep(NA,4), .905, .522, -.469, -.959, NA, .828, .375, -.357, -.079, -.817),7,5,byrow=TRUE) b <- matrix(c( 2.3, rep(NA,4), -1.1, rep(NA,4), -.2, rep(NA,4), -.69, -1.11, -.04, 1.14, NA, 1.49, -1.43, -.09, .41, 1.11, .126, -.206, -.257, .336, NA, .565, .865, -1.186, -1.199, .993),7,5,byrow=TRUE) c <- c(.14, .19, .26, rep(NA,4)) pars <- list(a,b,c) cat <- c(2,2,2,4,5,4,5) pm <- as.poly.mod(7, c("drm","gpcm","nrm"), list(1:3,4:5,6:7)) x <- mixed(pars, cat, pm, location=TRUE) plot(x) ###### Multidimensional Example ###### # Compute response probabilities for four dichotomous items # modeled using the M2PL and three polytomous items modeled # using the multidimensional graded response model. For the # later items, cumulative probabilities are computed. a <- matrix(c(1.66,1.72,.69,.19,.88,1.12,.68,1.21, .873, .226, .516, .380, .613, .286 ),7,2,byrow=TRUE) d <- matrix(c(-.38,NA,NA,NA,NA, -.68,NA,NA,NA,NA, -.91,NA,NA,NA,NA, -1.08,NA,NA,NA,NA, 2.255, 1.334, -.503, -2.051, -3.082, 1.917, 1.074, -.497, -1.521, -2.589, 1.624, .994, -.656, -1.978, NA),7,5,byrow=TRUE) cat <- c(2,2,2,2,6,6,5) pars <- cbind(a,d) pm <- as.poly.mod(7,c("drm","grm"),list(1:4,5:7)) x <- mixed(pars, cat, pm, dimensions=2, catprob=TRUE) plot(x)
This function computes the probability of responding in a specific category for one or more items for a given set of theta values using the nominal response model or multidimensional nominal response model.
nrm(x, cat, theta, dimensions = 1, items, information = FALSE, angle, ...) ## S4 method for signature 'matrix', 'numeric' nrm(x, cat, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'data.frame', 'numeric' nrm(x, cat, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'list', 'numeric' nrm(x, cat, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'irt.pars', 'ANY' nrm(x, cat, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'sep.pars', 'ANY' nrm(x, cat, theta, dimensions, items, information, angle, ...)nrm(x, cat, theta, dimensions = 1, items, information = FALSE, angle, ...) ## S4 method for signature 'matrix', 'numeric' nrm(x, cat, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'data.frame', 'numeric' nrm(x, cat, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'list', 'numeric' nrm(x, cat, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'irt.pars', 'ANY' nrm(x, cat, theta, dimensions, items, information, angle, ...) ## S4 method for signature 'sep.pars', 'ANY' nrm(x, cat, theta, dimensions, items, information, angle, ...)
x |
Object containing item parameters. See below for more details. |
cat |
vector identifying the number of response categories for each item |
theta |
vector, matrix, or list of theta values for which probabilities will be computed.
If |
dimensions |
number of modeled dimensions |
items |
numeric vector identifying the items for which probabilities should be computed |
information |
logical value. If |
angle |
vector or matrix of angles between the dimension 1 axis and the corresponding axes for each
of the other dimensions for which information will be computed. When there are more than two dimensions
and |
... |
further arguments passed to or from other methods |
theta can be specified as a vector, matrix, or list. For the unidimensional case, theta
should be a vector. If a matrix or list of values is supplied, they will be converted to a single vector
of theta values. For the multidimensional case, if a vector of values is supplied it will be assumed
that this same set of values should be used for each dimension. Probabilities will be computed for each
combination of theta values. Similarly, if a list is supplied, probabilities will be computed for each
combination of theta values. In instances where probabilities are desired for specific combinations of
theta values, a j x m matrix should be specified for j ability points and m dimensions where the columns
are ordered from dimension 1 to m.
Returns an object of class irt.prob
This method allows one to specify an n x (m x 2k)
matrix for n items, m dimensions, and k equal to the maximum number of response
categories across items. The first (m x k) columns are for category slope parameters
and the remaining columns are for the category difficulty parameters. For any items
with fewer categories than the maximum, the remaining cells in each block of (m x k)
columns should be NA.
Say we have one four category item and one
five category item, the first four columns of the four response item would include
the slope parameters. The fifth column for this item would be NA. The next
four columns would include the category difficulty values, and the last column would
be NA.
In the multidimensional case, the columns for
the slope and difficulty parameters should be grouped first by dimension and then by
category. Using the same example for the two items with two dimensions there will be
20 columns. The first four columns for the four category item would include the slope
parameters associated with the first dimension for each of the four categories
respectively. Columns 9-10 would be NA. Columns 11-14 would include the
category difficulties associated with the first dimension and columns 19-20 would
be NA.
See the method for x = "matrix"
This method is for a list with two elements. The
first element is an n x (m x k) matrix of category slope values for n items, m
dimensions, and k equal to the maximum number of response categories across items.
The second list element is an n x (m x k) matrix of category difficulty parameters.
For either element, for items with fewer categories than the maximum, the remaining
cells in the rows should be NA (see the examples for method x = "matrix" for
specification details).
This method can be used to compute probabilities
for the nrm items in an object of class "irt.pars". If x contains
dichotomous items or items associated with another polytomous model, a warning
will be displayed stating that probabilities will be computed for the nrm
items only. If x contains parameters for multiple groups, a list of
"irt.prob" objects will be returned.
This method can be used to compute probabilities
for the mcm items in an object of class sep.pars. If x contains
dichotomous items or items associated with another polytomous model, a warning
will be displayed stating that probabilities will be computed for the nrm
items only.
Jonathan P. Weeks [email protected]
Bock, R.D. (1972) Estimating item parameters and latent ability when responses are scored in two or more nominal categories. Psychometrika, 37(1), 29-51.
Bock, R.D. (1996) The nominal categories model. In W.J. van der Linden & Hambleton, R. K. (Eds.) Handbook of Modern Item Response Theory. New York: Springer-Verlag
Bolt, D. M. & Johnson, T. J. (in press) Applications of a MIRT model to self-report measures: Addressing score bias and DIF due to individual differences in response style. Applied Psychological Measurement.
Kolen, M. J., & Brennan, R. L. (2004) Test Equating, Scaling, and Linking. New York: Springer
Takane, Y., & De Leeuw, J. (1987) On the relationship between item response theory and factor analysis of discretized variables. Psychometrika, 52(3), 393-408.
Weeks, J. P. (2010) plink: An R package for linking mixed-format tests using IRT-based methods. Journal of Statistical Software, 35(12), 1–33. URL http://www.jstatsoft.org/v35/i12/
mixed: compute probabilities for mixed-format itemsplot: plot item characteristic/category curvesirt.prob, irt.pars, sep.pars: classes
###### Unidimensional Example ###### ## Item parameters from Bock (1972, p. 46,47) a <- matrix(c(.905, .522, -.469, -.959, NA, .828, .375, -.357, -.079, -.817), 2,5,byrow=TRUE) c <- matrix(c(.126, -.206, -.257, .336, NA, .565, .865, -1.186, -1.199, .993), 2,5,byrow=TRUE) pars <- cbind(a,c) x <- nrm(pars, c(4,5)) plot(x,auto.key=list(space="right")) ###### Multidimensional Example ###### # From Bolt & Johnson (in press) pars <- matrix(c(-1.28, -1.029, -0.537, 0.015, 0.519, 0.969, 1.343, 1.473, -0.585, -0.561, -0.445, -0.741, -0.584, 1.444, 0.29, 0.01, 0.04, 0.34, 0, -0.04, -0.63), 1,21) x <- nrm(pars, cat=7, dimensions=2) # Plot separated surfaces plot(x,separate=TRUE,drape=TRUE)###### Unidimensional Example ###### ## Item parameters from Bock (1972, p. 46,47) a <- matrix(c(.905, .522, -.469, -.959, NA, .828, .375, -.357, -.079, -.817), 2,5,byrow=TRUE) c <- matrix(c(.126, -.206, -.257, .336, NA, .565, .865, -1.186, -1.199, .993), 2,5,byrow=TRUE) pars <- cbind(a,c) x <- nrm(pars, c(4,5)) plot(x,auto.key=list(space="right")) ###### Multidimensional Example ###### # From Bolt & Johnson (in press) pars <- matrix(c(-1.28, -1.029, -0.537, 0.015, 0.519, 0.969, 1.343, 1.473, -0.585, -0.561, -0.445, -0.741, -0.584, 1.444, 0.29, 0.01, 0.04, 0.34, 0, -0.04, -0.63), 1,21) x <- nrm(pars, cat=7, dimensions=2) # Plot separated surfaces plot(x,separate=TRUE,drape=TRUE)
This function conducts separate calibration of unidimensional or multidimensional IRT single-format or mixed-format item parameters for multiple groups. The unidimensional methods include the Mean/Mean, Mean/Sigma, Haebara, and Stocking-Lord methods. The multidimensional methods include a least squares approach (Li & Lissitz, 2000; Min, 2003; Reckase & Martineau, 2004) and extensions of the Haebara and Stocking-Lord method using a single dilation parameter (Li & Lissitz, 2000), multiple dilation parameters (Min, 2003), or an oblique Procrustes approach (Oshima, Davey, & Lee, 2000; Reckase, 2009). The methods allow for symmetric and non-symmetric optimization and chain-linked rescaling of item and ability parameters.
