pcm.to.dichotomous: Convert Partial Credit Model (PCM) thresholds to equivalent...

In CambridgeAssessmentResearch/unimirt: Applications of Unidimensional IRT using the R Package 'mirt'

Convert Partial Credit Model (PCM) thresholds to equivalent set of n Rasch difficulties from dichotomous items

Description

Given a single polytomous item with n thresholds, calculate an equivalent set of Rasch difficulties from n dichomtomous items. The aim is to provide a different way of interpreting the thresholds from a partial credit model by seeing what set of difficulties from dichotomous items would lead to the same result.

Usage

pcm.to.dichotomous(pcm_thresh)

Arguments

pcm_thresh

A vector of thresholds from the partial credit model.

Details

Note that, in practice, this function rarely works in the way we might hope. Except in special circumstances (effectively when thresholds are correctly ordered and widely spaced), it is likely that some (or all) of the identified Rasch difficulties will be imaginary numbers. If a mix of imaginary and real numbers are returned in may be that the imaginary ones can be combined to create PCM thresholds for an item with fewer categories than started with. I have not investigated this fully.

Method here is based on numerically solving the equation for Fn(x) on page 115 of Huynh, H. (1994). On equivalence between a partial credit item and a set of independent Rasch binary items. Psychometrika, 59(1), 111-119. (see https://doi.org/10.1007/BF02294270). We use the expressions directly in terms of Si (above equation 6 in the paper) rather than those in terms of ai.

If the solution contains any non-zero imaginary parts it implies that an exactly equivalent set of dichotomoues difficulties cannot be found. In other words, the polytomous item cannot be considered as being the sum of multiple independent dichotomous items.

Value

A data.frame with the real and imaginary parts of the equivalent dichotomous Rasch difficulties. Very small imaginary numbers are rounded down to zero as they are considered to likely be part of estimation error.

Examples

## Not run: 

dichotomous.to.pcm(c(0,0))
pcm.to.dichotomous(c(-0.6931472,0.6931472))

dichotomous.to.pcm(c(-2,-4,1))
pcm.to.dichotomous(c(-4.132845,-1.922140,1.054985))

dichotomous.to.pcm(c(-1,-1,1,1))
pcm.to.dichotomous(c(-1.8200752,-0.6243906,0.6243906,1.8200752))

dich_diffs=pcm.to.dichotomous(c(-2,0,2,3.4))
dich_diffs
dichotomous.to.pcm(dich_diffs$real_part)

#and one that doesn't work (disordered thresholds)
pcm.to.dichotomous(c(1,0,3))
#investigate PCM thresholds for the imaginary bit
dichotomous.to.pcm(c(0.5228831-1.301217i,0.5228831+1.301217i))
#so equivalent to a 2-category PCM with disordered thresholds and a dichotomous item


## End(Not run)

CambridgeAssessmentResearch/unimirt documentation built on June 10, 2025, 6:03 a.m.