mcmc.2pnoh.Rd
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).
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)
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 |
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? |
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
\( \tau_k=\xi_k \cdot \omega_k \).
A list of class mcmc.sirt
with following entries:
Object of class mcmc.list
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.
Number of burnin iterations
Total number of iterations
Sampled values of \(\alpha_{ik}\) parameters
Sampled values of \(\beta_{ik}\) parameters
Sampled values of \(\xi_{k}\) parameters
Sampled values of \(\omega_{k}\) parameters
Sampled values of \(\sigma\) parameter
Sampled values of \(\nu\) parameter
Sampled values of \(\theta_p\) parameters
Sampled values of Deviance values
EAP reliability
Data frame with EAP person parameter estimates for \(\theta_p\) and their corresponding posterior standard deviations
Used data frame
Used student weights
Further values
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.
S3 methods: summary.mcmc.sirt
, plot.mcmc.sirt
The two-parameter normal ogive model can be estimated with
mcmc.2pno
.
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" ) }