Fits the Graded Response model for ordinal polytomous data, under the Item Response Theory approach.
grm(data, constrained = FALSE, IRT.param = TRUE, Hessian = FALSE,
start.val = NULL, na.action = NULL, control = list())a data.frame (that will be converted to a numeric matrix using
data.matrix()) or a numeric matrix of manifest variables.
logical; if TRUE the model with equal discrimination parameters across items is fitted.
See Examples for more info.
logical; if TRUE then the coefficients' estimates are reported under the
usual IRT parameterization. See Details for more info.
logical; if TRUE the Hessian matrix is computed.
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
extremity parameters and discrimination parameter; even if constrained = TRUE the discrimination
parameter should be provided for all the items. If "random" random starting values are computed.
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)..
a list of control values,
the number of quasi-Newton iterations. Default 150.
the number of Gauss-Hermite quadrature points. Default 21.
the optimization method to be used in optim(). Default "BFGS".
logical; if TRUE info about the optimization procedure are printed.
numeric value indicating the number of digits used in abbreviating the Item's names. Default 6.
An object of class grm with components,
a named list with components the parameter values at convergence for each item. These are always
the estimates of \(\beta_{ik}, \beta_i\) parameters, even if IRT.param = TRUE.
the log-likelihood value at convergence.
the convergence identifier returned by optim().
the approximate Hessian matrix at convergence returned by optim(); returned
only if Hessian = TRUE.
the number of function and gradient evaluations used by the quasi-Newton algorithm.
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.
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.
the maximum absolute value of the score vector at convergence.
the value of the constrained argument.
the value of the IRT.param argument.
a copy of the response data matrix.
the values used in the control argument.
the value of the na.action argument.
the matched call.
In case the Hessian matrix at convergence is not positive definite try to re-fit the model,
using start.val = "random".
The Graded Response Model is a type of polytomous IRT model, specifically designed for ordinal manifest variables. This model was first discussed by Samejima (1969) and it is mainly used in cases where the assumption of ordinal levels of response options is plausible.
The model is defined as follows $$\log\left(\frac{\gamma_{ik}}{1-\gamma_{ik}}\right) = \beta_i z -
\beta_{ik},$$ where \(\gamma_{ik}\) denotes the cumulative
probability of a response in category \(k\)th or lower to the \(i\)th item, given the latent ability \(z\).
If constrained = TRUE it is assumed that \(\beta_i = \beta\) for all \(i\).
If IRT.param = TRUE, then the parameters estimates are reported under the usual IRT parameterization,
i.e., $$\log\left(\frac{\gamma_{ik}}{1-\gamma_{ik}}\right) = \beta_i (z - \beta_{ik}^*),$$ where \(\beta_{ik}^* = \beta_{ik} / \beta_i\).
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.
Baker, F. and Kim, S-H. (2004) Item Response Theory, 2nd ed. New York: Marcel Dekker.
Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph Supplement, 34, 100--114.
Rizopoulos, D. (2006) ltm: An R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17(5), 1--25. URL http://www.jstatsoft.org/v17/i05/
coef.grm,
fitted.grm,
summary.grm,
anova.grm,
plot.grm,
vcov.grm,
margins,
factor.scores
# NOT RUN {
## The Graded Response model for the Science data:
grm(Science[c(1,3,4,7)])
## The Graded Response model for the Science data,
## assuming equal discrimination parameters across items:
grm(Science[c(1,3,4,7)], constrained = TRUE)
## The Graded Response model for the Environment data
grm(Environment)
# }
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