Joint Maximum Likelihood Estimation for the Partial Credit Model with a Design Matrix for Item Parameters and \(\varepsilon\)-Adjustment Bias Correction

Estimates the partial credit model with a design matrix for item parameters with joint maximum likelihood (JML). The \(\varepsilon\)-adjustment bias correction is implemented with reduces bias of the JML estimation method (Bertoli-Barsotti, Lando & Punzo, 2014).

Usage

immer_jml(dat, A=NULL, maxK=NULL, center_theta=TRUE, b_fixed=NULL, weights=NULL,
     irtmodel="PCM", pid=NULL, rater=NULL, eps=0.3, est_method="eps_adj", maxiter=1000,
     conv=1e-05, max_incr=3, incr_fac=1.1, maxiter_update=10, maxiter_line_search=6,
     conv_update=1e-05, verbose=TRUE, use_Rcpp=TRUE, shortcut=TRUE)

# S3 method for immer_jml
summary(object, digits=3, file=NULL, ...)

# S3 method for immer_jml
logLik(object, ...)

# S3 method for immer_jml
IRT.likelihood(object, theta=seq(-9,9,len=41), ...)

Arguments

dat: Data frame with polytomous item responses \(0,1,\ldots, K\)
A: Design matrix (items \(\times\) categories \(\times\) basis parameters). Entries for categories are for \(1,\ldots,K\)
maxK: Optional vector with maximum category per item
center_theta: Logical indicating whether the trait estimates should be centered
b_fixed: Matrix with fixed \(b\) parameters
irtmodel: Specified item response model. Can be one of the two partial credit model parametrizations PCM and PCM2.
weights: Optional vector of sampling weights
pid: Person identifier
rater: Optional rater identifier
eps: Adjustment parameter \(\varepsilon\)
est_method: Estimation method. Can be 'eps_adj' for the \(\varepsilon\)-adjustment, 'jml' for the JML without bias correction and 'jml_bc' for JML with bias correction.
maxiter: Maximum number of iterations
conv: Convergence criterion
max_incr: Maximum increment
incr_fac: Factor for shrinking increments from max_incr in every iteration
maxiter_update: Maximum number of iterations for parameter updates
maxiter_line_search: Maximum number of iterations within line search
conv_update: Convergence criterion for updates
verbose: Logical indicating whether convergence progress should be displayed
use_Rcpp: Logical indicating whether Rcpp package should be used for computation.
shortcut: Logical indicating whether a computational shortcut should be used for efficiency reasons
object: Object of class immer_jml
digits: Number of digits after decimal to print
file: Name of a file in which the output should be sunk
theta: Grid of \(\theta\) values
...: Further arguments to be passed.

Details

The function uses the partial credit model as \(P(X_i=h | \theta ) \propto \exp( h \theta - b_{ih} )\) with \(b_{ih}=\sum_l a_{ihl} \xi_l\) where the values \(a_{ihl}\) are included in the design matrix A and \(\xi_l\) denotes basis item parameters.

The adjustment parameter \(\varepsilon\) is applied to the sum score as the sufficient statistic for the person parameter. In more detail, extreme scores \(S_p=0\) (minimum score) or \(S_p=M_p\) (maximum score) are adjusted to \(S_p^\ast=\varepsilon\) or \(S_p^\ast=M_p - \varepsilon\), respectively. Therefore, the adjustment possesses more influence on parameter estimation for datasets with a small number of items.

Value

List with following entries

b: Item parameters \(b_{ih}\)
theta: Person parameters
theta_se: Standard errors for person parameters
xsi: Basis parameters
xsi_se: Standard errors for bias parameters
probs: Predicted item response probabilities
person: Data frame with person scores
dat_score: Scoring matrix
score_pers: Sufficient statistics for persons
score_items: Sufficient statistics for items
loglike: Log-likelihood value

References

Bertoli-Barsotti, L., Lando, T., & Punzo, A. (2014). Estimating a Rasch Model via fuzzy empirical probability functions. In D. Vicari, A. Okada, G. Ragozini & C. Weihs (Eds.). Analysis and Modeling of Complex Data in Behavioral and Social Sciences, Springer.

Examples

#############################################################################
# EXAMPLE 1: Rasch model
#############################################################################

data(data.read, package="sirt")
dat <- data.read

#---  Model 1: Rasch model with JML and epsilon-adjustment
mod1a <- immer::immer_jml(dat)
summary(mod1a)

if (FALSE) {
#- JML estimation, only handling extreme scores
mod1b <- immer::immer_jml( dat, est_method="jml")
summary(mod1b)

#- JML estimation with (I-1)/I bias correction
mod1c <- immer::immer_jml( dat, est_method="jml_bc" )
summary(mod1c)

# compare different estimators
round( cbind( eps=mod1a$xsi, JML=mod1b$xsi, BC=mod1c$xsi ), 2 )

#---  Model 2: LLTM by defining a design matrix for item difficulties
A <- array(0, dim=c(12,1,3) )
A[1:4,1,1] <- 1
A[5:8,1,2] <- 1
A[9:12,1,3] <- 1

mod2 <- immer::immer_jml(dat, A=A)
summary(mod2)

#############################################################################
# EXAMPLE 2: Partial credit model
#############################################################################

library(TAM)
data(data.gpcm, package="TAM")
dat <- data.gpcm

#-- JML estimation in TAM
mod0 <- TAM::tam.jml(resp=dat, bias=FALSE)
summary(mod0)

# extract design matrix
A <- mod0$A
A <- A[,-1,]

#-- JML estimation
mod1 <- immer::immer_jml(dat, A=A, est_method="jml")
summary(mod1)

#-- JML estimation with epsilon-adjusted bias correction
mod2 <- immer::immer_jml(dat, A=A, est_method="eps_adj")
summary(mod2)

#############################################################################
# EXAMPLE 3: Rating scale model with raters | Use design matrix from TAM
#############################################################################

data(data.ratings1, package="sirt")
dat <- data.ratings1

facets <- dat[,"rater", drop=FALSE]
resp <- dat[,paste0("k",1:5)]

#* Model 1: Rating scale model in TAM
formulaA <- ~ item + rater + step
mod1 <- TAM::tam.mml.mfr(resp=resp, facets=facets, formulaA=formulaA,
                pid=dat$idstud)
summary(mod1)

#* Model 2: Same model estimated with JML
resp0 <- mod1$resp
A0 <- mod1$A[,-1,]
mod2 <- immer::immer_jml(dat=resp0, A=A0, est_method="eps_adj")
summary(mod2)
}