rpf (version 0.5)

rpf.nrm: Create a nominal response model

Description

This function instantiates a nominal response model.

Usage

rpf.nrm(outcomes = 3, factors = 1, T.a = "trend", T.c = "trend")

Arguments

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

Value

• an item model

Details

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

$$a = T_a \alpha$$ $$c = T_c \gamma$$ $$\mathrm P(\mathrm{pick}=k|s,a_k,c_k,\theta) = C\ \frac{1}{1+\exp(-(s \theta a_k + c_k))}$$

where $a_k$ and $c_k$ are the result of multiplying two vectors of free parameters $\alpha$ and $\gamma$ 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 $\sum_k \mathrm P(\mathrm{pick}=k) = 1$.

References

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.

Examples

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)