anova: Anova method for fitted IRT models

Description

Performs a Likelihood Ratio Test between two nested IRT models.

Usage

# S3 method for gpcm
anova(object, object2, simulate.p.value = FALSE, 
    B = 200, verbose = getOption("verbose"), seed = NULL, …)
# S3 method for grm
anova(object, object2, …)
# S3 method for ltm
anova(object, object2, …)
# S3 method for rasch
anova(object, object2, …)
# S3 method for tpm
anova(object, object2, …)

Arguments

object

an object inheriting from either class gpcm, class grm, class ltm, class rasch or class tpm, representing the model under the null hypothesis.

object2

an object inheriting from either class gpcm, class grm, class ltm, class rasch, or class tpm, representing the model under the alternative hypothesis.

simulate.p.value

logical; if TRUE, the reported $p$ -value is based on a parametric Bootstrap approach.

the number of Bootstrap samples.

verbose

logical; if TRUE, information is printed in the console during the parametric Bootstrap.

seed

the seed to be used during the parametric Bootstrap; if NULL, a random seed is used.

…

additional arguments; currently none is used.

Value

An object of either class aov.gpcm, aov.grm, class aov.ltm or class aov.rasch with components,

nam0

the name of object.

the log-likelihood under the null hypothesis (object).

nb0

the number of parameter in object; returned only in aov.gpcm.

aic0

the AIC value for the model given by object.

bic0

the BIC value for the model given by object.

nam1

the name of object2.

the log-likelihood under the alternative hypothesis (object2).

nb1

the number of parameter in object; returned only in aov.gpcm.

aic1

the AIC value for the model given by object2.

bic1

the BIC value for the model given by object2.

LRT

the value of the Likelihood Ratio Test statistic.

the degrees of freedom for the test (i.e., the difference in the number of parameters).

p.value

the $p$ -value of the test.

Warning

The code does not check if the models are nested! The user is responsible to supply nested models in order the LRT to be valid.

When object2 represents a three parameter model, note that the null hypothesis in on the boundary of the parameter space for the guessing parameters. Thus, the Chi-squared reference distribution used by these function might not be totally appropriate.

Details

anova.gpcm() also includes the option to estimate the $p$ -value of the LRT using a parametric Bootstrap approach. In particular, B data sets are simulated under the null hypothesis (i.e., under the generalized partial credit model object), and both the null and alternative models are fitted and the value of LRT is computed. Then the $p$ -value is approximate using $[1 + \sum_{i = 1}^{B} I (T_{i} > T_{o b s})] / (B + 1),$ where $T_{o b s}$ is the value of the likelihood ratio statistic in the original data set, and $T_{i}$ the value of the statistic in the $i$ th Bootstrap sample.

In addition, when simulate.p.value = TRUE objects of class aov.gpcm have a method for the plot() generic function that produces a QQ plot comparing the Bootstrap sample of likelihood ration statistic with the asymptotic chi-squared distribution. For instance, you can use something like the following: lrt <- anova(obj1, obj2, simulate.p.value = TRUE); plot(lrt).

Examples

Run this code

# NOT RUN {
## LRT between the constrained and unconstrained GRMs 
## for the Science data:
fit0 <- grm(Science[c(1,3,4,7)], constrained = TRUE)
fit1 <- grm(Science[c(1,3,4,7)])
anova(fit0, fit1)


## LRT between the one- and two-factor models 
## for the WIRS data:
anova(ltm(WIRS ~ z1), ltm(WIRS ~ z1 + z2))


## An LRT between the Rasch and a constrained 
## two-parameter logistic model for the WIRS data: 
fit0 <- rasch(WIRS)
fit1 <- ltm(WIRS ~ z1, constraint = cbind(c(1, 3, 5), 2, 1))
anova(fit0, fit1)


## An LRT between the constrained (discrimination 
## parameter equals 1) and the unconstrained Rasch
## model for the LSAT data: 
fit0 <- rasch(LSAT, constraint = rbind(c(6, 1)))
fit1 <- rasch(LSAT)
anova(fit0, fit1)


## An LRT between the Rasch and the two-parameter 
## logistic model for the LSAT data: 
anova(rasch(LSAT), ltm(LSAT ~ z1))

# }

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