truescore.irt: Conversion of Trait Scores theta into True Scores tau ( theta...

In sirt: Supplementary Item Response Theory Models

Conversion of Trait Scores \theta into True Scores \tau ( \theta )

Description

This function computes the true score \tau=\tau(\theta)=\sum_{i=1}^I P_i(\theta) in a unidimensional item response model with I items. In addition, it also transforms conditional standard errors if they are provided.

Usage

truescore.irt(A, B, c=NULL, d=NULL, theta=seq(-3, 3, len=21),
    error=NULL, pid=NULL, h=0.001)

Arguments

A	Matrix or vector of item slopes. See Examples for polytomous responses.
B	Matrix or vector of item intercepts. Note that the entries in B refer to item intercepts and not to item difficulties.
c	Optional vector of guessing parameters
d	Optional vector of slipping parameters
theta	Vector of trait values
error	Optional vector of standard errors of trait
pid	Optional vector of person identifiers
h	Numerical differentiation parameter

Details

In addition, the function \pi(\theta)=\frac{1}{I} \cdot \tau( \theta) of the expected percent score is approximated by a logistic function

\pi ( \theta ) \approx l + ( u - l ) \cdot invlogit ( a \theta + b )

Value

A data frame with following columns:

truescore	True scores \tau=\tau ( \theta )
truescore.error	Standard errors of true scores
percscore	Expected correct scores which is \tau divided by the maximum true score
percscore.error	Standard errors of expected correct scores
lower	The l parameter
upper	The u parameter
a	The a parameter
b	The b parameter

Examples

#############################################################################
# EXAMPLE 1: Dataset with mixed dichotomous and polytomous responses
#############################################################################

data(data.mixed1)
dat <- data.mixed1

#****
# Model 1: Partial credit model
# estimate model with TAM package
library(TAM)
mod1 <- TAM::tam.mml( dat )
# estimate person parameter estimates
wmod1 <- TAM::tam.wle( mod1 )
wmod1 <- wmod1[ order(wmod1$theta), ]
# extract item parameters
A <- mod1$B[,-1,1]
B <- mod1$AXsi[,-1]
# person parameters and standard errors
theta <- wmod1$theta
error <- wmod1$error

# estimate true score transformation
dfr <- sirt::truescore.irt( A=A, B=B, theta=theta, error=error )

# plot different person parameter estimates and standard errors
par(mfrow=c(2,2))
plot( theta, dfr$truescore, pch=16, cex=.6, xlab=expression(theta), type="l",
    ylab=expression(paste( tau, "(",theta, ")" )), main="True Score Transformation" )
plot( theta, dfr$percscore, pch=16, cex=.6, xlab=expression(theta), type="l",
    ylab=expression(paste( pi, "(",theta, ")" )), main="Percent Score Transformation" )
points( theta, dfr$lower + (dfr$upper-dfr$lower)*
                stats::plogis(dfr$a*theta+dfr$b), col=2, lty=2)
plot( theta, error, pch=16, cex=.6, xlab=expression(theta), type="l",
    ylab=expression(paste("SE(",theta, ")" )), main="Standard Error Theta" )
plot( dfr$truescore, dfr$truescore.error, pch=16, cex=.6, xlab=expression(tau),
    ylab=expression(paste("SE(",tau, ")" ) ), main="Standard Error True Score Tau",
    type="l")
par(mfrow=c(1,1))

## Not run: 
#****
# Model 2: Generalized partial credit model
mod2 <- TAM::tam.mml.2pl( dat, irtmodel="GPCM")
# estimate person parameter estimates
wmod2 <- TAM::tam.wle( mod2 )
# extract item parameters
A <- mod2$B[,-1,1]
B <- mod2$AXsi[,-1]
# person parameters and standard errors
theta <- wmod2$theta
error <- wmod2$error
# estimate true score transformation
dfr <- sirt::truescore.irt( A=A, B=B, theta=theta, error=error )

#############################################################################
# EXAMPLE 2: Dataset Reading data.read
#############################################################################
data(data.read)

#****
# Model 1: estimate difficulty + guessing model
mod1 <- sirt::rasch.mml2( data.read, fixed.c=rep(.25,12) )
mod1$person <- mod1$person[ order( mod1$person$EAP), ]
# person parameters and standard errors
theta <- mod1$person$EAP
error <- mod1$person$SE.EAP
A <- rep(1,12)
B <- - mod1$item$b
c <- rep(.25,12)
# estimate true score transformation
dfr <- sirt::truescore.irt( A=A, B=B, theta=theta, error=error,c=c)

plot( theta, dfr$percscore, pch=16, cex=.6, xlab=expression(theta), type="l",
    ylab=expression(paste( pi, "(",theta, ")" )), main="Percent Score Transformation" )
points( theta, dfr$lower + (dfr$upper-dfr$lower)*
             stats::plogis(dfr$a*theta+dfr$b), col=2, lty=2)

#****
# Model 2: Rasch model
mod2 <- sirt::rasch.mml2( data.read  )
# person parameters and standard errors
theta <- mod2$person$EAP
error <- mod2$person$SE.EAP
A <- rep(1,12)
B <- - mod2$item$b
# estimate true score transformation
dfr <- sirt::truescore.irt( A=A, B=B, theta=theta, error=error )

## End(Not run)

sirt documentation built on May 29, 2024, 8:43 a.m.