ltm (version 1.1-0)

# gpcm: Generalized Partial Credit Model - Polytomous IRT

## Description

Fits the Generalized Partial Credit model for ordinal polytomous data, under the Item Response Theory approach.

## Usage

gpcm(data, constraint = c("gpcm", "1PL", "rasch"), IRT.param = TRUE,
start.val = NULL, na.action = NULL, control = list())

## Arguments

data

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.

## 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".

## 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).

The model is defined as follows $$P_{ik}(z) = \frac{\exp \sum \limits_{c = 0}^k \beta_i (z - \beta_{ic}^*)}{ \sum \limits_{r = 0}^{m_i} \exp \sum \limits_{c = 0}^r \beta_i (z - \beta_{ic}^*)},$$ where $$P_{ik}(z)$$ denotes the probability of responding in category $$k$$ for item $$i$$, given the latent ability $$z$$, $$\beta_{ic}^*$$ are the item-category parameters, $$\beta_i$$ is the discrimination parameter, $$m_i$$ is the number of categories for item $$i$$, and $$\sum \limits_{c = 0}^0 \beta_i (z - \beta_{ic}^*) \equiv 0.$$

If constraint = "rasch", then the discrimination parameter $$\beta_i$$ is assumed equal for all items and fixed at one. If constraint = "1PL", then the discrimination parameter $$\beta_i$$ is assumed equal for all items but is estimated. If constraint = "gpcm", then each item has its one discrimination parameter $$\beta_i$$ that is estimated. See Examples for more info.

If IRT.param = FALSE, then the linear predictor is of the form $$\beta_i z + \beta_{ic}$$.

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.

## References

Masters, G. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149--174.

Muraki, E. (1992). A generalized partial credit model: application of an EM algorithm. Applied Psychological Measurement, 16, 159--176.

## See Also

coef.gpcm, fitted.gpcm, summary.gpcm, anova.gpcm, plot.gpcm, vcov.gpcm, GoF.gpcm, margins, factor.scores

## Examples

Run this code
# NOT RUN {
## 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

# }


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