plink(x, common, rescale, ability, method, weights.t, weights.f, startvals, exclude, score = 1, base.grp = 1, symmetric = FALSE, rescale.com = TRUE, grp.names = NULL, dilation = "oblique", md.center = FALSE, dim.order = NULL, ...) ## S4 method for signature 'list', 'matrix' plink(x, common, rescale, ability, method, weights.t, weights.f, startvals, exclude, score, base.grp, symmetric, rescale.com, grp.names, dilation, md.center, dim.order, ...) ## S4 method for signature 'list', 'data.frame' plink(x, common, rescale, ability, method, weights.t, weights.f, startvals, exclude, score, base.grp, symmetric, rescale.com, grp.names, dilation, md.center, dim.order, ...) ## S4 method for signature 'list', 'list' plink(x, common, rescale, ability, method, weights.t, weights.f, startvals, exclude, score, base.grp, symmetric, rescale.com, grp.names, dilation, md.center, dim.order, ...) ## S4 method for signature 'irt.pars', 'ANY' plink(x, common, rescale, ability, method, weights.t, weights.f, startvals, exclude, score, base.grp, symmetric, rescale.com, grp.names, dilation, md.center, dim.order, ...)plink(x, common, rescale, ability, method, weights.t, weights.f, startvals, exclude, score = 1, base.grp = 1, symmetric = FALSE, rescale.com = TRUE, grp.names = NULL, dilation = "oblique", md.center = FALSE, dim.order = NULL, ...) ## S4 method for signature 'list', 'matrix' plink(x, common, rescale, ability, method, weights.t, weights.f, startvals, exclude, score, base.grp, symmetric, rescale.com, grp.names, dilation, md.center, dim.order, ...) ## S4 method for signature 'list', 'data.frame' plink(x, common, rescale, ability, method, weights.t, weights.f, startvals, exclude, score, base.grp, symmetric, rescale.com, grp.names, dilation, md.center, dim.order, ...) ## S4 method for signature 'list', 'list' plink(x, common, rescale, ability, method, weights.t, weights.f, startvals, exclude, score, base.grp, symmetric, rescale.com, grp.names, dilation, md.center, dim.order, ...) ## S4 method for signature 'irt.pars', 'ANY' plink(x, common, rescale, ability, method, weights.t, weights.f, startvals, exclude, score, base.grp, symmetric, rescale.com, grp.names, dilation, md.center, dim.order, ...)
x |
an object of class |
common |
matrix or list of common items. See below for more details. |
rescale |
if missing (default), the parameters in |
ability |
list of theta values with length equal to the number of groups.
If supplied, these values will be transformed to the base scale using the
constants identified in |
method |
character vector identifying the linking methods to use. Values can
include |
weights.t |
list containing information about the theta values and weights on the To scale for use with the characteristic curve methods. See below for more details. |
weights.f |
list containing information about the theta values and weights
on the From scale for use with the characteristic curve methods. This
argument will be ignored if |
startvals |
vector or list of slope and intercept starting values for the characteristic curve methods or character value(s) of "MM", "MS", or "LS" to identify that values from the Mean/Mean, Mean/Sigma, or least squares method should be used for the starting values. See below for more details. |
exclude |
character vector or list identifying common items that should be excluded when estimating the linking constants. See below for more details. |
score |
if |
base.grp |
integer identifying the group for the base scale |
symmetric |
if |
rescale.com |
if |
grp.names |
character vector of group names |
dilation |
character value identifying whether an oblique Procrustes approach
(Oshima, Davey, & Lee, 2000; Reckase 2009) |
md.center |
if |
dim.order |
matrix for identifying the ordering of factors across groups. See below for details. |
... |
further arguments passed to or from other methods. See below for details. |
If x contains only two elements, common should be a matrix. If x
contains more than two elements, common should be a list. In any of the common
matrices the first column identifies the common items for the first group of two adjacent
list elements in x. The second column in common identifies the corresponding
set of common items from the next list element in x. For example, if x
contains only two list elements, a single set of common items links them together. If
item 4 in group one (row 4 in slot pars) is the same as item 6 in group two, the
first row of common would be (4,6).
startvals can be a vector or list of starting values for the slope(s) and intercept(s).
For unidimensional methods, when there are only two groups, this argument should be a vector of
length of two with the first value for the slope and the second value for the intercept or a
character value equal to "MM" or"MS". When there are more than two groups a vector of starting
values or a character value can be supplied (the same numeric values, if a vector is supplied,
will be used for all pairs, or the corresponding MM/MS values for each pair of tests will be
used) or a list of vectors/character values can be supplied with the number of list elements
equal to the number of groups minus one. For the multidimensional methods, the same general
structure applies (a vector or character value for a single group or a vector, character value
or list for multiple groups); however, the length of the vector will vary depending on the
dilation approach used. If dilation is "obliquw", the first m*m values in startvals,
for m dimensions, should correspond to the values in the transformation matrix (starting with
the value in the upper-left corner, then the next value in the column, ..., then the first value
in the next column, etc.). The remaining m values should be for the translation vector. If
dilation is "orth.fd", the first value will be the slope parameter and the remaining m values
will be for the translation vector. If dilation is "orth.vd", the first m values are the
slopes for each dimension and the remaining m values are for the translation vector.
weights.t can be a list or a list of lists. The purpose of this object is to specify
the theta values on the To scale to integrate over in the characteristic curve methods
as well as any weights associated with the theta values. See Kim and Lee (2006) or Kolen
and Brennan (2004) for more information of these weights. The function as.weight
can be used to facilitate the creation of this object. If weights.t is missing, the
default–in the unidimensional case–is to use equally spaced theta values ranging from -4 to 4
with an increment of 0.05 and theta weights equal to one for all theta values. In the
multidimensional case, the default is to use 1000 randomly sampled values from a multivariate
normal distribution with correlations equal to 0.6 for all dimensions. The theta weights are
equal to the normal distribution weights at these points.
To better understand the elements of weights.t, let us assume for a moment that x
has parameters for only two groups and that we are using non-symmetric linking. In this instance,
weights.t would be a single list with length two. The first element should be a vector
of theta values corresponding to points on the To scale. The second list element should
be a vector of weights corresponding the theta values. If x contains multiple
groups, a single weights.t object can be supplied, and the same set of thetas and weights
will be used for all adjacent groups. However, a separate list of theta values and theta weights
for each adjacent group in x can be supplied.
The specification of weights.f is the same as that for weights.t, although the
theta values and weights for this object correspond to theta values on the From scale.
This argument is only used when symmetric=TRUE. If weights.f is missing and
symmetric=TRUE, the same theta values and weights used for weights.t will be
used for weights.f.
In general, all of the common items identified in x or common will be used
in estimating linking constants; however, there are instances where there is a need to exclude
certain common items (e.g., NRM or MCM items or items exhibiting parameter drift). Instead of
creating a new matrix or list of matrices for common, the exclude argument can
be used. exclude can be specified as a character vector or a list. In the former case,
a vector of model names (i.e., "drm", "grm", "gpcm", "nrm", "mcm") would be supplied, indicating
that any common item on any test associated with the given model(s) would be excluded from the
set of items used to estimate the linking constants. If the argument is specified as a list,
exclude should have as many elements as groups in x. Each list element can include
model names and/or item numbers corresponding to the common items that should be excluded for
the given group. If no items need to be excluded for a given group, the list element should be
NULL or NA. For example, say we have two groups and we would like to exclude the NRM items and
item 23 from the first group, we would specify exclude as exclude <- list(c("nrm",23),NA).
Notice that the item number corresponding item 23 in group 2 does not need to be specified.
The argument dim.order is a k x r matrix for k groups and r unique dimensions across
groups. This object identifies the common dimensions across groups. The elements in the
matrix should correspond to the dimension (i.e., the column in the matrix of slope parameters)
for a given group. For example, say there are four unique dimensions across two groups,
each group only measures three dimensions, and there are only two common dimensions. We might
specify a matrix as follows dim.order <- matrix(c(1:3,NA,NA,1:3),2,4). In words, this means
that dimensions 2 and 3 in the first group correspond to dimensions 1 and 2 in the second
group respectively. If no dim.order is specified, it is assumed that all of the
dimensions are common, or in instances with different numbers of factors, that the first
m dimensions for each group are common and the remaining r-m dimensions for a given group
are unique.
For the characteristic curve methods, response probabilities are computed using the functions
drm, grm, gpcm, nrm, and mcm for the associated models
respectively. Various arguments from these functions can be passed to plink. Specifically,
the argument incorrect can be passed to drm and catprob can be passed to
grm. In the functions drm, grm, and gpcm there is an argument D
for the value of a scaling constant. In plink, a single argument D can be passed
that will be applied to all applicable models, or arguments D.drm, D.grm, and
D.gpcm can be specified for each model respectively. If an argument is specified for D
and, say D.drm, the values for D.grm and D.gpcm (if applicable) will be
set equal to D. If only D.drm is specified, the values for D.grm and
D.gpcm (if applicable) will be set to 1.
Returns an object of class link. The labels for the linking constants are
specified in the following manner "group1/group2", meaning the group1 parameters were transformed
to the group2 test. The base group is indicated by an asterisk.
This method is used when x contains only
two list elements. If either of the list elements is of class irt.pars, they
can include multiple groups. common is the matrix of common items between
the two groups in x. See details for more information on common.
See the method for signature x="list",
common="matrix".
