Adjustment and Approximation of Individual Likelihood Functions
likelihood.adjustment.Rd
Approximates individual likelihood functions \(L(\bold{X}_p | \theta)\)
by normal distributions (see Mislevy, 1990). Extreme response patterns
are handled by adding pseudo-observations of items with extreme item
difficulties (see argument extreme.item
. The individual standard
deviations of the likelihood, used in the normal approximation, can be
modified by individual adjustment factors which are specified in adjfac
.
In addition, a reliability of the adjusted likelihood can be specified
in target.EAP.rel
.
Arguments
- likelihood
A matrix containing the individual likelihood \(L(\bold{X}_p | \theta)\) or an object of class
IRT.likelihood
.- theta
Optional vector of (unidimensional) \(\theta\) values
- prob.theta
Optional vector of probabilities of \(\theta\) trait distribution
- adjfac
Vector with individual adjustment factors of the standard deviations of the likelihood
- extreme.item
Item difficulties of two extreme pseudo items which are added as additional observed data to the likelihood. A large number (e.g.
extreme.item=15
) leaves the likelihood almost unaffected. See also Mislevy (1990).- target.EAP.rel
Target EAP reliability. An additional tuning parameter is estimated which adjusts the likelihood to obtain a pre-specified reliability.
- min_tuning
Minimum value of tuning parameter (if
! is.null(target.EAP.rel)
)- max_tuning
Maximum value of tuning parameter (if
! is.null(target.EAP.rel)
)- maxiter
Maximum number of iterations (if
! is.null(target.EAP.rel)
)- conv
Convergence criterion (if
! is.null(target.EAP.rel)
)- trait.normal
Optional logical indicating whether the trait distribution should be normally distributed (if
! is.null(target.EAP.rel)
).
References
Mislevy, R. (1990). Scaling procedures. In E. Johnson & R. Zwick (Eds.), Focusing the new design: The NAEP 1988 technical report (ETS RR 19-20). Princeton, NJ: Educational Testing Service.
Examples
if (FALSE) {
#############################################################################
# EXAMPLE 1: Adjustment of the likelihood | data.read
#############################################################################
library(CDM)
library(TAM)
data(data.read)
dat <- data.read
# define theta grid
theta.k <- seq(-6,6,len=41)
#*** Model 1: fit Rasch model in TAM
mod1 <- TAM::tam.mml( dat, control=list( nodes=theta.k) )
summary(mod1)
#*** Model 2: fit Rasch copula model
testlets <- substring( colnames(dat), 1, 1 )
mod2 <- sirt::rasch.copula2( dat, itemcluster=testlets, theta.k=theta.k)
summary(mod2)
# model comparison
IRT.compareModels( mod1, mod2 )
# extract EAP reliabilities
rel1 <- mod1$EAP.rel
rel2 <- mod2$EAP.Rel
# variance inflation factor
vif <- (1-rel2) / (1-rel1)
## > vif
## [1] 1.211644
# extract individual likelihood
like1 <- IRT.likelihood( mod1 )
# adjust likelihood from Model 1 to obtain a target EAP reliability of .599
like1b <- sirt::likelihood.adjustment( like1, target.EAP.rel=.599 )
# compare estimated latent regressions
lmod1a <- TAM::tam.latreg( like1, Y=NULL )
lmod1b <- TAM::tam.latreg( like1b, Y=NULL )
summary(lmod1a)
summary(lmod1b)
}