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)

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 d

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

Should theta values be saved?

Value

A list of class mcmc.sirt with following entries:
mcmcobjObject of class mcmc.list containing item parameters (b_marg and a_marg denote marginal item parameters) and person parameters (if requested)
summary.mcmcobjSummary 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.
icInformation criteria (DIC)
burninNumber of burnin iterations
iterTotal number of iterations
theta.chainSampled values of $\theta_p$ parameters
deviance.chainSampled values of deviance values
EAP.relEAP reliability
personData frame with EAP person parameter estimates for $\theta_p$ and their corresponding posterior standard deviations and for all testlet effects
datUsed data frame
weightsUsed student weights
...Further values

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

Examples

Run this code

#############################################################################
# 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 <- 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 <- 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 <- 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 <- mcmc.3pno.testlet( dat ,  guess.prior=1000*c(25,75) , testlets=testlets , 
              burnin=burnin, iter=iter )
summary(mod4)
plot(mod4, ask=TRUE, layout=2)

#############################################################################
# SIMULATED 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( rnorm( ITT , mean=0 , sd = 1 ) , 2 )
sd0 <- 1 # sd trait
sdt <- seq( 0 , 2 , len=TT ) # sd testlets
sdt <- sdt

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

#############################################################################
# SIMULATED 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( rnorm( ITT , mean=0 , sd = 1 ) , 2 )
a <- round( runif( ITT , 0.5 , 2 ) ,2)
sdt <- seq( 0 , 2 , len=TT ) # sd testlets
sdt <- sdt

# simulate theta
theta <- rnorm( N , sd = sd0 )
# simulate testlets
ut <- matrix(0,nrow=N , ncol=TT )
for (tt in 1:TT){ ut[,tt] <- 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( pnorm( aM*theta + ut + bM ) , N, ITT)
Y <- (matrix( 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 <- 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" ] )

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