This method is used when x includes two or
more list elements. When x has length two, common (although a single
matrix) should be a list with length one. If x has more than two list elements
common identifies the common items between adjacent list elements. If objects
of class irt.pars are included with multiple groups, common should
identify the common items between the first or last group in the irt.pars object,
depending on its location in x, and the adjacent list element(s) in x.
For example, if x has three elements: an irt.pars object with one group,
an irt.pars object with four groups, and a sep.pars object, common
will be a list with length two. The first element in common is a matrix
identifying the common items between the items in the first irt.pars object
and the first group in the second irt.pars object. The second element in
common should identify the common items between the fourth group in the
second irt.pars object and the items in the sep.pars object.
This method is intended for an irt.pars
object with multiple groups.
The translation vector for the multidimensional Stocking-Lord
method may converge to odd values. The least squares method has been
shown to produce more accurate constants than the characteristic curve
methods (in less time); however, if use of a characteristic curve
approach is desired, it is recommended that the Haebara method be used
and/or the relative tolerance for the optimization be lowered to 0.001
using the argument control=list(rel.tol=0.001).
Jonathan P. Weeks [email protected]
Haebara, T. (1980). Equating logistic ability scales by a weighted least squares method. Japanese Psychological Research, 22(3), 144-149.
Kim, S. & Lee, W.-C. (2006). An Extension of Four IRT Linking Methods for Mixed-Format Tests. Journal of Educational Measurement, 43(1), 53-76.
Kolen, M. J. & Brennan, R. L. (2004) Test Equating, Scaling, and Linking (2nd ed.). New York: Springer
Li, Y. H., & Lissitz, R. W. (2000). An evaluation of the accuracy of multidimensional IRT linking. Applied Psychological Measurement, 24(2), 115-138.
Loyd, B. H. & Hoover, H. D. (1980). Vertical Equating Using the Rasch Model. Journal of Educational Measurement, 17(3), 179-193.
Marco, G. L. (1977). Item Characteristic Curve Solutions to Three Intractable Testing Problems. Journal of Educational Measurement, 14(2), 139-160.
Min, K. -S. (2007). Evaluation of linking methods for multidimensional IRT calibrations. Asia Pacific Education Review, 8(1), 41-55.
Oshima, T. C., Davey, T., & Lee, K. (2000). Multidimensional linking: Four practical approaches. Journal of Educational Measurement, 37(4), 357-373.
Reckase, M. D. (2009). Multidimensional item response theory New York, Springer
Reckase, M. D., & Martineau, J. A. (2004). The vertical scaling of science achievement tests. Research report for the Center for Education and National Research Council.
Stocking, M. L. & Lord, F. M. (1983). Developing a common metric in item response theory. Applied Psychological Measurement, 7(2), 201-210.
Weeks, J. P. (2010) plink: An R package for linking mixed-format tests using IRT-based methods. Journal of Statistical Software, 35(12), 1–33. URL http://www.jstatsoft.org/v35/i12/
###### Unidimensional Examples ###### # Create irt.pars object with two groups (all dichotomous items), # rescale the item parameters using the Mean/Sigma linking constants, # and exclude item 27 from the common item set pm <- as.poly.mod(36) x <- as.irt.pars(KB04$pars, KB04$common, cat=list(rep(2,36),rep(2,36)), poly.mod=list(pm,pm)) out <- plink(x, rescale="MS", base.grp=2, D=1.7, exclude=list(27,NA)) summary(out, descrip=TRUE) pars.out <- link.pars(out) # Create object with six groups (all dichotomous items) pars <- TK07$pars common <- TK07$common cat <- list(rep(2,26),rep(2,34),rep(2,37),rep(2,40),rep(2,41),rep(2,43)) pm1 <- as.poly.mod(26) pm2 <- as.poly.mod(34) pm3 <- as.poly.mod(37) pm4 <- as.poly.mod(40) pm5 <- as.poly.mod(41) pm6 <- as.poly.mod(43) pm <- list(pm1, pm2, pm3, pm4, pm5, pm6) x <- as.irt.pars(pars, common, cat, pm, grp.names=paste("grade",3:8,sep="")) out <- plink(x) summary(out) constants <- link.con(out) # Extract linking constants # Create an irt.pars object and a sep.pars object for two groups of # nominal response model items. Compare non-symmetric and symmetric # minimization. Note: This example may take a minute or two to run ## Not run: pm <- as.poly.mod(60, "nrm", 1:60) pars1 <- as.irt.pars(act.nrm$yr97, cat=rep(5,60), poly.mod=pm) pars2 <- sep.pars(act.nrm$yr98, cat=rep(5,60), poly.mod=pm) out <- plink(list(pars1, pars2), matrix(1:60,60,2)) out1 <- plink(list(pars1, pars2), matrix(1:60,60,2), symmetric=TRUE) summary(out, descrip=TRUE) summary(out1, descrip=TRUE) ## End(Not run) # Compute linking constants for two groups with multiple-choice model # item parameters. Rescale theta values and item parameters using # the Haebara linking constants # Note: This example may take a minute or two to run ## Not run: theta <- rnorm(100) # In practice, estimated theta values would be used pm <- as.poly.mod(60, "mcm", 1:60) x <- as.irt.pars(act.mcm, common=matrix(1:60,60,2), cat=list(rep(6,60), rep(6,60)), poly.mod=list(pm,pm)) out <- plink(x, ability=list(theta,theta), rescale="HB") pars.out <- link.pars(out) ability.out <- link.ability(out) summary(out, descrip=TRUE) ## End(Not run) # Compute linking constants for two groups using mixed-format items and # a mixed placement of common items. Compare calibrations with the # inclusion or exclusion of NRM items. This example uses the dgn dataset. pm1 <- as.poly.mod(55,c("drm","gpcm","nrm"),dgn$items$group1) pm2 <- as.poly.mod(55,c("drm","gpcm","nrm"),dgn$items$group2) x <- as.irt.pars(dgn$pars,dgn$common,dgn$cat,list(pm1,pm2)) # Run with the NRM common items included out <- plink(x) # Run with the NRM common items excluded out1 <- plink(x,exclude="nrm") summary(out) summary(out1) # Compute linking constants for six groups using mixed-format items and # a mixed placement of common items. This example uses the reading dataset. # See the information on the dataset for an interpretation of the output. pm1 <- as.poly.mod(41, c("drm", "gpcm"), reading$items[[1]]) pm2 <- as.poly.mod(70, c("drm", "gpcm"), reading$items[[2]]) pm3 <- as.poly.mod(70, c("drm", "gpcm"), reading$items[[3]]) pm4 <- as.poly.mod(70, c("drm", "gpcm"), reading$items[[4]]) pm5 <- as.poly.mod(72, c("drm", "gpcm"), reading$items[[5]]) pm6 <- as.poly.mod(71, c("drm", "gpcm"), reading$items[[6]]) pm <- list(pm1, pm2, pm3, pm4, pm5, pm6) x <- as.irt.pars(reading$pars, reading$common, reading$cat, pm, base.grp=4) out <- plink(x) summary(out) ###### Multidimensional Examples ###### # Reckase Chapter 9 pm <- as.poly.mod(80, "drm", list(1:80)) x <- as.irt.pars(reckase9$pars, reckase9$common, list(rep(2,80),rep(2,80)), list(pm,pm), dimensions=2) # Compute constants using the least squares method and # the orthongal rotation with variable dilation. # Rescale the item parameters using the LS method out <- plink(x, dilation="orth.vd", rescale="LS") summary(out, descrip=TRUE) # Extract the rescaled item parameters pars.out <- link.pars(out) # Compute constants using an oblique Procrustes method # Display output and descriptives out <- plink(x, dilation="oblique") summary(out, descrip=TRUE) # Compute constants using and orthogonal rotation with # a fixed dilation parameter # Rescale the item parameters and ability estimates # using the "HB" and "LS" methods. # Display the optimization iterations ability <- matrix(rnorm(40),20,2) out <- plink(x, method=c("HB","LS"), dilation="orth.fd", rescale="HB", ability=list(ability,ability), control=list(trace=1,rel.tol=0.001)) summary(out) # Extract rescaled ability estimates ability.out <- out$ability # Compute linking constants for two groups using mixed-format items # modeled with the M3PL and MGPCM. Only compute constants using the # orth.vd dilation. pm <- as.poly.mod(60,c("drm","gpcm"), list(c(1:60)[md.mixed$cat==2], c(1:60)[md.mixed$cat>2])) x <- as.irt.pars(md.mixed$pars, matrix(1:60,60,2), list(md.mixed$cat, md.mixed$cat), list(pm, pm), dimensions=4, grp.names=c("Form.X","Form.Y")) out <- plink(x,dilation="orth.vd") summary(out, descrip=TRUE) # Illustration of construct shift and how to specify common dimensions pm <- as.poly.mod(80, "drm", list(1:80)) pars <- cbind(round(runif(80),2),reckase9$pars[[1]]) x <- as.irt.pars(list(pars,reckase9$pars[[2]]), reckase9$common, list(rep(2,80),rep(2,80)), list(pm,pm), dimensions=c(3,2)) dim.order <- matrix(c(1,2,3,NA,1,2),2,3,byrow=TRUE) out <- plink(x, dilation="oblique", dim.order=dim.order, rescale="LS") summary(out) pars.out <- link.pars(out)###### Unidimensional Examples ###### # Create irt.pars object with two groups (all dichotomous items), # rescale the item parameters using the