ltm (version 0.9-3)

anova: Anova method for fitted IRT models

Description

Performs a Likelihood Ratio Test between two nested IRT models.

Usage

## S3 method for class 'gpcm':
anova(object, object2, simulate.p.value = FALSE, 
    B = 200, verbose = getOption("verbose"), seed = NULL, ...)

## S3 method for class 'grm':
anova(object, object2, \dots)

## S3 method for class 'ltm':
anova(object, object2, \dots)

## S3 method for class 'rasch':
anova(object, object2, \dots)

## S3 method for class 'tpm':
anova(object, object2, \dots)

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.

B

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.
• L0
  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.
• L1
  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.
• df
  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 $[1 + \sum\limits_{i=1}^B I(T_i > T_{obs})] / (B + 1),$ where $T_{obs}$ 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).

See Also

GoF.gpcm, GoF.rasch, gpcm, grm, ltm, rasch, tpm

Examples

## 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))