a data.frame or a numeric matrix of manifest variables.
constraint
a character string specifying which version of the Generalized Partial Credit Model to fit. See Details and
Examples for more info.
IRT.param
logical; if TRUE then the coefficients' estimates are reported under the
usual IRT parameterization. See Details for more info.
start.val
a list of starting values or the character string "random". If a list, each one of its
elements corresponds to each item and should contain a numeric vector with initial values for the
threshold parameters and discrimination parameter; even if constraint = "rasch" or constraint = "1PL",
the discrimination parameter should be provided for all the items. If "random", random starting values are computed.
na.action
the na.action to be used on data; default NULL the model uses the available
cases, i.e., it takes into account the observed part of sample units with missing values (valid under MAR
mechanisms if the model is correctly specified).
control
a named list of control values with components,
iter.qN
the number of quasi-Newton iterations. Default 150.
GHk
the number of Gauss-Hermite quadrature points. Default 21.
optimizer
which optimization routine to use; options are "optim" and "nlminb", the latter being the default.
optimMethod
the optimization method to be used in optim(). Default is "BFGS".
numrDeriv
which numerical derivative algorithm to use to approximate the Hessian matrix; options are "fd" for
forward difference approximation and "cd" for central difference approximation. Default is "fd".
epsHes
step size to be used in the numerical derivative. Default is 1e-06. If you choose numrDeriv = "cd", then
change this to a larger value, e.g., 1e-03 or 1e-04.
parscale
the parscale control argument of optim(). Default is 0.5 for all parameters.
verbose
logical; if TRUE info about the optimization procedure are printed.
Details
The Generalized Partial Credit Model is an IRT model, that can handle ordinal manifest variables.
This model was discussed by Masters (1982) and it was extended by Muraki (1992).
where P_{ik}(z) denotes the
probability of responding in category k for item i, given the latent ability z, β_{ic}^* are the item-category
parameters, β_i is the discrimination parameter, m_i is the number of categories for item i, and
∑ limits_{c = 0}^0 β_i (z - β_{ic}^*) equiv 0.
If constraint = "rasch", then the discrimination parameter β_i is assumed equal for all items and fixed at one. If
constraint = "1PL", then the discrimination parameter β_i is assumed equal for all items but is estimated.
If constraint = "gpcm", then each item has its one discrimination parameter β_i that is estimated. See
Examples for more info.
If IRT.param = FALSE, then the linear predictor is of the form β_i z + β_{ic}.
The fit of the model is based on approximate marginal Maximum Likelihood, using the Gauss-Hermite quadrature rule
for the approximation of the required integrals.
Value
An object of class gpcm with components,
coefficients
a named list with components the parameter values at convergence for each item.
log.Lik
the log-likelihood value at convergence.
convergence
the convergence identifier returned by optim() or nlminb().
hessian
the approximate Hessian matrix at convergence.
counts
the number of function and gradient evaluations used by the quasi-Newton algorithm.
patterns
a list with two components: (i) X: a numeric matrix
that contains the observed response patterns, and (ii) obs: a numeric vector that contains the observed
frequencies for each observed response pattern.
GH
a list with two components used in the Gauss-Hermite rule: (i) Z: a numeric matrix that contains
the abscissas, and (ii) GHw: a numeric vector that contains the corresponding weights.
max.sc
the maximum absolute value of the score vector at convergence.
constraint
the value of the constraint argument.
IRT.param
the value of the IRT.param argument.
X
a copy of the response data matrix.
control
the values used in the control argument.
na.action
the value of the na.action argument.
call
the matched call.
Warning
In case the Hessian matrix at convergence is not positive definite try to re-fit the model by specifying the starting values or using
start.val = "random".
Note
gpcm() can also handle binary items and can be used instead of rasch and ltm though it is less
efficient. However, gpcm() can handle a mix of dichotomous and polytomous items that neither rasch nor
ltm can.
## The Generalized Partial Credit Model for the Science data:
gpcm(Science[c(1,3,4,7)])
## The Generalized Partial Credit Model for the Science data,
## assuming equal discrimination parameters across items:
gpcm(Science[c(1,3,4,7)], constraint = "1PL")
## The Generalized Partial Credit Model for the Science data,
## assuming equal discrimination parameters across items
## fixed at 1:
gpcm(Science[c(1,3,4,7)], constraint = "rasch")
## more examples can be found at:
## http://wiki.r-project.org/rwiki/doku.php?id=packages:cran:ltm#sample_analyses