likelihood.adjustment.RdApproximates 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.
likelihood.adjustment(likelihood, theta=NULL, prob.theta=NULL, adjfac=rep(1, nrow(likelihood)), extreme.item=5, target.EAP.rel=NULL, min_tuning=0.2, max_tuning=3, maxiter=100, conv=1e-04, trait.normal=TRUE)
| likelihood | A matrix containing the individual likelihood \(L(\bold{X}_p | \theta)\) or
an object of class |
|---|---|
| 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. |
| 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 |
| max_tuning | Maximum value of tuning parameter (if |
| maxiter | Maximum number of iterations (if |
| conv | Convergence criterion (if |
| trait.normal | Optional logical indicating whether the trait distribution should be
normally distributed (if |
Object of class IRT.likelihood.
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.
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) }