sirt (version 1.9-0)

mcmc.2pnoh: 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?

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

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 modelled, 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 $\tau_k = \xi_k \cdot \omega_k$.

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

#############################################################################
# SIMULATED 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 ) + rnorm(I , sd = sqrt(sig2) )
a <- rep( omega.k , Ik ) + rnorm(I , sd = sqrt(nu2) )
theta1 <- rnorm(N)
t1 <- rep(1,N)
p1 <- pnorm( outer(t1,a) * ( theta1 - outer(t1,b) ) )
dat <- 1  * ( p1 > runif(N*I)  )
itemgroups <- rep( paste0("A" , 1:K ) , Ik )

# estimate model
mod <- 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" )