mcmc.2pnoh: MCMC Estimation of the Hierarchical IRT Model for...

In sirt: Supplementary Item Response Theory Models

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

Description

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.

See Also

S3 methods: summary.mcmc.sirt, plot.mcmc.sirt

The two-parameter normal ogive model can be estimated with mcmc.2pno.

Examples

## Not run: 
#############################################################################
# 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" )

## End(Not run)

sirt documentation built on May 29, 2024, 8:43 a.m.