# immer_ccml: Composite Conditional Maximum Likelihood Estimation for the... In alexanderrobitzsch/immer: Item Response Models for Multiple Ratings

## 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

  1 2 3 4 5 6 7 8 9 10 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 sirt::rasch.pairwise.itemcluster of an implementation of the composite conditional maximum likelihood approach for the Rasch model.
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 ############################################################################# # 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)