Converts a \(\theta\) Score into a True Score \(\tau ( \theta)\)

Converts a \(\theta\) score into an unweighted true score \(\tau ( \theta)=\sum_i \sum_h h P_i ( \theta ) \). In addition, a weighted true score \(\tau ( \theta)=\sum_i \sum_h q_{ih} P_i ( \theta ) \) can also be computed by specifying item-category weights \(q_{ih}\) in the matrix Q.

IRT.truescore(object, iIndex=NULL, theta=NULL, Q=NULL)

Arguments

object

Object for which the CDM::IRT.irfprob S3 method is defined

iIndex

Optional vector with item indices

theta

Optional vector with \(\theta\) values

Q

Optional weighting matrix

Value

Data frame containing \(\theta\) values and corresponding true scores \(\tau( \theta ) \).

See also

See also sirt::truescore.irt for a conversion function for generalized partial credit models.

Examples

#############################################################################
# EXAMPLE 1: True score conversion for a test with polytomous items
#############################################################################

data(data.Students, package="CDM")
dat <- data.Students[, paste0("mj",1:4) ]

# fit partial credit model
mod1 <- TAM::tam.mml( dat,control=list(maxiter=20) )
summary(mod1)

# true score conversion
tmod1 <- TAM::IRT.truescore( mod1 )
round( tmod1, 4 )
# true score conversion with user-defined theta grid
tmod1b <- TAM::IRT.truescore( mod1, theta=seq( -8,8, len=33 ) )
# plot results
plot( tmod1$theta, tmod1$truescore, type="l",
            xlab=expression(theta), ylab=expression(tau( theta ) ) )
points( tmod1b$theta, tmod1b$truescore, pch=16, col="brown" )

if (FALSE) {
#############################################################################
# EXAMPLE 2: True scores with different category weightings
#############################################################################

data(data.timssAusTwn.scored)
dat <- data.timssAusTwn.scored
# extract item response data
dat <- dat[, grep("M03", colnames(dat) ) ]
# select items with do have maximum score of 2 (polytomous items)
ind <- which( apply( dat,  2, max, na.rm=TRUE )==2 )
I <- ncol(dat)
# define Q-matrix with scoring variant
Q <- matrix( 1, nrow=I, ncol=1 )
Q[ ind, 1 ] <- .5    # score of 0.5 for polyomous items

# estimate model
mod1 <- TAM::tam.mml( dat, Q=Q, irtmodel="PCM2", control=list( nodes=seq(-10,10,len=61) ) )
summary(mod1)

# true score with scoring (0,1,2) which is the default of the function
tmod1 <- TAM::IRT.truescore(mod1)
# true score with user specified weighting matrix
Q <- mod1$B[,,1]
tmod2 <- TAM::IRT.truescore(mod1, Q=Q)
}