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
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
|
\name{elementary_symmetric_functions}
\alias{elementary_symmetric_functions}
\title{Calculation of the Elementary Symmetric Functions and Their
Derivatives}
\description{
Calculation of \code{elementary_symmetric_functions} (ESFs), their first and,
in the case of dichotomous items, second derivatives with sum or
difference algorithm for the Rasch, rating scale and partial credit
model.
}
\usage{
elementary_symmetric_functions(par, order = 0L, log = TRUE,
diff = FALSE, engine = NULL)
}
\arguments{
\item{par}{numeric vector or a list. Either a vector of item difficulty
parameters of dichotomous items (Rasch model) or a list of
item-category parameters of polytomous items (rating scale and
partial credit model).}
\item{order}{integer between 0 and 2, specifying up to which derivative
the ESFs should be calculated. Please note, second order derivatives
are currently only possible for dichtomous items in an R
implementation \code{engine == "R".}}
\item{log}{logical. Are the parameters given in \code{par} on log
scale? Primarily used for internal recursive calls of
\code{elementary_symmetric_functions}.}
\item{diff}{logical. Should the first and second derivatives (if
requested) of the ESFs calculated with sum (\code{FALSE})
or difference algorithm (\code{TRUE}).}
\item{engine}{character, either \code{"C"} or \code{"R"}. If the
former, a C implementation is used to calculcate the ESFs and their
derivatives, otherwise (\code{"R"}) pure R code is used.}
}
\value{
\code{elementary_symmetric_function} returns a list of length 1 + \code{order}.
If \code{order = 0}, then the first (and only) element is a numeric
vector with the ESFs of order 0 to the maximum score possible with
the given parameters.
If \code{order = 1}, the second element of the list contains a
matrix, with the rows corresponding to the possible scores and the
columns corresponding to the derivatives with respect to the i-th
parameter of \code{par}.
For dichotomous items and \code{order = 2}, the third element of the
list contains an array with the second derivatives with respect to
every possible combination of two parameters given in \code{par}. The
rows of the individual matrices still correspond to the possibles
scores (orders) starting from zero.
}
\details{
Depending on the type of \code{par}, the elementary symmetric
functions for dichotomous (\code{par} is a numeric vector) or
polytomous items (\code{par} is a list) are calculated.
For dichotomous items, the summation and difference algorithm
published in Liou (1994) is used. For calculating the second order
derivatives, the equations proposed by Jansens (1984) are employed.
For polytomous items, the summation and difference algorithm published
by Fischer and Pococny (1994) is used (see also Fischer and Pococny,
1995).
}
\references{
Liou M (1994).
More on the Computation of Higher-Order Derivatives of the Elementary Symmetric Functions in the Rasch Model.
\emph{Applied Psychological Measurement}, \bold{18}, 53--62.
Jansen PGW (1984).
Computing the Second-Order Derivatives of the Symmetric Functions in the Rasch Model.
\emph{Kwantitatieve Methoden}, \bold{13}, 131--147.
Fischer GH, and Ponocny I (1994).
An Extension of the Partial Credit Model with an Application to the Measurement of Change.
\emph{Psychometrika}, \bold{59}(2), 177--192.
Fischer GH, and Ponocny I (1995).
\dQuote{Extended Rating Scale and Partial Credit Models for Assessing Change.}
In Fischer GH, and Molenaar IW (eds.).
\emph{Rasch Models: Foundations, Recent Developments, and Applications.}
}
\examples{
\donttest{
## zero and first order derivatives of 100 dichotomous items
di <- rnorm(100)
system.time(esfC <- elementary_symmetric_functions(di, order = 1))
## again with R implementation
system.time(esfR <- elementary_symmetric_functions(di, order = 1,
engine = "R"))
## are the results equal?
all.equal(esfC, esfR)
}
## calculate zero and first order elementary symmetric functions
## for 10 polytomous items with three categories each.
pi <- split(rnorm(20), rep(1:10, each = 2))
x <- elementary_symmetric_functions(pi)
## use difference algorithm instead and compare results
y <- elementary_symmetric_functions(pi, diff = TRUE)
all.equal(x, y)
}
\keyword{misc}
|
