| Title: | Equating of Multiple Forms |
|---|---|
| Description: | Equating of multiple forms using Item Response Theory (IRT) methods (Battauz M. (2017) <doi:10.1007/s11336-016-9517-x>, Battauz and 'Leoncio' (2023) <doi:10.1177/01466216231151702>, Haberman S. J. (2009) <doi:10.1002/j.2333-8504.2009.tb02197.x>). |
| Authors: | Michela Battauz [aut, cre], Waldir Leoncio [ctb] |
| Maintainer: | Michela Battauz <[email protected]> |
| License: | GPL-3 |
| Version: | 1.1.1 |
| Built: | 2024-11-15 06:55:23 UTC |
| Source: | CRAN |
The EquateMultiple package implements IRT-based methods to equate simultaneously many forms calibrated separately. This package estimates the equating coefficients to convert the item parameters and the ability values to the scale of the base form. It can be applied to a large number of test forms, as well as to 2 forms. The computation of the equated scores is also implemented.
This package implements the methods proposed in Haberman (2009), Battauz (2017)
and Battauz and Leoncio (2023).
Function multiec computes the equating coefficients to convert the
item parameters and the ability values to the scale of the base form.
The methods implemented are: multiple mean-geometric mean (Haberman, 2009),
multiple mean-mean, multiple item response function, multiple test
response function (Battauz, 2017), and likelihood-based linking (Battauz and Leoncio, 2023).
The function provides the equating coefficients, estimates of item parameters on a common scale
and the standard errors of the equating coefficients and item parameters on a common scale.
Equated scores can be computed using true score equating and
observed score equating methods. Standard errors of equated scores
are also provided.
Michela Battauz [aut, cre], Waldir Leoncio [ctb]
Maintainer: Michela Battauz <[email protected]>
Battauz, M. (2017). Multiple equating of separate IRT calibrations. Psychometrika, 82, 610–636. doi:10.1007/s11336-016-9517-x.
Battauz, M., Leoncio, W. (2023). A Likelihood Approach to Item Response Theory Equating of Multiple Forms Applied Psychological Measurement, 47, 200-220. doi: 10.1177/01466216231151702.
Haberman, S. J. (2009). Linking parameter estimates derived from an item response model through separate calibrations. ETS Research Report Series, 2009, i-9. doi:10.1002/j.2333-8504.2009.tb02197.x.
data(est2pl) # prepare the data mods <- modIRT(coef = est2pl$coef, var = est2pl$var, display = FALSE) # Estimation of the equating coefficients with the multiple mean-mean method eqMM <- multiec(mods = mods, base = 1, method = "mean-mean") summary(eqMM) # Estimation of the equating coefficients with the # multiple mean-geometric mean method (Haberman, 2009) eqMGM <- multiec(mods = mods, base = 1, method = "mean-gmean") summary(eqMGM) # Estimation of the equating coefficients with the multiple item response function method eqIRF <- multiec(mods = mods, base = 1, method = "irf") summary(eqIRF) # Estimation of the equating coefficients with the multiple item response function method # using as initial values the estimates obtained with the multiple mean-geometric mean method eqMGM <- multiec(mods = mods, base = 1, method = "mean-gmean", se = FALSE) eqIRF <- multiec(mods = mods, base = 1, method = "irf", start = eqMGM) summary(eqIRF) # Estimation of the equating coefficients with the multiple test response function method eqTRF <- multiec(mods = mods, base = 1, method = "trf") summary(eqTRF) # Estimation of the equating coefficients with the likelihood-based method eqLIK <- multiec(mods = mods, base = 1, method = "lik") summary(eqLIK) # scoring using the true score equating method and equating coefficients # obtained with the multiple item response function method score(eqIRF)data(est2pl) # prepare the data mods <- modIRT(coef = est2pl$coef, var = est2pl$var, display = FALSE) # Estimation of the equating coefficients with the multiple mean-mean method eqMM <- multiec(mods = mods, base = 1, method = "mean-mean") summary(eqMM) # Estimation of the equating coefficients with the # multiple mean-geometric mean method (Haberman, 2009) eqMGM <- multiec(mods = mods, base = 1, method = "mean-gmean") summary(eqMGM) # Estimation of the equating coefficients with the multiple item response function method eqIRF <- multiec(mods = mods, base = 1, method = "irf") summary(eqIRF) # Estimation of the equating coefficients with the multiple item response function method # using as initial values the estimates obtained with the multiple mean-geometric mean method eqMGM <- multiec(mods = mods, base = 1, method = "mean-gmean", se = FALSE) eqIRF <- multiec(mods = mods, base = 1, method = "irf", start = eqMGM) summary(eqIRF) # Estimation of the equating coefficients with the multiple test response function method eqTRF <- multiec(mods = mods, base = 1, method = "trf") summary(eqTRF) # Estimation of the equating coefficients with the likelihood-based method eqLIK <- multiec(mods = mods, base = 1, method = "lik") summary(eqLIK) # scoring using the true score equating method and equating coefficients # obtained with the multiple item response function method score(eqIRF)
eqc is a generic function which extracts the equating coefficients.
