immer_jml: Joint Maximum Likelihood Estimation for the Partial Credit...

In immer: Item Response Models for Multiple Ratings

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

Description

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 class 'immer_jml'
summary(object, digits=3, file=NULL, ...)

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

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

Arguments

dat	Data frame with polytomous item responses 0,1,…, K
A	Design matrix (items \times categories \times basis parameters). Entries for categories are for 1,…,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 θ values
...	Further arguments to be passed.

Details

The function uses the partial credit model as P(X_i=h | θ ) \propto \exp( h θ - b_{ih} ) with b_{ih}=∑_l a_{ihl} ξ_l where the values a_{ihl} are included in the design matrix A and ξ_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.

See Also

See TAM::tam.jml for joint maximum likelihood estimation. The varepsilon-adjustment is also implemented in sirt::mle.pcm.group.

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)

## Not run: 
#- 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)

## End(Not run)

immer documentation built on Aug. 22, 2022, 5:05 p.m.