gpcm {ltm} | R Documentation |
Fits the Generalized Partial Credit model for ordinal polytomous data, under the Item Response Theory approach.
gpcm(data, constraint = c("gpcm", "1PL", "rasch"), IRT.param = TRUE, start.val = NULL, na.action = NULL, control = list())
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,
|
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 β_i (z - β_{ic}^*)}{ sum limits_{r = 0}^{m_i} exp sum limits_{c = 0}^r β_i (z - β_{ic}^*)},
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
sum 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.
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. |
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"
.
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.
Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl
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.
coef.gpcm
,
fitted.gpcm
,
summary.gpcm
,
anova.gpcm
,
plot.gpcm
,
vcov.gpcm
,
GoF.gpcm
,
margins
,
factor.scores
## 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