Mean/Sigma linking constants, # and exclude item 27 from the common item set pm <- as.poly.mod(36) x <- as.irt.pars(KB04$pars, KB04$common, cat=list(rep(2,36),rep(2,36)), poly.mod=list(pm,pm)) out <- plink(x, rescale="MS", base.grp=2, D=1.7, exclude=list(27,NA)) summary(out, descrip=TRUE) pars.out <- link.pars(out) # Create object with six groups (all dichotomous items) pars <- TK07$pars common <- TK07$common cat <- list(rep(2,26),rep(2,34),rep(2,37),rep(2,40),rep(2,41),rep(2,43)) pm1 <- as.poly.mod(26) pm2 <- as.poly.mod(34) pm3 <- as.poly.mod(37) pm4 <- as.poly.mod(40) pm5 <- as.poly.mod(41) pm6 <- as.poly.mod(43) pm <- list(pm1, pm2, pm3, pm4, pm5, pm6) x <- as.irt.pars(pars, common, cat, pm, grp.names=paste("grade",3:8,sep="")) out <- plink(x) summary(out) constants <- link.con(out) # Extract linking constants # Create an irt.pars object and a sep.pars object for two groups of # nominal response model items. Compare non-symmetric and symmetric # minimization. Note: This example may take a minute or two to run ## Not run: pm <- as.poly.mod(60, "nrm", 1:60) pars1 <- as.irt.pars(act.nrm$yr97, cat=rep(5,60), poly.mod=pm) pars2 <- sep.pars(act.nrm$yr98, cat=rep(5,60), poly.mod=pm) out <- plink(list(pars1, pars2), matrix(1:60,60,2)) out1 <- plink(list(pars1, pars2), matrix(1:60,60,2), symmetric=TRUE) summary(out, descrip=TRUE) summary(out1, descrip=TRUE) ## End(Not run) # Compute linking constants for two groups with multiple-choice model # item parameters. Rescale theta values and item parameters using # the Haebara linking constants # Note: This example may take a minute or two to run ## Not run: theta <- rnorm(100) # In practice, estimated theta values would be used pm <- as.poly.mod(60, "mcm", 1:60) x <- as.irt.pars(act.mcm, common=matrix(1:60,60,2), cat=list(rep(6,60), rep(6,60)), poly.mod=list(pm,pm)) out <- plink(x, ability=list(theta,theta), rescale="HB") pars.out <- link.pars(out) ability.out <- link.ability(out) summary(out, descrip=TRUE) ## End(Not run) # Compute linking constants for two groups using mixed-format items and # a mixed placement of common items. Compare calibrations with the # inclusion or exclusion of NRM items. This example uses the dgn dataset. pm1 <- as.poly.mod(55,c("drm","gpcm","nrm"),dgn$items$group1) pm2 <- as.poly.mod(55,c("drm","gpcm","nrm"),dgn$items$group2) x <- as.irt.pars(dgn$pars,dgn$common,dgn$cat,list(pm1,pm2)) # Run with the NRM common items included out <- plink(x) # Run with the NRM common items excluded out1 <- plink(x,exclude="nrm") summary(out) summary(out1) # Compute linking constants for six groups using mixed-format items and # a mixed placement of common items. This example uses the reading dataset. # See the information on the dataset for an interpretation of the output. pm1 <- as.poly.mod(41, c("drm", "gpcm"), reading$items[[1]]) pm2 <- as.poly.mod(70, c("drm", "gpcm"), reading$items[[2]]) pm3 <- as.poly.mod(70, c("drm", "gpcm"), reading$items[[3]]) pm4 <- as.poly.mod(70, c("drm", "gpcm"), reading$items[[4]]) pm5 <- as.poly.mod(72, c("drm", "gpcm"), reading$items[[5]]) pm6 <- as.poly.mod(71, c("drm", "gpcm"), reading$items[[6]]) pm <- list(pm1, pm2, pm3, pm4, pm5, pm6) x <- as.irt.pars(reading$pars, reading$common, reading$cat, pm, base.grp=4) out <- plink(x) summary(out) ###### Multidimensional Examples ###### # Reckase Chapter 9 pm <- as.poly.mod(80, "drm", list(1:80)) x <- as.irt.pars(reckase9$pars, reckase9$common, list(rep(2,80),rep(2,80)), list(pm,pm), dimensions=2) # Compute constants using the least squares method and # the orthongal rotation with variable dilation. # Rescale the item parameters using the LS method out <- plink(x, dilation="orth.vd", rescale="LS") summary(out, descrip=TRUE) # Extract the rescaled item parameters pars.out <- link.pars(out) # Compute constants using an oblique Procrustes method # Display output and descriptives out <- plink(x, dilation="oblique") summary(out, descrip=TRUE) # Compute constants using and orthogonal rotation with # a fixed dilation parameter # Rescale the item parameters and ability estimates # using the "HB" and "LS" methods. # Display the optimization iterations ability <- matrix(rnorm(40),20,2) out <- plink(x, method=c("HB","LS"), dilation="orth.fd", rescale="HB", ability=list(ability,ability), control=list(trace=1,rel.tol=0.001)) summary(out) # Extract rescaled ability estimates ability.out <- out$ability # Compute linking constants for two groups using mixed-format items # modeled with the M3PL and MGPCM. Only compute constants using the # orth.vd dilation. pm <- as.poly.mod(60,c("drm","gpcm"), list(c(1:60)[md.mixed$cat==2], c(1:60)[md.mixed$cat>2])) x <- as.irt.pars(md.mixed$pars, matrix(1:60,60,2), list(md.mixed$cat, md.mixed$cat), list(pm, pm), dimensions=4, grp.names=c("Form.X","Form.Y")) out <- plink(x,dilation="orth.vd") summary(out, descrip=TRUE) # Illustration of construct shift and how to specify common dimensions pm <- as.poly.mod(80, "drm", list(1:80)) pars <- cbind(round(runif(80),2),reckase9$pars[[1]]) x <- as.irt.pars(list(pars,reckase9$pars[[2]]), reckase9$common, list(rep(2,80),rep(2,80)), list(pm,pm), dimensions=c(3,2)) dim.order <- matrix(c(1,2,3,NA,1,2),2,3,byrow=TRUE) out <- plink(x, dilation="oblique", dim.order=dim.order, rescale="LS") summary(out) pars.out <- link.pars(out)
This function plots item response curves/surfaces using the lattice
package in addition to vector plots, contour plots, and level plots for
multidimensional items, and comparison plots for examining parameter drift.
## S3 method for signature 'irt.pars' ## S3 method for class 'irt.pars' plot(x, y, ..., type, separate, combine, items, item.names, item.nums, panels, drift, groups, grp.names, sep.mod, drift.sd, save.hist) ## S3 method for signature 'irt.prob' ## S3 method for class 'irt.prob' plot(x, y, ..., type, separate, combine, items, item.names, item.nums, panels, save.hist) ## S3 method for signature 'sep.pars' ## S3 method for class 'sep.pars' plot(x, y, ..., type, separate, combine, items, item.names, item.nums, panels, save.hist) ## S3 method for signature 'list' ## S3 method for class 'irt.pars' plot(x, y, ..., type, separate, combine, items, item.names, item.nums, panels, drift, groups, grp.names, sep.mod, drift.sd, save.hist)## S3 method for signature 'irt.pars' ## S3 method for class 'irt.pars' plot(x, y, ..., type, separate, combine, items, item.names, item.nums, panels, drift, groups, grp.names, sep.mod, drift.sd, save.hist) ## S3 method for signature 'irt.prob' ## S3 method for class 'irt.prob' plot(x, y, ..., type, separate, combine, items, item.names, item.nums, panels, save.hist) ## S3 method for signature 'sep.pars' ## S3 method for class 'sep.pars' plot(x, y, ..., type, separate, combine, items, item.names, item.nums, panels, save.hist) ## S3 method for signature 'list' ## S3 method for class 'irt.pars' plot(x, y, ..., type, separate, combine, items, item.names, item.nums, panels, drift, groups, grp.names, sep.mod, drift.sd, save.hist)
x |
object of class |
y |
this is an argument in the generic plot function, but is not used for these methods |
... |
further arguments passed to or from other methods |
type |
type of plot to produce (for multidimensional models). Values can include
"wireframe", "contourplot", "levelplot", "vectorplot1", "vectorplot2", "vectorplot3".
The default if |
separate |
logical value. If |
combine |
vector identifying the number of response categories to plot in each
panel. If |
items |
numeric vector identifying the items to plot |
item.names |
vector of item names for use in labeling the panels |
item.nums |
logical value. If |
panels |
number of panels to display in the output window. If the number of items
is greater than |
drift |
character vector identifying the plots to create to examine item parameter drift. Acceptable values are "a", "b", "c" for the various parameters respectively, "pars" to compare all of these parameters, "TCC" to compare test characteristic curves, "ICC" to compare item characteristic curves, or "all" to produce all of these plots. |
groups |
numeric vector identifying the groups in |
grp.names |
character vector of group names to use when creating the drift plots |
sep.mod |
logical value. If |
drift.sd |
numeric value identifying the number of standard deviations to use when
creating the perpendicular confidence region for the drift comparison plots. If
missing, |
save.hist |
logical value. If |
All of the plots, with the exception of the vector plots, are based on the lattice package.
For the unidimensional plots, any arguments associated with the xyplot function can be
included. For the multidimensional plots, any arguments associated with the wireframe,
contourplot, or levelplot functions (corresponding to the argument specified in
type) can be passed as well.