## S3 method for class 'mlteqc' eqc(x, ...)## S3 method for class 'mlteqc' eqc(x, ...)
x |
object of the class |
... |
further arguments passed to or from other methods. |
A data frame containing the equating coefficients.
Michela Battauz
data(est2pl) # prepare the data mods <- modIRT(coef = est2pl$coef, var = est2pl$var, display = FALSE) # Estimation of the equating coefficients with the multiple item response function method eqIRF <- multiec(mods = mods, base = 1, method = "irf") # extract equating coefficients eqc(eqIRF)data(est2pl) # prepare the data mods <- modIRT(coef = est2pl$coef, var = est2pl$var, display = FALSE) # Estimation of the equating coefficients with the multiple item response function method eqIRF <- multiec(mods = mods, base = 1, method = "irf") # extract equating coefficients eqc(eqIRF)
item.common is a generic function which extracts the estimates of the
item parameters on the scale of the base form, which is obtained using
the item parameter estimates across all forms.
item.common(x, ...) ## S3 method for class 'mlteqc' item.common(x, ...)item.common(x, ...) ## S3 method for class 'mlteqc' item.common(x, ...)
x |
object of the class |
... |
further arguments passed to or from other methods. |
A data frame containing item names (Item),
estimated value of the item parameter (Estimate)
and its standard error (StdErr).
Michela Battauz
data(est2pl) # prepare the data mods <- modIRT(coef = est2pl$coef, var = est2pl$var, display = FALSE) # Estimation of the equating coefficients with the multiple item response function method eqIRF <- multiec(mods = mods, base = 1, method = "irf") # estimates of the item parameters on the scale of the base form item.common(eqIRF)data(est2pl) # prepare the data mods <- modIRT(coef = est2pl$coef, var = est2pl$var, display = FALSE) # Estimation of the equating coefficients with the multiple item response function method eqIRF <- multiec(mods = mods, base = 1, method = "irf") # estimates of the item parameters on the scale of the base form item.common(eqIRF)
itm is a generic function which extracts a data frame containing the
item parameters of multiple forms in the original scale and
the item parameters converted to the scale of the base form.
## S3 method for class 'mlteqc' itm(x, ...)## S3 method for class 'mlteqc' itm(x, ...)
x |
object of the class |
... |
further arguments passed to or from other methods. |
A data frame containing item names (Item),
item parameters of all the forms (e.g. T1, ..., T3),
and item parameters of all the forms converted
in the scale of the base form (e.g. T3.as.T1).
Michela Battauz
data(est2pl) # prepare the data mods <- modIRT(coef = est2pl$coef, var = est2pl$var, display = FALSE) # Estimation of the equating coefficients with the multiple item response function method eqIRF <- multiec(mods = mods, base = 1, method = "irf") # extract item parameters itm(eqIRF)data(est2pl) # prepare the data mods <- modIRT(coef = est2pl$coef, var = est2pl$var, display = FALSE) # Estimation of the equating coefficients with the multiple item response function method eqIRF <- multiec(mods = mods, base = 1, method = "irf") # extract item parameters itm(eqIRF)
List of 5 datasets with the binary responses (correct/wrong) to 5 forms of a math test.
data("mathTest")data("mathTest")
A list of length 5, containing 5 data frames. Each dataset contains rows of responses from individuals to various items, with the item labels as the column headers.
data(mathTest) mathTest[[1]][1:3,]data(mathTest) mathTest[[1]][1:3,]
Calculates the equating coefficients between multiple forms.
multiec(mods, base = 1, method = "mean-mean", se = TRUE, nq = 30, start = NULL, iter.max = 100000, obsinf = TRUE, trace = TRUE)multiec(mods, base = 1, method = "mean-mean", se = TRUE, nq = 30, start = NULL, iter.max = 100000, obsinf = TRUE, trace = TRUE)
mods |
an object of the class |
base |
integer value indicating the base form. |
method |
the method used to compute the equating coefficients.
