MCMC Estimation of the Hierarchical IRT Model for Criterion-Referenced Measurement

This function estimates the hierarchical IRT model for criterion-referenced measurement which is based on a two-parameter normal ogive response function (Janssen, Tuerlinckx, Meulders & de Boeck, 2000).

Usage

mcmc.2pnoh(dat, itemgroups, prob.mastery=c(.5,.8), weights=NULL,
      burnin=500, iter=1000, N.sampvalues=1000,
      progress.iter=50, prior.variance=c(1,1), save.theta=FALSE)

Arguments

dat: Data frame with dichotomous item responses
itemgroups: Vector with characters or integers which define the criterion to which an item is associated.
prob.mastery: Probability levels which define nonmastery, transition and mastery stage (see Details)
weights: An optional vector with student sample weights
burnin: Number of burnin iterations
iter: Total number of iterations
N.sampvalues: Maximum number of sampled values to save
progress.iter: Display progress every progress.iter-th iteration. If no progress display is wanted, then choose progress.iter larger than iter.
prior.variance: Scale parameter of the inverse gamma distribution for the $\sigma^2$ and $\nu^2$ item variance parameters
save.theta: Should theta values be saved?

Details

The hierarchical IRT model for criterion-referenced measurement (Janssen et al., 2000) assumes that every item $i$ intends to measure a criterion $k$. The item response function is defined as $$ P(X_{pik}=1 | \theta_p )= \Phi [ \alpha_{ik} ( \theta_p - \beta_{ik} ) ] \quad, \quad \theta_p \sim N(0,1) $$ Item parameters $(\alpha_{ik},\beta_{ik})$ are hierarchically modeled, i.e. $$ \beta_{ik} \sim N( \xi_k, \sigma^2 ) \quad \mbox{and} \quad \alpha_{ik} \sim N( \omega_k, \nu^2 ) $$ In the mcmc.list output object, also the derived parameters $d_{ik}=\alpha_{ik} \beta_{ik}$ and $\tau_k=\xi_k \omega_k$ are calculated. Mastery and nonmastery probabilities are based on a reference item $Y_{k}$ of criterion $k$ and a response function $$ P(Y_{pk}=1 | \theta_p )= \Phi [ \omega_{k} ( \theta_p - \xi_{k} ) ] \quad, \quad \theta_p \sim N(0,1) $$ With known item parameters and person parameters, response probabilities of criterion $k$ are calculated. If a response probability of criterion $k$ is larger than prob.mastery[2], then a student is defined as a master. If this probability is smaller than prob.mastery[1], then a student is a nonmaster. In all other cases, students are in a transition stage.

In the mcmcobj output object, the parameters d[i] are defined by $d_{ik}=\alpha_{ik} \cdot \beta_{ik}$ while tau[k] are defined by $ \tau_k=\xi_k \cdot \omega_k $.

Value

A list of class mcmc.sirt with following entries:

mcmcobj: Object of class mcmc.list
summary.mcmcobj: Summary of the mcmcobj object. In this summary the Rhat statistic and the mode estimate MAP is included. The variable PercSEratio indicates the proportion of the Monte Carlo standard error in relation to the total standard deviation of the posterior distribution.
burnin: Number of burnin iterations
iter: Total number of iterations
alpha.chain: Sampled values of $\alpha_{ik}$ parameters
beta.chain: Sampled values of $\beta_{ik}$ parameters
xi.chain: Sampled values of $\xi_{k}$ parameters
omega.chain: Sampled values of $\omega_{k}$ parameters
sigma.chain: Sampled values of $\sigma$ parameter
nu.chain: Sampled values of $\nu$ parameter
theta.chain: Sampled values of $\theta_p$ parameters
deviance.chain: Sampled values of Deviance values
EAP.rel: EAP reliability
person: Data frame with EAP person parameter estimates for $\theta_p$ and their corresponding posterior standard deviations
dat: Used data frame
weights: Used student weights
...: Further values

References

Janssen, R., Tuerlinckx, F., Meulders, M., & De Boeck, P. (2000). A hierarchical IRT model for criterion-referenced measurement. Journal of Educational and Behavioral Statistics, 25, 285-306.

Examples

if (FALSE) {
#############################################################################
# EXAMPLE 1: Simulated data according to Janssen et al. (2000, Table 2)
#############################################################################

N <- 1000
Ik <- c(4,6,8,5,9,6,8,6,5)
xi.k <- c( -.89, -1.13, -1.23, .06, -1.41, -.66, -1.09, .57, -2.44)
omega.k <- c(.98, .91, .76, .74, .71, .80, .79, .82, .54)

# select 4 attributes
K <- 4
Ik <- Ik[1:K] ; xi.k <- xi.k[1:K] ; omega.k <- omega.k[1:K]
sig2 <- 3.02
nu2 <- .09
I <- sum(Ik)
b <- rep( xi.k, Ik ) + stats::rnorm(I, sd=sqrt(sig2) )
a <- rep( omega.k, Ik ) + stats::rnorm(I, sd=sqrt(nu2) )
theta1 <- stats::rnorm(N)
t1 <- rep(1,N)
p1 <- stats::pnorm( outer(t1,a) * ( theta1 - outer(t1,b) ) )
dat <- 1  * ( p1 > stats::runif(N*I)  )
itemgroups <- rep( paste0("A", 1:K ), Ik )

# estimate model
mod <- sirt::mcmc.2pnoh(dat, itemgroups, burnin=200, iter=1000 )
# summary
summary(mod)
# plot
plot(mod$mcmcobj, ask=TRUE)
# write coda files
mcmclist2coda( mod$mcmcobj, name="simul_2pnoh" )
}