3PNO Testlet Model
This function estimates the 3PNO testlet model (Wang, Bradlow & Wainer, 2002, 2007) by Markov Chain Monte Carlo methods (Glas, 2012).
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 )
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 |
weights |
An optional vector with student sample weights |
est.slope |
Should item slopes be estimated? The default is |
est.guess |
Should guessing parameters be estimated? The default is |
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 |
burnin |
Number of burnin iterations |
iter |
Number of iterations |
N.sampvalues |
Maximum number of sampled values to save |
progress.iter |
Display progress every |
save.theta |
Logical indicating whether theta values should be saved |
save.gamma.testlet |
Logical indicating whether gamma values should be saved |
The testlet response model for person p at item i is defined as
P(X_{pi}=1 )=c_i + ( 1 - c_i ) Φ ( a_i θ_p + γ_{p,t(i)} + b_i ) \quad, \quad θ_p \sim N ( 0,1 ), γ_{p,t(i)} \sim N( 0, σ^2_t )
In case of est.slope=FALSE, all item slopes a_i are set to 1. Then
a variance σ^2 of the θ_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 γ_{p,t(i)}) according the defining response equation
P(X_{pi}=1 )=c_i + ( 1 - c_i ) Φ ( a_i^\ast θ_p + b_i^\ast )
A list of class mcmc.sirt with following entries:
mcmcobj |
Object of class |
summary.mcmcobj |
Summary of the |
ic |
Information criteria (DIC) |
burnin |
Number of burnin iterations |
iter |
Total number of iterations |
theta.chain |
Sampled values of θ_p parameters |
deviance.chain |
Sampled values of deviance values |
EAP.rel |
EAP reliability |
person |
Data frame with EAP person parameter estimates for θ_p and their corresponding posterior standard deviations and for all testlet effects |
dat |
Used data frame |
weights |
Used student weights |
... |
Further values |
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.
S3 methods: summary.mcmc.sirt, plot.mcmc.sirt
## 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" ] )
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