R: MCMC Estimation of the Hierarchical IRT Model for...

mcmc.2pnoh {sirt}

R Documentation

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.

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)

[Package sirt version 4.1-15 Index]