gpcmodel | R Documentation |
gpcmodel
is a basic fitting function for generalized partial credit
models providing a wrapper around mirt
and
multipleGroup
relying on marginal maximum likelihood (MML)
estimation via the standard EM algorithm.
gpcmodel(y, weights = NULL, impact = NULL, type = c("GPCM", "PCM"),
grouppars = FALSE, vcov = TRUE, nullcats = "downcode",
start = NULL, method = "BFGS", maxit = 500, reltol = 1e-5, ...)
y |
item response object that can be coerced (via |
weights |
an optional vector of weights (interpreted as case weights) |
.
impact |
an optional |
type |
character string, specifying the type of model to be estimated. In addition to the default GPCM (generalized partial credit model) it is also possible to estimate a standard PCM (partial credit model) by marginal maximum likelihood (MML). |
grouppars |
logical. Should the estimated distributional group parameters of a multiple group model be included in the model parameters? |
vcov |
logical or character specifying the type of variance-covariance
matrix (if any) computed for the final model. The default |
nullcats |
character string, specifying how items with
null categories (i.e., categories not observed) should be treated. Currently
only |
start |
an optional vector or list of starting values (see examples below). |
method , maxit , reltol |
control parameters for the optimizer employed by
|
... |
further arguments passed to |
gpcmodel
provides a basic fitting function for generalized partial
credit models (GPCMs) providing a wrapper around mirt
(and
multipleGroup
, respectively) relying on MML estimation via
the standard EM algorithm (Bock & Aitkin, 1981). Models are estimated under the
slope/intercept parametrization, see e.g. Chalmers (2012). The probability of
person i
falling into category x_{ij}
of item j
out of all
categories p_{j}
is modelled as:
P(X_{ij} = x_{ij}|\theta_{i},a_{j},\boldsymbol{d_{j}}) =
\frac{\exp{(x_{ij}a_{j}\theta_{i} + d_{jx_{ij}})}}{\displaystyle\sum_{k = 0} ^
{p_{j}}\exp{(ka_{j}\theta_{i} + d_{jk})}}
Note that all d_{j0}
are fixed at 0. A reparametrization of the
intercepts to the classical IRT parametrization, see e.g. Muraki (1992), is
provided via threshpar
.
If an optional impact
variable is supplied, a multiple-group model of
the following form is being fitted: Item parameters are fixed to be equal
across the whole sample. For the first group of the impact
variable the
person parameters are fixed to follow the standard normal distribution. In the
remaining impact
groups, the distributional parameters (mean and
variance of a normal distribution) of the person parameters are
estimated freely. See e.g. Baker & Kim (2004, Chapter 11), Debelak & Strobl
(2019), or Schneider et al. (2022) for further details. To improve convergence of the model fitting
algorithm, the first level of the impact
variable should always correspond
to the largest group. If this is not the case, levels are re-ordered internally.
If grouppars
is set to TRUE
the freely estimated distributional
group parameters (if any) are returned as part of the model parameters.
Instead of the default GPCM, a standard partial credit model (PCM) can also
be estimated via MML by setting type = "PCM"
. In this case all slopes
are restricted to be equal across all items.
gpcmodel
returns an object of class "gpcmodel"
for which
several basic methods are available, including print
, plot
,
summary
, coef
, vcov
, logLik
, estfun
,
discrpar
, itempar
, threshpar
, and
personpar
.
gpcmodel
returns an S3 object of class "gpcmodel"
,
i.e., a list of the following components:
coefficients |
estimated model parameters in slope/intercept parametrization, |
vcov |
covariance matrix of the model parameters, |
data |
modified data, used for model-fitting, i.e., centralized so
that the first category is zero for all items, treated null categories as
specified via argument |
items |
logical vector of length |
categories |
list of length |
n |
number of observations (with non-zero weights), |
n_org |
original number of observations in |
weights |
the weights used (if any), |
na |
logical indicating whether the data contain |
nullcats |
currently always |
impact |
either |
loglik |
log-likelihood of the fitted model, |
df |
number of estimated (more precisely, returned) model parameters, |
code |
convergence code from |
iterations |
number of iterations used by |
reltol |
convergence threshold passed to |
grouppars |
the logical |
type |
the |
mirt |
the |
call |
original function call. |
Baker FB, Kim SH (2004). Item Response Theory: Parameter Estimation Techniques. Chapman & Hall/CRC, Boca Raton.
Bock RD, Aitkin M (1981). Marginal Maximum Likelihood Estimation of Item Parameters: Application of an EM Algorithm. Psychometrika, 46(4), 443–459.
Chalmers RP (2012). mirt: A Multidimensional Item Response Theory Package for the R Environment. Journal of Statistical Software, 48(6), 1–29. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.18637/jss.v048.i06")}
Debelak R, Strobl C (2019). Investigating Measurement Invariance by Means of Parameter Instability Tests for 2PL and 3PL Models. Educational and Psychological Measurement, 79(2), 385–398. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1177/0013164418777784")}
Muraki E (1992). A Generalized Partial Credit Model: Application of an EM Algorithm. Applied Psychological Measurement, 16(2), 159–176.
Schneider L, Strobl C, Zeileis A, Debelak R (2022). An R Toolbox for Score-Based Measurement Invariance Tests in IRT Models. Behavior Research Methods, forthcoming. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.3758/s13428-021-01689-0")}
pcmodel
, rsmodel
, nplmodel
,
raschmodel
, btmodel
if(requireNamespace("mirt")) {
o <- options(digits = 4)
## mathematics 101 exam results
data("MathExam14W", package = "psychotools")
## generalized partial credit model
gpcm <- gpcmodel(y = MathExam14W$credit)
summary(gpcm)
## how to specify starting values as a vector of model parameters
st <- coef(gpcm)
gpcm <- gpcmodel(y = MathExam14W$credit, start = st)
## or a list containing a vector of slopes and a list of intercept vectors
## itemwise
set.seed(0)
st <- list(a = rlnorm(13, 0, 0.0625), d = replicate(13, rnorm(2, 0, 1), FALSE))
gpcm <- gpcmodel(y = MathExam14W$credit, start = st)
## visualizations
plot(gpcm, type = "profile")
plot(gpcm, type = "regions")
plot(gpcm, type = "piplot")
plot(gpcm, type = "curves", xlim = c(-6, 6))
plot(gpcm, type = "information", xlim = c(-6, 6))
## visualizing the IRT parametrization
plot(gpcm, type = "curves", xlim = c(-6, 6), items = 1)
abline(v = threshpar(gpcm)[[1]])
abline(v = itempar(gpcm)[1], lty = 2)
options(digits = o$digits)
}
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