1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
|
% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/nrm.R
\name{rpf.nrm}
\alias{rpf.nrm}
\title{Create a nominal response model}
\usage{
rpf.nrm(outcomes = 3, factors = 1, T.a = "trend", T.c = "trend")
}
\arguments{
\item{outcomes}{The number of choices available}
\item{factors}{the number of factors}
\item{T.a}{the T matrix for slope parameters}
\item{T.c}{the T matrix for intercept parameters}
}
\value{
an item model
}
\description{
This function instantiates a nominal response 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
\deqn{a = T_a \alpha}
\deqn{c = T_c \gamma}
\deqn{\mathrm P(\mathrm{pick}=k|s,a_k,c_k,\theta) = C\ \frac{1}{1+\exp(-(s \theta a_k + c_k))}}
where \eqn{a_k} and \eqn{c_k} are the result of multiplying two vectors
of free parameters \eqn{\alpha} and \eqn{\gamma} by fixed matrices \eqn{T_a} and \eqn{T_c}, respectively;
\eqn{a_0} and \eqn{c_0} are fixed to 0 for identification;
and \eqn{C} is a normalizing factor to ensure that \eqn{\sum_k \mathrm P(\mathrm{pick}=k) = 1}.
}
\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)
}
\references{
Thissen, D., Cai, L., & Bock, R. D. (2010). The
Nominal Categories Item Response Model. In M. L. Nering &
R. Ostini (Eds.), \emph{Handbook of Polytomous Item Response
Theory Models} (pp. 43--75). Routledge.
}
\seealso{
Other response model:
\code{\link{rpf.drm}()},
\code{\link{rpf.gpcmp}()},
\code{\link{rpf.grmp}()},
\code{\link{rpf.grm}()},
\code{\link{rpf.lmp}()},
\code{\link{rpf.mcm}()}
}
\concept{response model}
|
