immer (version 1.5-13)

immer_ccml: Composite Conditional Maximum Likelihood Estimation for the Partial Credit Model with a Design Matrix for Item Parameters

Description

Estimates the partial credit model with a design matrix for item parameters with composite conditional maximum likelihood estimation. The estimation uses pairs of items \(X_i\) and \(X_j\) and considers conditional likelihoods \(P(X_i=k, X_j=h | \theta) / P( X_i + X_j=k+h| \theta )\). By using this strategy, the trait \(\theta\) cancels out (like in conditional maximum likelihood estimation). The proposed strategy is a generalization of the Zwinderman (1995) composite conditional maximum likelihood approach of the Rasch model to the partial credit model. See Varin, Reid and Firth (2011) for a general introduction to composite conditional maximum likelihood estimation.

Usage

immer_ccml( dat, weights=NULL, irtmodel="PCM", A=NULL, b_fixed=NULL, control=NULL )
# S3 method for immer_ccml
summary(object, digits=3, file=NULL, ...)
# S3 method for immer_ccml
coef(object, ...)
# S3 method for immer_ccml
vcov(object, ...)

Value

List with following entries (selection)

coef

Item parameters

vcov

Covariance matrix for item parameters

se

Standard errors for item parameters

nlminb_result

Output from optimization with stats::nlminb

suff_stat

Used sufficient statistics

ic

Information criteria

Arguments

dat

Data frame with polytomous item responses \(0,1,\ldots, K\)

weights

Optional vector of sampling weights

irtmodel

Model string for specifying the item response model

A

Design matrix (items \(\times\) categories \(\times\) basis parameters). Entries for categories are for \(1,\ldots,K\)

b_fixed

Matrix with fixed \(b\) parameters

control

Control arguments for optimization function stats::nlminb

object

Object of class immer_ccml

digits

Number of digits after decimal to print

file

Name of a file in which the output should be sunk

...

Further arguments to be passed.

Details

The function estimates the partial credit model as \(P(X_i=h | \theta ) \propto \exp( h \theta - b_{ih} )\) with \(b_{ih}=\sum_l a_{ihl} \xi_l\) where the values \(a_{ihl}\) are included in the design matrix A and \(\xi_l\) denotes basis item parameters.

References

Varin, C., Reid, N., & Firth, D. (2011). An overview of composite likelihood methods. Statistica Sinica, 21, 5-42.

Zwinderman, A. H. (1995). Pairwise parameter estimation in Rasch models. Applied Psychological Measurement, 19(4), 369-375.

See Also

See sirt::rasch.pairwise.itemcluster of an implementation of the composite conditional maximum likelihood approach for the Rasch model.

Examples

#############################################################################
# EXAMPLE 1: Partial credit model with CCML estimation
#############################################################################

library(TAM)

data(data.gpcm, package="TAM")
dat <- data.gpcm

#-- initial MML estimation in TAM to create a design matrix
mod1a <- TAM::tam.mml(dat, irtmodel="PCM2")
summary(mod1a)

#* define design matrix
A <- - mod1a$A[,-1,-1]
A <- A[,,-1]
str(A)

#-- estimate model
mod1b <- immer::immer_ccml( dat, A=A)
summary(mod1b)