mcmc.3pno.testlet: 3PNO Testlet Model

Description

This function estimates the 3PNO testlet model (Wang, Bradlow & Wainer, 2002, 2007) by Markov Chain Monte Carlo methods (Glas, 2012).

Usage

mcmc.3pno.testlet(dat, testlets=rep(NA, ncol(dat)),
   weights=NULL, est.slope=TRUE, est.guess=TRUE, guess.prior=NULL,
   testlet.variance.prior=c(1, 0.2), burnin=500, iter=1000,
   N.sampvalues=1000, progress.iter=50, save.theta=FALSE, save.gamma.testlet=FALSE )

Value

A list of class mcmc.sirt with following entries:

mcmcobj: Object of class mcmc.list containing item parameters (b_marg and a_marg denote marginal item parameters) and person parameters (if requested)
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.
ic: Information criteria (DIC)
burnin: Number of burnin iterations
iter: Total number of iterations
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 and for all testlet effects
dat: Used data frame
weights: Used student weights
...: Further values

Arguments

dat: Data frame with dichotomous item responses for $N$ persons and $I$ items
testlets: An integer or character vector which indicates the allocation of items to testlets. Same entries corresponds to same testlets. If an entry is NA, then this item does not belong to any testlet.
weights: An optional vector with student sample weights
est.slope: Should item slopes be estimated? The default is TRUE.
est.guess: Should guessing parameters be estimated? The default is TRUE.
guess.prior: A vector of length two or a matrix with $I$ items and two columns which defines the beta prior distribution of guessing parameters. The default is a non-informative prior, i.e. the Beta(1,1) distribution.
testlet.variance.prior: A vector of length two which defines the (joint) prior for testlet variances assuming an inverse chi-squared distribution. The first entry is the effective sample size of the prior while the second entry defines the prior variance of the testlet. The default of c(1,.2) means that the prior sample size is 1 and the prior testlet variance is .2.
burnin: Number of burnin iterations
iter: 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.
save.theta: Logical indicating whether theta values should be saved
save.gamma.testlet: Logical indicating whether gamma values should be saved

Details

The testlet response model for person $p$ at item $i$ is defined as $$ P(X_{pi}=1 )=c_i + ( 1 - c_i ) \Phi ( a_i \theta_p + \gamma_{p,t(i)} + b_i ) \quad, \quad \theta_p \sim N ( 0,1 ), \gamma_{p,t(i)} \sim N( 0, \sigma^2_t ) $$

In case of est.slope=FALSE, all item slopes $a_i$ are set to 1. Then a variance $\sigma^2$ of the $\theta_p$ distribution is estimated which is called the Rasch testlet model in the literature (Wang & Wilson, 2005).

In case of est.guess=FALSE, all guessing parameters $c_i$ are set to 0.

After fitting the testlet model, marginal item parameters are calculated (integrating out testlet effects $\gamma_{p,t(i)}$) according the defining response equation $$ P(X_{pi}=1 )=c_i + ( 1 - c_i ) \Phi ( a_i^\ast \theta_p + b_i^\ast ) $$

References

Glas, C. A. W. (2012). Estimating and testing the extended testlet model. LSAC Research Report Series, RR 12-03.

Wainer, H., Bradlow, E. T., & Wang, X. (2007). Testlet response theory and its applications. Cambridge: Cambridge University Press.

Wang, W.-C., & Wilson, M. (2005). The Rasch testlet model. Applied Psychological Measurement, 29, 126-149.