For the multidimensional plots, vector plots are limited to two dimensions. The other three
types of plots can be used with multiple dimensions by plotting the results for the first
two dimensions conditional on each combination of theta values for the other dimensions. The
function can handle up to ten dimensions; however, the usefulness of the plots will likely
diminish if more than four dimensions are modeled.
There are three types of vector plots. The first two types create a set of vectors that
originate at the location of the multidimensional difficulty and characterize the length
of the vectors at the appropriate angles, relative to the axes, based on the multidimensional
discrimination. Setting type equal to "vectorplot2" will also include the reference
composite based on an eigenvalue decomposition of the slope parameters. The third type of
vector plot is similar to the first with the key distinction that the lines are drawn from
the origin to the points of multidimensional difficulty. When there are more than two dimensions,
a panel of vectorplots is created for each pair of dimensions.
When multiple pages are created, the PgUp and PgDn buttons can be used to view the plots on different pages on a Windows machine. For other operating systems, it is suggested that the multipage plot be sent to an external graphics file.
Jonathan P. Weeks [email protected], Adam Wyse (Contributing Author)
Weeks, J. P. (2010) plink: An R package for linking mixed-format tests using IRT-based methods. Journal of Statistical Software, 35(12), 1–33. URL http://www.jstatsoft.org/v35/i12/
###### Unidimensional Examples ###### # Compute probabilites for three dichotomous (3PL) items and two polytomous # (gpcm) items then plot the item characteristic/category curves dichot <- matrix(c(1.2, .8, .9, 2.3, -1.1, -.2, .24, .19, .13),3,3) poly <- matrix(c(.64, -1.8, -.73, .45, NA, .88, .06, 1.4, 1.9, 2.6), 2,5,byrow=TRUE) pars <- rbind(cbind(dichot,matrix(NA,3,2)),poly) cat <- c(2,2,2,4,5) pm <- as.poly.mod(5, c("drm","gpcm"), list(1:3,4:5)) x <- mixed(pars, cat, pm) plot(x) # Compute probabilites for six items using the 3PL and plot all of the item # characteristic curves on a single panel using the combine argument a <- c(.71,.96,.36,1.05,1.76,.64) b <- c(-.16,1.18,2.03,1.09,.82,-1.56) c <- c(.22,.17,.24,.08,.20,.13) theta <- seq(-3,3,.2) pars <- cbind(a,b,c) x <- drm(pars,theta) plot(x,combine=6,item.names="Items 1-6",auto.key=list(space="right")) # Compute probabilties for the nominal response model using the ACT # mathematics data. Plot the item category curves for a subset of 9 items. x <- nrm(act.nrm[[1]], rep(5,60)) plot(x, items=c(2,7,12,20,35,41,48,57,60),auto.key=list(space="right")) # Create irt.pars object with two groups (all dichotomous items) # rescale the item parameters using the Stocking-Lord linking constants # Create drift plots pm <- as.poly.mod(36) x <- as.irt.pars(KB04$pars, KB04$common, cat=list(rep(2,36),rep(2,36)), poly.mod=list(pm,pm)) out <- plink(x, rescale="SL", base.grp=2, D=1.7) plot(out$pars, drift="all", grp.names=c("Form X","Form Y"), item.nums=TRUE) ## Not run: # Compute linking constants for six groups using mixed-format items # Create drift plots with distinct markers for the two response models pm1 <- as.poly.mod(41, c("drm", "gpcm"), reading$items[[1]]) pm2 <- as.poly.mod(70, c("drm", "gpcm"), reading$items[[2]]) pm3 <- as.poly.mod(70, c("drm", "gpcm"), reading$items[[3]]) pm4 <- as.poly.mod(70, c("drm", "gpcm"), reading$items[[4]]) pm5 <- as.poly.mod(72, c("drm", "gpcm"), reading$items[[5]]) pm6 <- as.poly.mod(71, c("drm", "gpcm"), reading$items[[6]]) pm <- list(pm1, pm2, pm3, pm4, pm5, pm6) x <- as.irt.pars(reading$pars, reading$common, reading$cat, pm, base.grp=4) out <- plink(x, rescale="HB") plot(out$pars, drift=c("a","b"), sep.mod=TRUE) ## End(Not run) ###### Multidimensional Examples ###### # Compute response probabilities for 20 items from the # reckase9 data. Probabilites are modeled using the M2PL. # Each of the six plot types will be created. x <- drm(reckase9[[1]][[1]][31:50,],dimensions=2) # Wireframe plot plot(x, drape=TRUE, item.names=paste("Item",31:50)) # Contour Plot plot(x, type="contourplot",labels=FALSE,cuts=20) # Level Plot plot(x, type="levelplot",cuts=20) # Use all the items for the vector plots x <- drm(reckase9[[1]][[1]],dimensions=2) # Vector Plot 1 plot(x, type="vectorplot1",item.nums=FALSE) # Vector Plot 2 plot(x, type="vectorplot2",item.nums=FALSE) # Vector Plot 3 plot(x, type="vectorplot3",xlim=c(-1.5,1.5),ylim=c(-1.5,1.5)) # Compute response probabilities for a single three-category item using # the multidimensional generalized partial credit model for three # dimensions. Plot the conditional item category surfaces in a single # panel and then the separated item category surfaces in separate panels pars <- matrix(c(1.1999,0.5997,0.8087,2.1730,-1.4576),1,5) x <- gpcm(pars,3,dimensions=3,theta=-4:4) # plot combined item category surfaces. Rotate the plots using the # screen argument plot(x, screen=list(z=-30,x=-60)) ## Not run: # plot separated item category surfaces x <- gpcm(pars,3,dimensions=3) plot(x,separate=TRUE,drape=TRUE,panels=1) ## End(Not run)###### Unidimensional Examples ###### # Compute probabilites for three dichotomous (3PL) items and two polytomous # (gpcm) items then plot the item characteristic/category curves dichot <- matrix(c(1.2, .8, .9, 2.3, -1.1, -.2, .24, .19, .13),3,3) poly <- matrix(c(.64, -1.8, -.73, .45, NA, .88, .06, 1.4, 1.9, 2.6), 2,5,byrow=TRUE) pars <- rbind(cbind(dichot,matrix(NA,3,2)),poly) cat <- c(2,2,2,4,5) pm <- as.poly.mod(5, c("drm","gpcm"), list(1:3,4:5)) x <- mixed(pars, cat, pm) plot(x) # Compute probabilites for six items using the 3PL and plot all of the item # characteristic curves on a single panel using the combine argument a <- c(.71,.96,.36,1.05,1.76,.64) b <- c(-.16,1.18,2.03,1.09,.82,-1.56) c <- c(.22,.17,.24,.08,.20,.13) theta <- seq(-3,3,.2) pars <- cbind(a,b,c) x <- drm(pars,theta) plot(x,combine=6,item.names="Items 1-6",auto.key=list(space="right")) # Compute probabilties for the nominal response model using the ACT # mathematics data. Plot the item category curves for a subset of 9 items. x <- nrm(act.nrm[[1]], rep(5,60)) plot(x, items=c(2,7,12,20,35,41,48,57,60),auto.key=list(space="right")) # Create irt.pars object with two groups (all dichotomous items) # rescale the item parameters using the Stocking-Lord linking constants # Create drift plots pm <- as.poly.mod(36) x <- as.irt.pars(KB04$pars, KB04$common, cat=list(rep(2,36),rep(2,36)), poly.mod=list(pm,pm)) out <- plink(x, rescale="SL", base.grp=2, D=1.7) plot(out$pars, drift="all", grp.names=c("Form X","Form Y"), item.nums=TRUE) ## Not run: # Compute linking constants for six groups using mixed-format items # Create drift plots with distinct markers for the two response models pm1 <- as.poly.mod(41, c("drm", "gpcm"), reading$items[[1]]) pm2 <- as.poly.mod(70, c("drm", "gpcm"), reading$items[[2]]) pm3 <- as.poly.mod(70, c("drm", "gpcm"), reading$items[[3]]) pm4 <- as.poly.mod(70, c("drm", "gpcm"), reading$items[[4]]) pm5 <- as.poly.mod(72, c("drm", "gpcm"), reading$items[[5]]) pm6 <- as.poly.mod(71, c("drm", "gpcm"), reading$items[[6]]) pm <- list(pm1, pm2, pm3, pm4, pm5, pm6) x <- as.irt.pars(reading$pars, reading$common, reading$cat, pm, base.grp=4) out <- plink(x, rescale="HB") plot(out$pars, drift=c("a","b"), sep.mod=TRUE) ## End(Not run) ###### Multidimensional Examples ###### # Compute response probabilities for 20 items from the # reckase9 data. Probabilites are modeled using the M2PL. # Each of the six plot types will be created. x <- drm(reckase9[[1]][[1]][31:50,],dimensions=2) # Wireframe plot plot(x, drape=TRUE, item.names=paste("Item",31:50)) # Contour Plot plot(x, type="contourplot",labels=FALSE,cuts=20) # Level Plot plot(x, type="levelplot",cuts=20) # Use all the items for the vector plots x <- drm(reckase9[[1]][[1]],dimensions=2) # Vector Plot 1 plot(x, type="vectorplot1",item.nums=FALSE) # Vector Plot 2 plot(x, type="vectorplot2",item.nums=FALSE) # Vector Plot 3 plot(x, type="vectorplot3",xlim=c(-1.5,1.5),ylim=c(-1.5,1.5)) # Compute response probabilities for a single three-category item using # the multidimensional generalized partial credit model for three # dimensions. Plot the conditional item category surfaces in a single # panel and then the separated item category surfaces in separate panels pars <- matrix(c(1.1999,0.5997,0.8087,2.1730,-1.4576),1,5) x <- gpcm(pars,3,dimensions=3,theta=-4:4) # plot combined item category surfaces. Rotate the plots using the # screen argument plot(x, screen=list(z=-30,x=-60)) ## Not run: # plot separated item category surfaces x <- gpcm(pars,3,dimensions=3) plot(x,separate=TRUE,drape=TRUE,panels=1) ## End(Not run)
The formal S4 class for poly.mod. This class characterizes the models and associated items for a set of item parameters.
