immer_ccml: Composite Conditional Maximum Likelihood Estimation for the...

In immer: Item Response Models for Multiple Ratings

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 | θ) / P( X_i + X_j=k+h| θ ). By using this strategy, the trait θ 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 class 'immer_ccml'
summary(object, digits=3, file=NULL, ...)

## S3 method for class 'immer_ccml'
coef(object, ...)

## S3 method for class 'immer_ccml'
vcov(object, ...)

Arguments

dat	Data frame with polytomous item responses 0,1,…, 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,…,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 | θ ) \propto \exp( h θ - b_{ih} ) with b_{ih}=∑_l a_{ihl} ξ_l where the values a_{ihl} are included in the design matrix A and ξ_l denotes basis item parameters.

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

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)

immer documentation built on Aug. 22, 2022, 5:05 p.m.