3PNO Testlet Model

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 )

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 $$ P(X_{pi}=1 )=c_i + ( 1 - c_i ) \Phi ( a_i^\ast \theta_p + b_i^\ast ) $$

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

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.

Wang, X., Bradlow, E. T., & Wainer, H. (2002). A general Bayesian model for testlets: Theory and applications. Applied Psychological Measurement, 26, 109-128.

See also

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

Examples