Create a nominal response model
This function instantiates a nominal response model.
rpf.nrm(outcomes = 3, factors = 1, T.a = "trend", T.c = "trend")
outcomes |
The number of choices available |
factors |
the number of factors |
T.a |
the T matrix for slope parameters |
T.c |
the T matrix for intercept parameters |
The transformation matrices T.a and T.c are chosen by the analyst and not estimated. The T matrices must be invertible square matrices of size outcomes-1. As a shortcut, either T matrix can be specified as "trend" for a Fourier basis or as "id" for an identity basis. The response probability function is
a = T_a α
c = T_c γ
\mathrm P(\mathrm{pick}=k|s,a_k,c_k,θ) = C\ \frac{1}{1+\exp(-(s θ a_k + c_k))}
where a_k and c_k are the result of multiplying two vectors of free parameters α and γ by fixed matrices T_a and T_c, respectively; a_0 and c_0 are fixed to 0 for identification; and C is a normalizing factor to ensure that ∑_k \mathrm P(\mathrm{pick}=k) = 1.
an item model
Thissen, D., Cai, L., & Bock, R. D. (2010). The Nominal Categories Item Response Model. In M. L. Nering & R. Ostini (Eds.), Handbook of Polytomous Item Response Theory Models (pp. 43–75). Routledge.
spec <- rpf.nrm() rpf.prob(spec, rpf.rparam(spec), 0) # typical parameterization for the Generalized Partial Credit Model gpcm <- function(outcomes) rpf.nrm(outcomes, T.c=lower.tri(diag(outcomes-1),TRUE) * -1) spec <- gpcm(4) rpf.prob(spec, rpf.rparam(spec), 0)
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