gpcmodel

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Generalized Partial Credit Model Fitting Function

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.

Keywords
regression
Usage
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, ...)
Arguments
y

item response object that can be coerced (via as.matrix) to a numeric matrix with scores 0, 1, … Typically, either already a matrix, data frame, or dedicated object of class itemresp.

weights

an optional vector of weights (interpreted as case weights)

impact

an optional factor allowing for grouping the subjects (rows). If specified, a multiple-group model is fitted to account for impact (see details below). By default, no impact is modelled, i.e., a single-group model is used.

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 vcov = TRUE corresponds to vcov = "Oakes", see mirt for further options. If set to vcov = FALSE (or vcov = "none"), vcov() will return a matrix of NAs only.

nullcats

character string, specifying how items with null categories (i.e., categories not observed) should be treated. Currently only "downcode" is available, i.e., all categories above a null category are shifted down to close the observed gap(s).

start

an optional vector or list of starting values (see examples below).

method, maxit, reltol

control parameters for the optimizer employed by mirt for the EM algorithm.

further arguments passed to mirt or multipleGroup, respectively.

Details

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) or Debelak & Strobl (2018) 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.

Value

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 "nullcats" and without observations with zero weight,

items

logical vector of length ncol(y), indicating which items were used during estimation,

categories

list of length ncol(y), containing integer vectors starting from one to the number of categories minus one per item,

n

number of observations (with non-zero weights),

n_org

original number of observations in y,

weights

the weights used (if any),

na

logical indicating whether the data contain NAs,

nullcats

currently always NULL as eventual items with null categories are handled via "downcode",

impact

either NULL or the supplied impact variable with the levels reordered in decreasing order (if this has not been the case prior to fitting the model),

loglik

log-likelihood of the fitted model,

df

number of estimated (more precisely, returned) model parameters,

code

convergence code from mirt,

iterations

number of iterations used by mirt,

reltol

convergence threshold passed to mirt,

grouppars

the logical grouppars value,

type

the type of model restriction specified,

mirt

the mirt object fitted internally.

References

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. 10.18637/jss.v048.i06

Debelak R, Strobl C (2018). Investigating Measurement Invariance by Means of Parameter Instability Tests for 2PL and 3PL Models. Educational and Psychological Measurement, forthcoming. 10.1177/0013164418777784

Muraki E (1992). A Generalized Partial Credit Model: Application of an EM Algorithm. Applied Psychological Measurement, 16(2), 159--176.

See Also

pcmodel, rsmodel, plmodel, raschmodel, btmodel

Aliases
  • gpcmodel
  • print.gpcmodel
  • summary.gpcmodel
  • print.summary.gpcmodel
  • coef.gpcmodel
  • bread.gpcmodel
  • estfun.gpcmodel
  • logLik.gpcmodel
  • vcov.gpcmodel
Examples
# NOT RUN {
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)
}
# }
Documentation reproduced from package psychotools, version 0.5-1, License: GPL-2 | GPL-3

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