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.
cmp_probs(alpha, scale, pa1, pa2, thRaw)A vector of probabilities of observing each outcome
discrimination parameter
scale correction factor
first latent worth
second latent worth
vector of positive thresholds
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.
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).
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
# 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))
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