mcmc.3pno.testlet: 3PNO Testlet Model
In sirt: Supplementary Item Response Theory Models

mcmc.3pno.testlet

R Documentation

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 )

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.

Examples

## Not run: 
#############################################################################
# EXAMPLE 1: Dataset Reading
#############################################################################
data(data.read)
dat <- data.read
I <- ncol(dat)

# set burnin and total number of iterations here (CHANGE THIS!)
burnin <- 200
iter <- 500

#***
# Model 1: 1PNO model
mod1 <- sirt::mcmc.3pno.testlet( dat,  est.slope=FALSE, est.guess=FALSE,
            burnin=burnin, iter=iter )
summary(mod1)
plot(mod1,ask=TRUE) # plot MCMC chains in coda style
plot(mod1,ask=TRUE, layout=2) # plot MCMC output in different layout

#***
# Model 2: 3PNO model with Beta(5,17) prior for guessing parameters
mod2 <- sirt::mcmc.3pno.testlet( dat,  guess.prior=c(5,17),
               burnin=burnin, iter=iter )
summary(mod2)

#***
# Model 3: Rasch (1PNO) testlet model
testlets <- substring( colnames(dat), 1, 1 )
mod3 <- sirt::mcmc.3pno.testlet( dat,  testlets=testlets,  est.slope=FALSE,
           est.guess=FALSE, burnin=burnin, iter=iter )
summary(mod3)

#***
# Model 4: 3PNO testlet model with (almost) fixed guessing parameters .25
mod4 <- sirt::mcmc.3pno.testlet( dat,  guess.prior=1000*c(25,75), testlets=testlets,
              burnin=burnin, iter=iter )
summary(mod4)
plot(mod4, ask=TRUE, layout=2)

#############################################################################
# EXAMPLE 2: Simulated data according to the Rasch testlet model
#############################################################################
set.seed(678)

N <- 3000   # number of persons
I <- 4      # number of items per testlet
TT <- 3     # number of testlets

ITT <- I*TT
b <- round( stats::rnorm( ITT, mean=0, sd=1 ), 2 )
sd0 <- 1 # sd trait
sdt <- seq( 0, 2, len=TT ) # sd testlets

# simulate theta
theta <- stats::rnorm( N, sd=sd0 )
# simulate testlets
ut <- matrix(0,nrow=N, ncol=TT )
for (tt in 1:TT){
    ut[,tt] <- stats::rnorm( N, sd=sdt[tt] )
}
ut <- ut[, rep(1:TT,each=I) ]
# calculate response probability
prob <- matrix( stats::pnorm( theta + ut + matrix( b, nrow=N, ncol=ITT,
            byrow=TRUE ) ), N, ITT)
Y <- (matrix( stats::runif(N*ITT), N, ITT) < prob )*1
colMeans(Y)

# define testlets
testlets <- rep(1:TT, each=I )

burnin <- 300
iter <- 1000

#***
# Model 1: 1PNO model (without testlet structure)
mod1 <- sirt::mcmc.3pno.testlet( dat=Y,  est.slope=FALSE, est.guess=FALSE,
            burnin=burnin, iter=iter, testlets=testlets )
summary(mod1)

summ1 <- mod1$summary.mcmcobj
# compare item parameters
cbind( b, summ1[ grep("b", summ1$parameter ), "Mean" ] )
# Testlet standard deviations
cbind( sdt, summ1[ grep("sigma\.testlet", summ1$parameter ), "Mean" ] )

#***
# Model 2: 1PNO model (without testlet structure)
mod2 <- sirt::mcmc.3pno.testlet( dat=Y,  est.slope=TRUE, est.guess=FALSE,
           burnin=burnin, iter=iter, testlets=testlets )
summary(mod2)

summ2 <- mod2$summary.mcmcobj
# compare item parameters
cbind( b, summ2[ grep("b\[", summ2$parameter ), "Mean" ] )
# item discriminations
cbind( sd0, summ2[ grep("a\[", summ2$parameter ), "Mean" ] )
# Testlet standard deviations
cbind( sdt, summ2[ grep("sigma\.testlet", summ2$parameter ), "Mean" ] )

#############################################################################
# EXAMPLE 3: Simulated data according to the 2PNO testlet model
#############################################################################
set.seed(678)

N <- 3000    # number of persons
I <- 3      # number of items per testlet
TT <- 5    # number of testlets

ITT <- I*TT
b <- round( stats::rnorm( ITT, mean=0, sd=1 ), 2 )
a <- round( stats::runif( ITT, 0.5, 2 ),2)
sdt <- seq( 0, 2, len=TT ) # sd testlets
sd0 <- 1

# simulate theta
theta <- stats::rnorm( N, sd=sd0 )
# simulate testlets
ut <- matrix(0,nrow=N, ncol=TT )
for (tt in 1:TT){
   ut[,tt] <- stats::rnorm( N, sd=sdt[tt] )
}
ut <- ut[, rep(1:TT,each=I) ]
# calculate response probability
bM <- matrix( b, nrow=N, ncol=ITT, byrow=TRUE )
aM <- matrix( a, nrow=N, ncol=ITT, byrow=TRUE )
prob <- matrix( stats::pnorm( aM*theta + ut + bM ), N, ITT)
Y <- (matrix( stats::runif(N*ITT), N, ITT) < prob )*1
colMeans(Y)

# define testlets
testlets <- rep(1:TT, each=I )

burnin <- 500
iter <- 1500

#***
# Model 1: 2PNO model
mod1 <- sirt::mcmc.3pno.testlet( dat=Y,  est.slope=TRUE, est.guess=FALSE,
             burnin=burnin, iter=iter, testlets=testlets )
summary(mod1)

summ1 <- mod1$summary.mcmcobj
# compare item parameters
cbind( b, summ1[ grep("b\[", summ1$parameter ), "Mean" ] )
# item discriminations
cbind( a, summ1[ grep("a\[", summ1$parameter ), "Mean" ] )
# Testlet standard deviations
cbind( sdt, summ1[ grep("sigma\.testlet", summ1$parameter ), "Mean" ] )

## End(Not run)

sirt documentation built on May 29, 2024, 8:43 a.m.