Multidimensional IRT Copula Model

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.

Usage

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,...)

Arguments

dat: An \(N \times I\) data frame. Cases with only missing responses are removed from the analysis.
itemcluster: 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.
weights: Optional vector of sampling weights
dims: A vector indicating to which dimension an item is allocated. The default is that all items load on the first dimension.
copula.type: 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).

progress: Print progress? Default is TRUE.
mmliter: Maximum number of iterations.
delta: An optional vector of starting values for the dependency parameter delta.
theta.k: Discretized trait distribution
alpha1: 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: alpha2 parameter in the generalized logistic item response model
numdiff.parm: Parameter for numerical differentiation
est.b: Integer vector of item difficulties to be estimated
est.a: Integer vector of item discriminations to be estimated
est.delta: Integer vector of length length(itemcluster). Nonzero integers correspond to delta parameters which are estimated. Equal integers indicate parameter equality constraints.
b.init: Initial \(b\) parameters
a.init: Initial \(a\) parameters
est.alpha: Should both alpha parameters be estimated? Default is FALSE.
glob.conv: Convergence criterion for all parameters
alpha.conv: Maximal change in alpha parameters for convergence
conv1: Maximal change in item parameters for convergence
dev.crit: Maximal change in the deviance. Default is .2.
rho.init: Initial value for off-diagonal elements in correlation matrix
increment.factor: 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: Object of class rasch.copula2 or rasch.copula3
file: Optional file name for summary output
digits: Number of digits after decimal in summary output
...: Further arguments to be passed

Value

A list with following entries

N.itemclusters: Number of item clusters
item: Estimated item parameters
iter: Number of iterations
dev: Deviance
delta: Estimated dependency parameters \(\delta\)
b: Estimated item difficulties
a: Estimated item slopes
mu: Mean
sigma: Standard deviation
alpha1: Parameter \(\alpha_1\) in the generalized item response model
alpha2: Parameter \(\alpha_2\) in the generalized item response model
ic: Information criteria
theta.k: Discretized ability distribution
pi.k: Fixed \(\theta\) distribution
deviance: Deviance
pattern: Item response patterns with frequencies and posterior distribution
person: Data frame with person parameters
datalist: List of generated data frames during estimation
EAP.rel: Reliability of the EAP
copula.type: Type of copula
summary.delta: Summary for estimated \(\delta\) parameters
f.qk.yi: Individual posterior
f.yi.qk: Individual likelihood
...: Further values

References

Braeken, J. (2011). A boundary mixture approach to violations of conditional independence. Psychometrika, 76(1), 57-76. 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. 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. 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. doi:10.1111/jedm.12054

Stukel, T. A. (1988). Generalized logistic models. Journal of the American Statistical Association, 83(402), 426-431. doi:10.1080/01621459.1988.10478613

Usage

Arguments

Value

References

See also

Examples