IRT.mle: Person Parameter Estimation

Description

Computes the maximum likelihood estimate (MLE), weighted likelihood estimate (WLE) and maximum aposterior estimate (MAP) of ability in unidimensional item response models (Penfield & Bergeron, 2005; Warm, 1989). Item response functions can be defined by the user.

Usage

IRT.mle(data, irffct, arg.list, theta=rep(0,nrow(data)), type="MLE",
     mu=0, sigma=1, maxiter=20, maxincr=3, h=0.001, convP=1e-04,
     maxval=9, progress=TRUE)

Arguments

data

Data frame with item responses

irffct

User defined item response (see Examples). Arguments must be specified in arg.list. The function must contain theta and ii (item index) as arguments.

theta

Initial ability estimate

arg.list

List of arguments for irffct.

type

Type of ability estimate. It can be "MLE" (the default), "WLE" or "MAP".

Mean of normal prior distribution (for type="MAP")

sigma

Standard deviation of normal prior distribution (for type="MAP")

maxiter

Maximum number of iterations

maxincr

Maximum increment

Numerical differentiation parameter

convP

Convergence criterion

maxval

Maximum ability value to be estimated

progress

Logical indicating whether iteration progress should be displayed

Value

Data frame with estimated abilities (est) and its standard error (se).

References

Penfield, R. D., & Bergeron, J. M. (2005). Applying a weighted maximum likelihood latent trait estimator to the generalized partial credit model. Applied Psychological Measurement, 29, 218-233.

Warm, T. A. (1989). Weighted likelihood estimation of ability in item response theory. Psychometrika, 54, 427-450.

Examples

Run this code

# NOT RUN {
#############################################################################
# EXAMPLE 1: Generalized partial credit model
#############################################################################

data(data.ratings1)
dat <- data.ratings1

# estimate model
mod1 <- sirt::rm.facets( dat[, paste0( "k",1:5) ], rater=dat$rater,
             pid=dat$idstud, maxiter=15)
# extract dataset and item parameters
data <- mod1$procdata$dat2.NA
a <- mod1$ipars.dat2$a
b <- mod1$ipars.dat2$b
theta0 <- mod1$person$EAP
# define item response function for item ii
calc.pcm <- function( theta, a, b, ii ){
    K <- ncol(b)
    N <- length(theta)
    matrK <- matrix( 0:K, nrow=N, ncol=K+1, byrow=TRUE)
    eta <- a[ii] * theta * matrK - matrix( c(0,b[ii,]), nrow=N, ncol=K+1, byrow=TRUE)
    eta <- exp(eta)
    probs <- eta / rowSums(eta, na.rm=TRUE)
    return(probs)
}
arg.list <- list("a"=a, "b"=b )

# MLE
abil1 <- sirt::IRT.mle( data, irffct=calc.pcm, theta=theta0, arg.list=arg.list )
str(abil1)
# WLE
abil2 <- sirt::IRT.mle( data, irffct=calc.pcm, theta=theta0, arg.list=arg.list, type="WLE")
str(abil2)
# MAP with prior distribution N(.2, 1.3)
abil3 <- sirt::IRT.mle( data, irffct=calc.pcm, theta=theta0, arg.list=arg.list,
              type="MAP", mu=.2, sigma=1.3 )
str(abil3)

#############################################################################
# EXAMPLE 2: Rasch model
#############################################################################

data(data.read)
dat <- data.read
I <- ncol(dat)

# estimate Rasch model
mod1 <- sirt::rasch.mml2( dat )
summary(mod1)

# define item response function
irffct <- function( theta, b, ii){
    eta <- exp( theta - b[ii] )
    probs <- eta / ( 1 + eta )
    probs <- cbind( 1 - probs, probs )
    return(probs)
}
# initial person parameters and item parameters
theta0 <- mod1$person$EAP
arg.list <- list( "b"=mod1$item$b  )

# estimate WLE
abil <- sirt::IRT.mle( data=dat, irffct=irffct, arg.list=arg.list,
            theta=theta0, type="WLE")
# compare with wle.rasch function
theta <- sirt::wle.rasch( dat, b=mod1$item$b )
cbind( abil[,1], theta$theta, abil[,2], theta$se.theta )

#############################################################################
# EXAMPLE 3: Ramsay quotient model
#############################################################################

data(data.read)
dat <- data.read
I <- ncol(dat)

# estimate Ramsay model
mod1 <- sirt::rasch.mml2( dat, irtmodel="ramsay.qm" )
summary(mod1)
# define item response function
irffct <- function( theta, b, K, ii){
    eta <- exp( theta / b[ii] )
    probs <- eta / ( K[ii] + eta )
    probs <- cbind( 1 - probs, probs )
    return(probs)
}
# initial person parameters and item parameters
theta0 <- exp( mod1$person$EAP )
arg.list <- list( "b"=mod1$item2$b, "K"=mod1$item2$K )
# estimate MLE
res <- sirt::IRT.mle( data=dat, irffct=irffct, arg.list=arg.list, theta=theta0,
            maxval=20, maxiter=50)
# }