The IRT models associated with the codes:
drm:dichotomous response models (includes 1PL, 2PL, and 3PL)
gpcm:generalized partial credit model (includes the partial credit model)
grm:graded response model
mcm:multiple-choice model
nrm:nominal response model
Objects can be created by calls of the form new("poly.mod", ...), but this is
not encouraged. Use the function as.poly.mod instead.
model:character vector identifying all the models associated with the
corresponding set of item parameters. The only acceptable models are
drm, gpcm, grm, mcm, and nrm. See below for more
details.
items:list with the same length as model, where each element
identifies the items(rows) in the corresponding set of item parameters associated
with the model(s) identified in model.
The names of the list elements for items must correspond to the elements in model.
For example, if the poly.mod object is pm and the first element in pm@model is
drm,one should be able to reference the associated items by pm@items$drm. If
the list elements are unnamed, some functions may not work properly.
Jonathan P. Weeks [email protected]
as.poly.mod, irt.pars, sep.pars,
irt.prob
This function imports item and/or ability parameters from BILOG-MG 3, PARSCALE 4, MULTILOG 7, TESTFACT 4, ICL, BMIRT, and ltm.
read.bilog(file, ability = FALSE, pars.only = TRUE, as.irt.pars = TRUE) read.parscale(file, ability = FALSE, loc.out = FALSE, pars.only = TRUE, as.irt.pars = TRUE) read.multilog(file, cat, poly.mod, ability = FALSE, contrast = "dev", drm.3PL = TRUE, loc.out = FALSE, as.irt.pars = TRUE) read.testfact(file, ability = FALSE, guessing = FALSE, bifactor = FALSE, as.irt.pars = TRUE) read.icl(file, poly.mod, ability = FALSE, loc.out = FALSE, as.irt.pars = TRUE) read.bmirt(file, ability = FALSE, sign.adjust = TRUE, loc.out = FALSE, pars.only = TRUE, as.irt.pars = TRUE) read.erm(x, loc.out = FALSE, as.irt.pars = TRUE) read.ltm(x, loc.out = FALSE, as.irt.pars = TRUE)read.bilog(file, ability = FALSE, pars.only = TRUE, as.irt.pars = TRUE) read.parscale(file, ability = FALSE, loc.out = FALSE, pars.only = TRUE, as.irt.pars = TRUE) read.multilog(file, cat, poly.mod, ability = FALSE, contrast = "dev", drm.3PL = TRUE, loc.out = FALSE, as.irt.pars = TRUE) read.testfact(file, ability = FALSE, guessing = FALSE, bifactor = FALSE, as.irt.pars = TRUE) read.icl(file, poly.mod, ability = FALSE, loc.out = FALSE, as.irt.pars = TRUE) read.bmirt(file, ability = FALSE, sign.adjust = TRUE, loc.out = FALSE, pars.only = TRUE, as.irt.pars = TRUE) read.erm(x, loc.out = FALSE, as.irt.pars = TRUE) read.ltm(x, loc.out = FALSE, as.irt.pars = TRUE)
file |
filename of file containing the item or ability parameters |
ability |
if |
pars.only |
if |
loc.out |
if |
as.irt.pars |
if |
cat |
vector with the number of response categories for each item. For
multiple-choice model items, |
poly.mod |
a |
contrast |
an object identifying the type of contrast(s) used to estimate the various parameters for each item. See below for more details. |
drm.3PL |
logical value indicating whether the dichotomous items (if applicable) were modeled using the three parameter logistic model (3PL) |
guessing |
logical value indicating whether a guessing parameter was modeled |
bifactor |
logical value indicating whether the bifactor model was used to estimate the item/ability parameters |
sign.adjust |
logical value indicating whether the difficulty/step parameters should be multiplied by -1 to make them consistent with common formulations of multidimensional response models |
x |
output object from one of the following functions in the |
The file extensions for the item parameter and ability files respectively are as follows:
.par and .sco for BILOG-MG, PARSCALE, and MULTILOG, .par and .fsc for TESTFACT, and
.par and .ss for BMIRT. For ICL, the file extensions are specified by the user,
and for ltm, the name of the output object is specified by the user.
When item parameters are estimated in MULTILOG for models other than the 1PL, 2PL, and GRM,
the program estimates (and returns) contrast parameters. MULTILOG implements three types
of contrasts: deviation, polynomial, and triangle (see Thissen & Steinberg, 1986 for more
information). A single type of contrast can be used for all parameters (a, b, and c) for all
items or different contrasts can be specified for individual parameters and individual items.
If a single type of contrast is used for all parameters for all items, a character value
can be specified for the contrast argument: "dev", "poly", or "tri" for the three
types of contrasts respectively. When different contrasts are used, contrast should
be a list of length nine. The list elements should be ordered as follows "dev.a","poly.a",
"tri.a","dev.c","poly.c","tri.c","dev.d", "poly.d","tri.d" where the first three elements
correspond to the various contrasts for the slope parameters, the next three elements
correspond to the contrasts for the category parameters, and the last three elements
correspond to the contrasts for the lower asymptote (guessing parameters). There are two
approaches that can be implemented using this list 1) character vectors with the model
names "drm", "grm", "gpcm", "nrm", and "mcm" indicating that the given parameters for
all items associated with the given model should be transformed using the specified contrast.
In instances where a model is not included for a given parameter (for any of the contrasts)
the parameters will be transformed using deviation contrasts. 2) numeric vectors
identifying the contrasts used for given parameters for given items can be specified.
It is only necessary to include item numbers for the various parameter/contrast combinations
when deviation contrasts are not used. See below for examples of how to formulate this argument.
Returns a data.frame or an object of class irt.pars if
as.irt.pars = TRUE.
These functions are currently unable to handle output generated when subtests are used.
Jonathan P. Weeks [email protected]
Hanson, B. A. (2002). IRT command language [Computer Program]. URL http://www.b-a-h.com/software/irt/icl/
Mair, P & Hatzinger, R. (2007). Extended Rasch modeling: The eRm package for the application of IRT models in R. URL http://www.jstatsoft.org/v20/i09
Muraki, E. & Bock, R. D. (2003). PARSCALE 4: IRT item analysis and test scoring for rating scale data [Computer Program]. Chicago, IL: Scientific Software International. URL http://www.ssicentral.com
Rizopoulos, D. (2006). ltm: An R package for latent variable modeling and item response theory analyses. Journal of Statistical Software, 17(5), 1-25. URL http://www.jstatsoft.org/v17/i05/
Thissen, D. (2003). MULTILOG 7: Multiple, categorical item analysis and test scoring using item response theory [Computer Program]. Chicago, IL: Scientific Software International. URL http://www.ssicentral.com
Thissen, D. & Steinberg, L. (1986). A taxonomy of item response models. Psychometrika, 51(4), 567-577.
Weeks, J. P. (2010) plink: An R package for linking mixed-format tests using IRT-based methods. Journal of Statistical Software, 35(12), 1–33. URL http://www.jstatsoft.org/v35/i12/
Wood, R., Wilson, D. T., Muraki, E., Schilling, S. G., Gibbons, R., & Bock, R. D. (2003). TESTFACT 4: Test scoring, item statistics, and item factor analysis [Computer Program]. Chicago, IL: Scientific Software International. URL http://www.ssicentral.com
Yao, L. (2008). BMIRT: Bayesian multivariate item response theory [Computer Program]. Monterey, CA: CTB/McGraw-Hill.
