cmp_probs: Item response function for pairwise comparisons

In pcFactorStan: Stan Models for the Paired Comparison Factor Model

Item response function for pairwise comparisons

Description

Use itemModelExplorer to explore the item model. In this shiny app, the discrimination parameter does what is customary in item response models. However, it is not difficult to show that discrimination is a function of thresholds and scale. That is, discrimination is not an independent parameter. In paired comparison models, discrimination and measurement error are confounded.

Usage

cmp_probs(alpha, scale, pa1, pa2, thRaw)

Arguments

alpha	discrimination parameter
scale	scale correction factor
pa1	first latent worth
pa2	second latent worth
thRaw	vector of positive thresholds

Details

The thresholds are parameterized as the difference from the previous threshold. For example, thresholds c(0.5, 0.6) are not at the same location but are at locations c(0.5, 1.1). Thresholds are symmetric. If there is one threshold then the model admits three possible response outcomes (e.g. win, tie, and lose). Responses are always stored centered with zero representing a tie. Therefore, it is necessary to add one plus the number of thresholds to response data to index into the vector returned by cmp_probs. For example, if our response data is (-1, 0, 1) and has one threshold then we would add 2 (1 + 1 threshold) to obtain the indices (1, 2, 3).

Value

A vector of probabilities of observing each outcome

Math

Up until version 1.4, the item response model was based on the partial credit model (Masters, 1982). In version 1.5, the graded response model is used instead (Samejima, 1969). The advantage of the graded response model is greater independence among threshold parameters and the ability to compute only the parts of the model that are actually needed given particular observations. The curves predicted by both models are similar and should obtain similar results in data analyses.

References

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph Supplement, 34(4, Pt. 2), 100.

Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174. doi: 10.1007/BF02296272

Examples

# Returns probabilities of
# c(pa1 > pa2, pa1 = pa2, pa1 < pa2)
cmp_probs(1,1,0,1,.8)

# Add another threshold for a symmtric 3 point Likert scale
cmp_probs(1,1,0,.5,c(.8, 1.6))

pcFactorStan documentation built on Sept. 14, 2023, 1:09 a.m.