rasch.copula2: Rasch Copula Model

Description

This function handles local dependence by specifying copulas for residuals in unidimensional item response models (Braeken, 2011; Braeken, Tuerlinckx & de Boeck, 2007). Estimation is allowed for item difficulties, item slopes and a generalized logistic link function (Stukel, 1988).

Usage

rasch.copula2(dat, itemcluster, 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 = 0.001, alpha.conv = 1e-04, conv1 = 0.001,
    dev.crit=.2)

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 r

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). The vector copula.type must mat

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 reponse 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 reponse 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.

Value

A list with following entries
N.itemclustersNumber of item clusters
itemEstimated item parameters
iterNumber of iterations
devDeviance
deltaEstimated dependency parameters $\delta$
bEstimated item difficulties
aEstimated item slopes
muMean
sigmaStandard deviation
alpha1Parameter $\alpha_1$ in the generalized item response model
alpha2Parameter $\alpha_2$ in the generalized item response model
icInformation criteria
theta.kDiscretized ability distribution
devianceDeviance
patternItem response patterns
datalistList of generated data frames during estimation
EAP.relReliability of the EAP
copula.typeType of copula
summary.deltaSummary for estimated $\delta$ parameters
f.qk.yiIndividual posterior
f.yi.qkIndividual likelihood
...

References

Braeken, J. (2011). A boundary mixture approach to violations of conditional independence. Psychometrika, 76, 57-76. Braeken, J., Tuerlinckx, F. & De Boeck, P. (2007). Copulas for residual dependencies. Psychometrika, 72, 393-411. Stukel, T. A. (1988). Generalized logistic models. Journal of the American Statistical Association, 83, 426-431.

Examples

Run this code

#############################################
# Example 1: Reading data
#############################################

data(data.read)
dat <- data.read

# define item clusters
itemcluster <- rep( 1:3 , each=4 )

# estimate Copula model
mod1 <- rasch.copula2( dat=dat , itemcluster=itemcluster)

# estimate Rasch model
mod2 <- rasch.copula2( dat=dat , itemcluster=itemcluster , 
        delta=rep(0,3) , est.delta=rep(0,3 ) )

summary(mod1)
summary(mod2)

############################################################
# Simulated Example 2:
# 11 Items: 2 itemclusters with 2 resp. 3 dependent items
#           and 6 independent items
############################################################

set.seed(5698)
I <- 11                             # number of items
n <- 1500                           # number of persons
b <- seq(-2,2, len=I)               # item difficulties
theta <- rnorm( n , sd = 1 )        # person abilities
# itemcluster
itemcluster <- rep(0,I)
itemcluster[ c(3,5 )] <- 1
itemcluster[c(2,4,9)] <- 2
# residual correlations
rho <- c( .7 , .5 )

# simulate data
dat <- sim.rasch.dep( theta , b , itemcluster , rho )
colnames(dat) <- paste("I" , seq(1,ncol(dat)) , sep="")

# estimate Rasch copula model
mod1 <- rasch.copula2( dat , itemcluster = itemcluster )
summary(mod1)

# both itemclusters have Cook-Johnson copula as dependency
mod1c <- rasch.copula2( dat , itemcluster = itemcluster ,
			copula.type ="cook.johnson")
summary(mod1c)

# first item boundary mixture and second item Cook-Johnson copula
mod1d <- rasch.copula2( dat , itemcluster = itemcluster ,
			copula.type = c( "bound.mixt" , "cook.johnson" ) )
summary(mod1d)

# compare result with Rasch model estimation in rasch.copula2
# delta must be set to zero
mod2 <- rasch.copula2( dat , itemcluster = itemcluster , delta = c(0,0) ,
            est.delta = c(0,0) )
summary(mod2)

############################################################
# Simulated Example 3:
# 12 Items: Cluster 1 -> Items 1,...,4
#           Cluster 2 -> Items 6,...,9
#           Cluster 3 -> Items 10,11,12
############################################################

set.seed(967)
I <- 12                             # number of items
n <- 450                            # number of persons
b <- seq(-2,2, len=I)               # item difficulties
b <- sample(b)                      # sample item difficulties
theta <- rnorm( n , sd = 1 )        # person abilities
# itemcluster
itemcluster <- rep(0,I)
itemcluster[ 1:4 ] <- 1
itemcluster[ 6:9 ] <- 2
itemcluster[ 10:12 ] <- 3
# residual correlations
rho <- c( .35 , .25 , .30 )

# simulate data
dat <- sim.rasch.dep( theta , b , itemcluster , rho )
colnames(dat) <- paste("I" , seq(1,ncol(dat)) , sep="")

# estimate Rasch copula model 
mod1 <- rasch.copula2( dat , itemcluster = itemcluster )
summary(mod1)

# person parameter estimation assuming the Rasch copula model
pmod1 <- person.parameter.rasch.copula(raschcopula.object = mod1 )

# Rasch model estimation
mod2 <- rasch.copula2( dat , itemcluster = itemcluster , 
        delta = rep(0,3) , est.delta = rep(0,3) )
summary(mod1)
summary(mod2)

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