This function handles local dependence by specifying copulas for residuals in multidimensional item response models for dichotomous item responses (Braeken, 2011; Braeken, Tuerlinckx & de Boeck, 2007; Schroeders, Robitzsch & Schipolowski, 2014). Estimation is allowed for item difficulties, item slopes and a generalized logistic link function (Stukel, 1988).
The function rasch.copula3 allows the estimation of multidimensional
models while rasch.copula2 only handles unidimensional models.
rasch.copula2(dat, itemcluster, weights=NULL, copula.type="bound.mixt",
progress=TRUE, mmliter=1000, delta=NULL,
theta.k=seq(-4, 4, len=21), alpha1=0, alpha2=0,
numdiff.parm=1e-06, est.b=seq(1, ncol(dat)),
est.a=rep(1, ncol(dat)), est.delta=NULL, b.init=NULL, a.init=NULL,
est.alpha=FALSE, glob.conv=0.0001, alpha.conv=1e-04, conv1=0.001,
dev.crit=.2, increment.factor=1.01)rasch.copula3(dat, itemcluster, dims=NULL, copula.type="bound.mixt",
progress=TRUE, mmliter=1000, delta=NULL,
theta.k=seq(-4, 4, len=21), alpha1=0, alpha2=0,
numdiff.parm=1e-06, est.b=seq(1, ncol(dat)),
est.a=rep(1, ncol(dat)), est.delta=NULL, b.init=NULL, a.init=NULL,
est.alpha=FALSE, glob.conv=0.0001, alpha.conv=1e-04, conv1=0.001,
dev.crit=.2, rho.init=.5, increment.factor=1.01)
# S3 method for rasch.copula2
summary(object, file=NULL, digits=3, ...)
# S3 method for rasch.copula3
summary(object, file=NULL, digits=3, ...)
# S3 method for rasch.copula2
anova(object,...)
# S3 method for rasch.copula3
anova(object,...)
# S3 method for rasch.copula2
logLik(object,...)
# S3 method for rasch.copula3
logLik(object,...)
# S3 method for rasch.copula2
IRT.likelihood(object,...)
# S3 method for rasch.copula3
IRT.likelihood(object,...)
# S3 method for rasch.copula2
IRT.posterior(object,...)
# S3 method for rasch.copula3
IRT.posterior(object,...)
A list with following entries
Number of item clusters
Estimated item parameters
Number of iterations
Deviance
Estimated dependency parameters \(\delta\)
Estimated item difficulties
Estimated item slopes
Mean
Standard deviation
Parameter \(\alpha_1\) in the generalized item response model
Parameter \(\alpha_2\) in the generalized item response model
Information criteria
Discretized ability distribution
Fixed \(\theta\) distribution
Deviance
Item response patterns with frequencies and posterior distribution
Data frame with person parameters
List of generated data frames during estimation
Reliability of the EAP
Type of copula
Summary for estimated \(\delta\) parameters
Individual posterior
Individual likelihood
Further values
An \(N \times I\) data frame. Cases with only missing responses are removed from the analysis.
An integer vector of length \(I\) (number of items). Items with the same integers define a joint item cluster of (positively) locally dependent items. Values of zero indicate that the corresponding item is not included in any item cluster of dependent responses.
Optional vector of sampling weights
A vector indicating to which dimension an item is allocated. The default is that all items load on the first dimension.
A character or a vector containing one of the following copula
types: bound.mixt (boundary mixture copula),
cook.johnson (Cook-Johnson copula) or frank (Frank copula)
(see Braeken, 2011).
The vector copula.type must match the number of different
itemclusters. For every itemcluster, a different copula type
may be specified (see Examples).
Print progress? Default is TRUE.
Maximum number of iterations.
An optional vector of starting values for the dependency parameter delta.
Discretized trait distribution
alpha1 parameter in the generalized logistic item response model
(Stukel, 1988). The default is 0 which leads together with alpha2=0
to the logistic link function.
alpha2 parameter in the generalized logistic item response model
Parameter for numerical differentiation
Integer vector of item difficulties to be estimated
Integer vector of item discriminations to be estimated
Integer vector of length length(itemcluster). Nonzero integers
correspond to delta parameters which are estimated.
Equal integers indicate parameter equality constraints.
Initial \(b\) parameters
Initial \(a\) parameters
Should both alpha parameters be estimated? Default is FALSE.
Convergence criterion for all parameters
Maximal change in alpha parameters for convergence
Maximal change in item parameters for convergence
Maximal change in the deviance. Default is .2.
Initial value for off-diagonal elements in correlation matrix
A numeric value larger than one which controls the size of increments in iterations. To stabilize convergence, choose values 1.05 or 1.1 in some situations.
Object of class rasch.copula2 or rasch.copula3
Optional file name for summary output
Number of digits after decimal in summary output
Further arguments to be passed
Braeken, J. (2011). A boundary mixture approach to violations of conditional independence. Psychometrika, 76(1), 57-76. tools:::Rd_expr_doi("10.1007/s11336-010-9190-4")
Braeken, J., Kuppens, P., De Boeck, P., & Tuerlinckx, F. (2013). Contextualized personality questionnaires: A case for copulas in structural equation models for categorical data. Multivariate Behavioral Research, 48(6), 845-870. tools:::Rd_expr_doi("10.1080/00273171.2013.827965")
Braeken, J., & Tuerlinckx, F. (2009). Investigating latent constructs with item response models: A MATLAB IRTm toolbox. Behavior Research Methods, 41(4), 1127-1137.
Braeken, J., Tuerlinckx, F., & De Boeck, P. (2007). Copula functions for residual dependency. Psychometrika, 72(3), 393-411. tools:::Rd_expr_doi("10.1007/s11336-007-9005-4")
Schroeders, U., Robitzsch, A., & Schipolowski, S. (2014). A comparison of different psychometric approaches to modeling testlet structures: An example with C-tests. Journal of Educational Measurement, 51(4), 400-418. tools:::Rd_expr_doi("10.1111/jedm.12054")
Stukel, T. A. (1988). Generalized logistic models. Journal of the American Statistical Association, 83(402), 426-431. tools:::Rd_expr_doi("10.1080/01621459.1988.10478613")
For a summary see summary.rasch.copula2.
For simulating locally dependent item responses see sim.rasch.dep.
Person parameters estimates are obtained by person.parameter.rasch.copula.
See rasch.mml2 for the generalized logistic link function.
See also Braeken and Tuerlinckx (2009) for alternative (and more expanded) copula models implemented in the MATLAB software. See https://ppw.kuleuven.be/okp/software/irtm/.
See Braeken, Kuppens, De Boeck and Tuerlinckx (2013) for an extension of the copula modeling approach to polytomous data.