Zimowski, M. F., Muraki, E., Mislevy, R. J., & Bock, R. D. (2003). BILOG-MG 3: Multiple-group IRT analysis and test maintenance for binary items [Computer Program]. Chicago, IL: Scientific Software International. URL http://www.ssicentral.com
# Illustration of how to formulate the contrast argument. Say that we # have 20 items where the first 15 are modeled using the 3PL and the # last five are modeled using the GPCM. For the 3PL items, deviation # contrasts are commonly used for all of the parameters, whereas with # the GPCM items, polynomial contrasts are typically used for the slope # parameters and triangle contrasts are used for the category parameters. # The contrast argument could be specified as follows contrast <- vector("list",9) # Note: the list elements do not need to be named for read.multilog names(contrast) <- c("dev.a","poly.a","tri.a","dev.c","poly.c","tri.c", "dev.d", "poly.d","tri.d") contrast$poly.a <- 16:20 contrast$tri.c <- 16:20 # The object could alternatively be formatted as follows contrast <- vector("list",9) names(contrast) <- c("dev.a","poly.a","tri.a","dev.c","poly.c","tri.c", "dev.d","poly.d","tri.d") contrast$dev.a <- 1:15 contrast$poly.a <- 16:20 contrast$dev.c <- 1:15 contrast$tri.c <- 16:20 contrast$dev.d <- 1:15# Illustration of how to formulate the contrast argument. Say that we # have 20 items where the first 15 are modeled using the 3PL and the # last five are modeled using the GPCM. For the 3PL items, deviation # contrasts are commonly used for all of the parameters, whereas with # the GPCM items, polynomial contrasts are typically used for the slope # parameters and triangle contrasts are used for the category parameters. # The contrast argument could be specified as follows contrast <- vector("list",9) # Note: the list elements do not need to be named for read.multilog names(contrast) <- c("dev.a","poly.a","tri.a","dev.c","poly.c","tri.c", "dev.d", "poly.d","tri.d") contrast$poly.a <- 16:20 contrast$tri.c <- 16:20 # The object could alternatively be formatted as follows contrast <- vector("list",9) names(contrast) <- c("dev.a","poly.a","tri.a","dev.c","poly.c","tri.c", "dev.d","poly.d","tri.d") contrast$dev.a <- 1:15 contrast$poly.a <- 16:20 contrast$dev.c <- 1:15 contrast$tri.c <- 16:20 contrast$dev.d <- 1:15
This (unidimensional) dataset includes item parameters from a large-scale reading assessment. The parameters were estimated using a combination of the three parameter logistic model (3PL) and the generalized partial credit model (GPCM). There are six sets of parameters which are based on tests administered in four grades over three years. In particular, the data include a grade 3 and grade 4 test in year 0, a grade 4 and grade 5 test in year 1, and a grade 5 and grade 6 test in year 2. The label for the information related to the grade 3 test is grade3.0 where the value after the decimal indicates the year. Similar labels are used for the other grade/year combinations. This dataset is used for illustrative purposes to show in a multi-group scenario how items from different response models can be mixed together and common items can be in different positions across groups.
readingreading
A list of length four. The first element is a list of length six with item parameter estimates for each grade/year. There is no location parameter for the GPCM items. The second list element is a list identifying the number of response categories for the six grade/year combinations. The third element specifies which items correspond to the different item response models for each grade/year respectively. The last element is a list of common item matrices for each adjacent grade/year group.
Briggs, D. C. & Weeks, J. P. (In Press) The Impact of Vertical Scaling Decisions on Growth Interpretations. Educational Measurement: Issues and Practices.
This dataset includes multidimensional two parameter logistic model (M2PL) item parameter estimates for two dimensions and two groups. The Form 1 items are from a "new" form and the Form 2 items are from an "old" form. There are 16 common items between the two forms.
reckase9reckase9
A list of length two. The first element pars is a list of length two.
Each of the list elements in pars is a 80 x 3 matrix of item parameters. Element
one is for Form 1, and element two is for Form 2. The first two columns
in each matrix are for the slope parameters and the third column is for the parameter
related to item difficulty.
The second list element in reckase9 is a matrix identifying the common items
between the two sets of parameters in pars. That is, the first column in
common identifies the rows in pars$form1 that are common items.
The second column identifies the corresponding set of common items in pars$form2.
Reckase, M. D. (2009). Multidimensional item response theory. New York: Springer.
The formal S4 class for sep.pars. This class stores a set of separated item parameters and characteristics of these parameters.
a will be an n x m matrix for n items and m dimensions if there are no nominal response model
or multiple-choice model items. Otherwise, if nrm or mcm items are included, a will be an
n x (k x m) matrix with k equal to the maximum number of response categories across items. If nrm or
mcm items are included, the discrimination/slope parameters for the dichotomous response models, the
graded response model, partial credit model and generalized partial credit model (and the multidimensional
extensions of these models) are listed in the first m columns with all other columns filled with NAs.
b is an n x (k x m) matrix of difficulty, threshold, step, or category parameters (depending on
the corresponding model) for n items and m dimensions with k equal to the maximum number of b
parameters across all items. For items with b less than k x m, the row is right-filled with NAs.
c will be an n x 1 matrix for n items if there are no multiple-choice model items. Otherwise,
if mcm items are included, c will be an n x k matrix with k equal to the maximum number of
response categories across items minus one. If mcm items are included, the lower asymptote parameters
for the dichotomous response models, are listed in the first column with all other columns filled with
NAs. The c values for the 1PL, 2PL, M1PL, and M2PL equal zero.
Objects can be created by calls of the form new("sep.pars", ...), but this is
not encouraged. Use the function sep.pars instead.
a:matrix of discrimination/slope parameters
b:matrix of difficulty, threshold, step, and category parameters (depending on the associated IRT model)
c:matrix of lower asymptote parameters and category proportions for the
dichotomous response models and multiple-choice model respectively. c is equal to
NA for each item for all other models.
cat:vector identifying the number of categories associated with each item.
All dichotomous items will have cat values equal to 2. Graded response model and
partial credit/generalized partial credit model items will have cat values equal
to the number of step/threshold parameters plus one. Nominal response model items will
have cat values equal to the number of categories, and multiple-choice model items
will have cat values equal to the number of categories plus one (the 'do not know'
category).
n:vector identifying the total number of items, the total number of dichotomous and polytomous items, and the number of items associated with each polytomous model.
mod.lab:character vector of labels for the model(s).
location:logical value. If TRUE, the step and/or threshold parameters in
slot b for the graded response model and generalized partial credit model include a
location parameter.
dimensions:numeric value identifying the number of modeled dimensions.
model:character vector identifying all the models associated with the
corresponding set of item parameters. The only acceptable models are
drm, gpcm, grm, mcm, and nrm (see class poly.mod
for more information).
items:list with the same length as model, where each element identifies
the items(rows) in the corresponding set of item parameters associated with the model(s)
identified in model.
Class poly.mod, directly.
Class list.poly, by class poly.mod, distance 2.
Jonathan P. Weeks [email protected]
This function splits the item parameters in the specified object into discrimination/slope parameters, difficulty/step/threshold/category parameters, and lower asymptote/category probability parameters.
sep.pars(x, cat, poly.mod, dimensions = 1, location = FALSE, loc.out = FALSE, ...) ## S4 method for signature 'numeric' sep.pars(x, cat, poly.mod, dimensions, location, loc.out, ...) ## S4 method for signature 'matrix' sep.pars(x, cat, poly.mod, dimensions, location, loc.out, ...) ## S4 method for signature 'data.frame' sep.pars(x, cat, poly.mod, dimensions, location, loc.out, ...) ## S4 method for signature 'irt.pars' sep.pars(x, cat, poly.mod, dimensions, location, loc.out, ...) ## S4 method for signature 'list' sep.pars(x, cat, poly.mod, dimensions, location, loc.out, ...)sep.pars(x, cat, poly.mod, dimensions = 1, location = FALSE, loc.out = FALSE, ...) ## S4 method for signature 'numeric' sep.pars(x, cat, poly.mod, dimensions, location, loc.out, ...) ## S4 method for signature 'matrix' sep.pars(x, cat, poly.mod, dimensions, location, loc.out, ...) ## S4 method for signature 'data.frame' sep.pars(x, cat, poly.mod, dimensions, location, loc.out, ...) ## S4 method for signature 'irt.pars' sep.pars(x, cat, poly.mod, dimensions, location, loc.out, ...) ## S4 method for signature 'list' sep.pars(x, cat, poly.mod, dimensions, location, loc.out, ...)
