Conversion of the Parameterization of the Partial Credit Model

Converts a parameterization of the partial credit model (see Details).

Usage

pcm.conversion(b)

Arguments

b: Matrix of item-category-wise intercepts $b_{ik}$ (see Details).

Details

Assume that the input matrix b containing parameters $b_{ik}$ is defined according to the following parametrization of the partial credit model $$ P( X_{pi}=k | \theta_p ) \propto exp ( k \theta_p - b_{ik} ) $$ if item $i$ possesses $K_i$ categories. The transformed parameterization is defined as $$b_{ik}=k \delta_i + \sum_{v=1}^{k} \tau_{iv} \quad \mbox{with} \quad \sum_{k=1}^{K_i} \tau_{ik}=0 $$ The function pcm.conversion has the $\delta$ and $\tau$ parameters as values. The $\delta$ parameter is simply $\delta_i=b_{iK_i} / K_i$.

Value

List with the following entries

delta: Vector of $\delta$ parameters
tau: Matrix of $\tau$ parameters

Examples

if (FALSE) {
#############################################################################
# EXAMPLE 1: Transformation PCM for data.mg
#############################################################################

library(CDM)
data(data.mg,package="CDM")
dat <- data.mg[ 1:1000, paste0("I",1:11) ]

#*** Model 1: estimate partial credit model in parameterization "PCM"
mod1a <- TAM::tam.mml( dat, irtmodel="PCM")
# use parameterization "PCM2"
mod1b <- TAM::tam.mml( dat, irtmodel="PCM2")
summary(mod1a)
summary(mod1b)

# convert parameterization of Model 1a into parameterization of Model 1b
b <- mod1a$item[, c("AXsi_.Cat1","AXsi_.Cat2","AXsi_.Cat3") ]
# compare results
pcm.conversion(b)
mod1b$xsi
}