Adjustment and Approximation of Individual Likelihood Functions
Approximates individual likelihood functions L(\bold{X}_p | θ)
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 | θ) or
an object of class |
theta |
Optional vector of (unidimensional) θ values |
prob.theta |
Optional vector of probabilities of θ 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.
## Not run: ############################################################################# # 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) ## End(Not run)
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