This should be one of " |
se |
logical; if |
nq |
number of quadrature points used for the Gauss-Hermite quadrature for
methods " |
start |
initial values. This can be a vector containing the A and B equating coefficients excluding
the base form, or an object of class |
iter.max |
maximum number of iterations allowed.
Used only with methods " |
obsinf |
logical; if |
trace |
logical; if |
The methods implemented for the computation of the multiple equating coefficients are
the multiple mean-mean method ("mean-mean"),
the multiple mean-geometric mean method ("mean-gmean"),
the multiple item response function method ("irf"),
the multiple test response function method ("trf"),
and likelihood-based equating ("lik").
An object of class mlteqc with components
A |
A equating coefficients. |
B |
B equating coefficients. |
se.A |
standard errors of A equating coefficients. |
se.B |
standard errors of B equating coefficients. |
varAB |
covariance matrix of equating coefficients. |
as |
discrimination parameters on a common scale |
bs |
difficulty parameters on a common scale |
se.as |
standard errors of discrimination parameters on a common scale. |
se.bs |
standard errors of difficulty parameters on a common scale. |
tab |
data frame containing item parameter names ( |
varFull |
list of covariance matrices of the item parameter estimates of every form. |
partial |
partial derivatives of equating coefficients with respect to the item parameters. |
itmp |
number of item parameters of the IRT model. |
method |
the equating method used. |
basename |
the name of the base form. |
convergence |
An integer code. 0 indicates successful convergence.
Returned only with methods " |
Michela Battauz, Waldir Leoncio [ctb]
Battauz, M. (2017). Multiple equating of separate IRT calibrations. Psychometrika, 82, 610–636.
Battauz, M., Leoncio, W. (2023). A Likelihood Approach to Item Response Theory Equating of Multiple Forms Applied Psychological Measurement, 47, 200-220. doi: 10.1177/01466216231151702.
Haberman, S. J. (2009). Linking parameter estimates derived from an item response model through separate calibrations. ETS Research Report Series, 2009, i-9.
data(est2pl) # prepare the data mods <- modIRT(coef = est2pl$coef, var = est2pl$var, display = FALSE) # Estimation of the equating coefficients with the multiple mean-mean method eqMM <- multiec(mods = mods, base = 1, method = "mean-mean") summary(eqMM) # Estimation of the equating coefficients with the # multiple mean-geometric mean method (Haberman, 2009) eqMGM <- multiec(mods = mods, base = 1, method = "mean-gmean") summary(eqMGM) # Estimation of the equating coefficients with the multiple item response function method eqIRF <- multiec(mods = mods, base = 1, method = "irf") summary(eqIRF) # Estimation of the equating coefficients with the multiple item response function method # using as initial values the estimates obtained with the multiple mean-geometric mean method eqMGM <- multiec(mods = mods, base = 1, method = "mean-gmean", se = FALSE) eqIRF <- multiec(mods = mods, base = 1, method = "irf", start = eqMGM) summary(eqIRF) # Estimation of the equating coefficients with the multiple test response function method eqTRF <- multiec(mods = mods, base = 1, method = "trf") summary(eqTRF) # Estimation of the equating coefficients with the likelihood-based method eqLIK <- multiec(mods = mods, base = 1, method = "lik") summary(eqLIK)data(est2pl) # prepare the data mods <- modIRT(coef = est2pl$coef, var = est2pl$var, display = FALSE) # Estimation of the equating coefficients with the multiple mean-mean method eqMM <- multiec(mods = mods, base = 1, method = "mean-mean") summary(eqMM) # Estimation of the equating coefficients with the # multiple mean-geometric mean method (Haberman, 2009) eqMGM <- multiec(mods = mods, base = 1, method = "mean-gmean") summary(eqMGM) # Estimation of the equating coefficients with the multiple item response function method eqIRF <- multiec(mods = mods, base = 1, method = "irf") summary(eqIRF) # Estimation of the equating coefficients with the multiple item response function method # using as initial values the estimates obtained with the multiple mean-geometric mean method eqMGM <- multiec(mods = mods, base = 1, method = "mean-gmean", se = FALSE) eqIRF <- multiec(mods = mods, base = 1, method = "irf", start = eqMGM) summary(eqIRF) # Estimation of the equating coefficients with the multiple test response function method eqTRF <- multiec(mods = mods, base = 1, method = "trf") summary(eqTRF) # Estimation of the equating coefficients with the likelihood-based method eqLIK <- multiec(mods = mods, base = 1, method = "lik") summary(eqLIK)
plot method for class mlteqc, which plots item parameter
estimates of one form agains the item parameter estimates of the base form
before and after conversion.