x |
Object containing item parameters. For details on the formatting of parameters
for specific item response models see the corresponding methods (i.e.,
|
cat |
vector identifying the number of response categories for each item. If
multiple-choice model items are included, |
poly.mod |
object of class |
dimensions |
number of modeled dimensions |
location |
if |
loc.out |
if |
... |
further arguments passed to or from other methods |
Returns an object of class sep.pars
Jonathan P. Weeks [email protected]
###### Unidimensional Examples ###### # Create object for three dichotomous (1PL) items with difficulties -1, 0, 1 x <- sep.pars(c(-1,0,1)) # Create object for three dichotomous (3PL) items and two polytomous # (gpcm) items without a location parameter (the parameters are # formatted as a matrix) dichot <- matrix(c(1.2, .8, .9, 2.3, -1.1, -.2, .24, .19, .13),3,3) poly <- matrix(c(.64, -1.8, -.73, .45, NA, .88, .06, 1.4, 1.9, 2.6), 2,5,byrow=TRUE) pars <- rbind(cbind(dichot,matrix(NA,3,2)),poly) cat <- c(2,2,2,4,5) pm <- as.poly.mod(5, c("drm","gpcm"), list(1:3,4:5)) x <- sep.pars(pars, cat, pm) summary(x) # Create object for three dichotomous (3PL) items and two polytomous # (gpcm) items without a location parameter (the parameters are # included in a list) a <- c(1.2, .8, .9, .64, .88) b <- matrix(c( 2.3, rep(NA,3), -1.1, rep(NA,3), -.2, rep(NA,3), -1.8, -.73, .45, NA, .06, 1.4, 1.9, 2.6),5,4,byrow=TRUE) c <- c(1.4, 1.9, 2.6, NA, NA) pars <- list(a,b,c) cat <- c(2,2,2,4,5) pm <- as.poly.mod(5, c("drm","gpcm"), list(1:3,4:5)) x <- sep.pars(pars, cat, pm) summary(x) # Create object for three dichotomous (3PL) items, four polytomous # items, two gpcm items and two nrm items. Include a location parameter # for the gpcm items. Maintain the location parameter in the output. a <- matrix(c( 1.2, rep(NA,4), .8, rep(NA,4), .9, rep(NA,4), .64, rep(NA,4), .88, rep(NA,4), .905, .522, -.469, -.959, NA, .828, .375, -.357, -.079, -.817),7,5,byrow=TRUE) b <- matrix(c( 2.3, rep(NA,4), -1.1, rep(NA,4), -.2, rep(NA,4), -.69, -1.11, -.04, 1.14, NA, 1.49, -1.43, -.09, .41, 1.11, .126, -.206, -.257, .336, NA, .565, .865, -1.186, -1.199, .993),7,5,byrow=TRUE) c <- c(.14, .19, .26, rep(NA,4)) pars <- list(a,b,c) cat <- c(2,2,2,4,5,4,5) pm <- as.poly.mod(7, c("drm","gpcm","nrm"), list(1:3,4:5,6:7)) x <- sep.pars(pars, cat, pm, location=TRUE, loc.out=TRUE) summary(x, descrip=TRUE) # Create irt.pars object with two groups then run sep.pars pm <- as.poly.mod(36) x <- as.irt.pars(KB04$pars, KB04$common, cat=list(rep(2,36),rep(2,36)), list(pm,pm), grp.names=c("form.x","form.y")) out <- sep.pars(x) summary(out, descrip=TRUE) ###### Multidimensional Examples ###### # Create object for three dichotomous (M1PL) items for two dimensions # with parameters related to item difficulties of -1, 0, 1 x <- sep.pars(c(-1,0,1), dimensions=2) # Create object for three dichotomous (M3PL) items and two polytomous # (MGPCM) items without a location parameter for four dimensions # (the parameters are included in a list) a <- matrix(c(0.5038, 2.1910, 1.1317, 0.2493, 2.9831, 0.4811, 0.3566, 0.4306, 0.2397, 0.2663, 1.5588, 0.5295, 0.2020, 0.2410, 1.2061, 0.5552, 0.2054, 0.6302, 0.3152, 0.2037),5,4,byrow=TRUE) b <- matrix(c(0.5240, rep(NA,3), -1.8841, rep(NA,3), 0.2570, rep(NA,3), -1.4207, 0.3041, -0.5450, NA, -2.1720, 0.0954, 0.6531, 0.9114),5,4,byrow=TRUE) c <- c(0.1022, 0.3528, 0.2498, NA, NA) pars <- list(a,b,c) cat <- c(2,2,2,4,5) pm <- as.poly.mod(5, c("drm","gpcm"), list(1:3,4:5)) x <- sep.pars(pars, cat, pm, dimensions=4) summary(x, descrip=TRUE)###### Unidimensional Examples ###### # Create object for three dichotomous (1PL) items with difficulties -1, 0, 1 x <- sep.pars(c(-1,0,1)) # Create object for three dichotomous (3PL) items and two polytomous # (gpcm) items without a location parameter (the parameters are # formatted as a matrix) dichot <- matrix(c(1.2, .8, .9, 2.3, -1.1, -.2, .24, .19, .13),3,3) poly <- matrix(c(.64, -1.8, -.73, .45, NA, .88, .06, 1.4, 1.9, 2.6), 2,5,byrow=TRUE) pars <- rbind(cbind(dichot,matrix(NA,3,2)),poly) cat <- c(2,2,2,4,5) pm <- as.poly.mod(5, c("drm","gpcm"), list(1:3,4:5)) x <- sep.pars(pars, cat, pm) summary(x) # Create object for three dichotomous (3PL) items and two polytomous # (gpcm) items without a location parameter (the parameters are # included in a list) a <- c(1.2, .8, .9, .64, .88) b <- matrix(c( 2.3, rep(NA,3), -1.1, rep(NA,3), -.2, rep(NA,3), -1.8, -.73, .45, NA, .06, 1.4, 1.9, 2.6),5,4,byrow=TRUE) c <- c(1.4, 1.9, 2.6, NA, NA) pars <- list(a,b,c) cat <- c(2,2,2,4,5) pm <- as.poly.mod(5, c("drm","gpcm"), list(1:3,4:5)) x <- sep.pars(pars, cat, pm) summary(x) # Create object for three dichotomous (3PL) items, four polytomous # items, two gpcm items and two nrm items. Include a location parameter # for the gpcm items. Maintain the location parameter in the output. a <- matrix(c( 1.2, rep(NA,4), .8, rep(NA,4), .9, rep(NA,4), .64, rep(NA,4), .88, rep(NA,4), .905, .522, -.469, -.959, NA, .828, .375, -.357, -.079, -.817),7,5,byrow=TRUE) b <- matrix(c( 2.3, rep(NA,4), -1.1, rep(NA,4), -.2, rep(NA,4), -.69, -1.11, -.04, 1.14, NA, 1.49, -1.43, -.09, .41, 1.11, .126, -.206, -.257, .336, NA, .565, .865, -1.186, -1.199, .993),7,5,byrow=TRUE) c <- c(.14, .19, .26, rep(NA,4)) pars <- list(a,b,c) cat <- c(2,2,2,4,5,4,5) pm <- as.poly.mod(7, c("drm","gpcm","nrm"), list(1:3,4:5,6:7)) x <- sep.pars(pars, cat, pm, location=TRUE, loc.out=TRUE) summary(x, descrip=TRUE) # Create irt.pars object with two groups then run sep.pars pm <- as.poly.mod(36) x <- as.irt.pars(KB04$pars, KB04$common, cat=list(rep(2,36),rep(2,36)), list(pm,pm), grp.names=c("form.x","form.y")) out <- sep.pars(x) summary(out, descrip=TRUE) ###### Multidimensional Examples ###### # Create object for three dichotomous (M1PL) items for two dimensions # with parameters related to item difficulties of -1, 0, 1 x <- sep.pars(c(-1,0,1), dimensions=2) # Create object for three dichotomous (M3PL) items and two polytomous # (MGPCM) items without a location parameter for four dimensions # (the parameters are included in a list) a <- matrix(c(0.5038, 2.1910, 1.1317, 0.2493, 2.9831, 0.4811, 0.3566, 0.4306, 0.2397, 0.2663, 1.5588, 0.5295, 0.2020, 0.2410, 1.2061, 0.5552, 0.2054, 0.6302, 0.3152, 0.2037),5,4,byrow=TRUE) b <- matrix(c(0.5240, rep(NA,3), -1.8841, rep(NA,3), 0.2570, rep(NA,3), -1.4207, 0.3041, -0.5450, NA, -2.1720, 0.0954, 0.6531, 0.9114),5,4,byrow=TRUE) c <- c(0.1022, 0.3528, 0.2498, NA, NA) pars <- list(a,b,c) cat <- c(2,2,2,4,5) pm <- as.poly.mod(5, c("drm","gpcm"), list(1:3,4:5)) x <- sep.pars(pars, cat, pm, dimensions=4) summary(x, descrip=TRUE)
This function summarizes the item parameters for a given object. If the object is
of class link, the function prints the linking constants
## S3 method for class 'irt.pars' summary(object, ..., descrip = FALSE) ## S3 method for class 'sep.pars' summary(object, ..., descrip = FALSE) ## S3 method for class 'list' summary(object, ..., descrip = FALSE) ## S3 method for class 'link' summary(object, ..., descrip = FALSE)## S3 method for class 'irt.pars' summary(object, ..., descrip = FALSE) ## S3 method for class 'sep.pars' summary(object, ..., descrip = FALSE) ## S3 method for class 'list' summary(object, ..., descrip = FALSE) ## S3 method for class 'link' summary(object, ..., descrip = FALSE)
object |
an |
... |
further arguments passed to or from other methods |
descrip |
if |
The method for objects of class irt.pars summarizes the parameters for a single
group or multiple groups. If multiple groups are included in the object, parameters will
be summarized for each group separately.
The method for objects of class sep.pars summarizes the parameters for a single
sep.pars object.
Objects of class list can include any combination of irt.pars or
sep.pars objects. The parameters will be summarized for each group separately.
The method for objects of class link prints the linking constants for n-1 groups.
If descrip = TRUE, the method summarizes the parameters for all groups
separately. The labels are specified in the following manner "group1/group2", meaning
the group1 parameters were transformed to the group2 test using the associated constants.
The base group is indicated by an asterisk.
Jonathan P. Weeks [email protected]
Weeks, J. P. (2010) plink: An R package for linking mixed-format tests using IRT-based methods. Journal of Statistical Software, 35(12), 1–33. URL http://www.jstatsoft.org/v35/i12/
as.irt.pars, irt.pars, sep.pars,
sep.pars, link
This (unidimensional) dataset includes three parameter logistic model (3PL) item parameter estimates from simulated dataset based on items from the 1992 Iowa Test of Basic Skills vocabulary test. The data are for six groups (grades 3-8) with varying numbers of common items between adjacent grades.
TK07TK07
A list of length two. The first element pars is a list of length six. Each
of the list elements in pars is a n x 3 matrix of item parameters where n
differs for each grade. The second list element in TK07 is a list of length
five identifying the common items between adjacent groups in pars. Each list
element in common is a j x 2 matrix for j common items. For example, there
are 13 common items between grades 3 and 4. The first list element in common
is for the common items between these grades. The first column in the matrix show that
items 14-26 in grade 3 are the same as items 1-13 in grade 4. Similar matrices are
identified for all other grade pairs.
The parameters were obtained from Dr. Ye Tong at Pearson Educational Measurement and were used in the following article:
Tong, Y. & Kolen, M. J. (2007) Comparisons of methodologies and results in vertical scaling for educational achievement tests. Applied Measurement in Education, 20(2), 227-253.