grm  R Documentation 
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())
data 
a 
constrained 
logical; if 
IRT.param 
logical; if 
Hessian 
logical; if 
start.val 
a list of starting values or the character string 
na.action 
the 
control 
a list of control values,

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
logit(γ_ik) = beta_i z  beta_ik,
where γ_{ik} denotes the cumulative
probability of a response in category kth or lower to the ith item, given the latent ability z.
If constrained = TRUE
it is assumed that β_i = β for all i.
If IRT.param = TRUE
, then the parameters estimates are reported under the usual IRT parameterization,
i.e.,
logit (γ_ik) = 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 GaussHermite quadrature rule for the approximation of the required integrals.
An object of class grm
with components,
coefficients 
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 
log.Lik 
the loglikelihood value at convergence. 
convergence 
the convergence identifier returned by 
hessian 
the approximate Hessian matrix at convergence returned by 
counts 
the number of function and gradient evaluations used by the quasiNewton algorithm. 
patterns 
a list with two components: (i) 
GH 
a list with two components used in the GaussHermite rule: (i) 
max.sc 
the maximum absolute value of the score vector at convergence. 
constrained 
the value of the 
IRT.param 
the value of the 
X 
a copy of the response data matrix. 
control 
the values used in the 
na.action 
the value of the 
call 
the matched call. 
In case the Hessian matrix at convergence is not positive definite try to refit the model,
using start.val = "random"
.
grm()
returns the parameter estimates such that the discrimination parameter for the first item
beta_1 is positive.
When the coefficients' estimates are reported under the usual IRT parameterization (i.e., IRT.param = TRUE
),
their standard errors are calculated using the Delta method.
grm()
can also handle binary items, which should be coded as ‘1, 2’ instead of ‘0, 1’.
Some parts of the code used for the calculation of the loglikelihood and the score vector have been based
on polr()
from package MASS.
Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl
Baker, F. and Kim, SH. (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 doi: 10.18637/jss.v017.i05
coef.grm
,
fitted.grm
,
summary.grm
,
anova.grm
,
plot.grm
,
vcov.grm
,
margins
,
factor.scores
## 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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