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)

Value

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

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".
mu: 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
h: Numerical differentiation parameter
convP: Convergence criterion
maxval: Maximum ability value to be estimated
progress: Logical indicating whether iteration progress should be displayed

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

if (FALSE) {
#############################################################################
# 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)
}