## S3 method for class 'mlteqc' plot(x, form = 2, ask = prod(par("mfcol")) < x$itmp*2 && dev.interactive(), ...)## S3 method for class 'mlteqc' plot(x, form = 2, ask = prod(par("mfcol")) < x$itmp*2 && dev.interactive(), ...)
x |
an object of the class |
form |
test form. It can be specified by name or number. |
ask |
logical; if |
... |
further arguments passed to or from other methods. |
Michela Battauz
data(est2pl) # prepare the data mods <- modIRT(coef = est2pl$coef, var = est2pl$var, display = FALSE) # Estimation of the equating coefficients with the multiple mean-mean method eqMM <- multiec(mods = mods, base = 1, method = "mean-mean") plot(eqMM, form = "T2")data(est2pl) # prepare the data mods <- modIRT(coef = est2pl$coef, var = est2pl$var, display = FALSE) # Estimation of the equating coefficients with the multiple mean-mean method eqMM <- multiec(mods = mods, base = 1, method = "mean-mean") plot(eqMM, form = "T2")
Relates number-correct scores on multiple forms.
## S3 method for class 'mlteqc' score(obj, method="TSE", D=1, scores=NULL, se=TRUE, nq=30, w=0.5, theta=NULL, weights=NULL, ...)## S3 method for class 'mlteqc' score(obj, method="TSE", D=1, scores=NULL, se=TRUE, nq=30, w=0.5, theta=NULL, weights=NULL, ...)
obj |
object of the class |
method |
the scoring method to be used. This should be one of " |
D |
constant D of the IRT model used to estimate item parameters. |
scores |
integer values to be converted. |
se |
logical; is |
nq |
number of quadrature points used to approximate integrals with observed score equating.
Used only if arguments |
w |
synthetic weight for population 1. It should be a number between 0 and 1. |
theta |
vector of ability values used to approximate integrals with observed score equating. |
weights |
vector of weights used to approximate integrals with observed score equating. |
... |
further arguments passed to or from other methods. |
In this function common items are internal, i.e. they are used for scoring the test.
A data frame containing theta values (only for true score equating), scores of the form chosen as base, equated scores of all other forms, and standard errors of equated scores.
Michela Battauz
Kolen, M.J. and Brennan, R.L. (2014). Test equating, scaling, and linking: methods and practices, 3nd ed., New York: Springer.
Ogasawara, H. (2001). Item response theory true score equatings and their standard errors. Journal of Educational and Behavioral Statistics, 26, 31–50.
Ogasawara, H. (2003). Asymptotic standard errors of IRT observed-score equating methods. Psychometrika, 68, 193–211.
data(est2pl) # prepare the data mods <- modIRT(coef = est2pl$coef, var = est2pl$var, display = FALSE) # Estimation of the equating coefficients with the multiple item response function method eqIRF<-multiec(mods = mods, base = 1, method = "irf") summary(eqIRF) # scoring using the true score equating method score(eqIRF) # scoring using observed score equating method, without standard errors score(eqIRF, method = "OSE", se = FALSE)data(est2pl) # prepare the data mods <- modIRT(coef = est2pl$coef, var = est2pl$var, display = FALSE) # Estimation of the equating coefficients with the multiple item response function method eqIRF<-multiec(mods = mods, base = 1, method = "irf") summary(eqIRF) # scoring using the true score equating method score(eqIRF) # scoring using observed score equating method, without standard errors score(eqIRF, method = "OSE", se = FALSE)
summary method for class mlteqc.
## S3 method for class 'mlteqc' summary(object, ...)## S3 method for class 'mlteqc' summary(object, ...)
object |
an object of the class |
... |
further arguments passed to or from other methods. |
Michela Battauz
data(est2pl) # prepare the data mods <- modIRT(coef = est2pl$coef, var = est2pl$var, display = FALSE) # Estimation of the equating coefficients with the multiple mean-mean method eqMM <- multiec(mods = mods, base = 1, method = "mean-mean") summary(eqMM)data(est2pl) # prepare the data mods <- modIRT(coef = est2pl$coef, var = est2pl$var, display = FALSE) # Estimation of the equating coefficients with the multiple mean-mean method eqMM <- multiec(mods = mods, base = 1, method = "mean-mean") summary(eqMM)