#############################################################################
## SIMULATED EXAMPLE 1: Noncompensatory and compensatory IRT models
#############################################################################
set.seed(997)
# (1) simulate data from a two-dimensional noncompensatory
# item response model
# -> increase number of iterations in all models!
N <- 1000 # number of persons
I <- 10 # number of items
theta0 <- rnorm( N , sd= 1 )
theta1 <- theta0 + rnorm(N , sd = .7 )
theta2 <- theta0 + rnorm(N , sd = .7 )
Q <- matrix( 1 , nrow=I,ncol=2 )
Q[ 1:(I/2) , 2 ] <- 0
Q[ I,1] <- 0
b <- matrix( rnorm( I*2 ) , I , 2 )
a <- matrix( 1 , I , 2 )
# simulate data
prob <- dat <- matrix(0 , nrow=N , ncol=I )
for (ii in 1:I){
prob[,ii] <- ( plogis( theta1 - b[ii,1] ) )^Q[ii,1]
prob[,ii] <- prob[,ii] * ( plogis( theta2 - b[ii,2] ) )^Q[ii,2]
}
dat[ prob > matrix(runif( N*I),N,I) ] <- 1
colnames(dat) <- paste0("I",1:I)
#***
# Model 1: Noncompensatory 1PL model
mod1 <- smirt(dat, Qmatrix=Q , maxiter=10 ) # change number of iterations
summary(mod1)
#***
# Model 2: Noncompensatory 2PL model
mod2 <- smirt(dat,Qmatrix=Q , est.a="2PL" , maxiter=15 )
summary(mod2)
# Model 2a: avoid convergence problems with increment.factor
mod2a <- smirt(dat,Qmatrix=Q , est.a="2PL" , maxiter=30 , increment.factor=1.03)
summary(mod2a)
#***
# Model 3: some fixed c and d parameters different from zero or one
c.init <- rep(0,I)
c.init[ c(3,7)] <- .2
d.init <- rep(1,I)
d.init[c(4,8)] <- .95
mod3 <- smirt( dat , Qmatrix=Q , c.init=c.init , d.init=d.init )
summary(mod3)
#***
# Model 4: some estimated c and d parameters (in parameter groups)
est.c <- c.init <- rep(0,I)
c.estpars <- c(3,6,7)
c.init[ c.estpars ] <- .2
est.c[c.estpars] <- 1
est.d <- rep(0,I)
d.init <- rep(1,I)
d.estpars <- c(6,9)
d.init[ d.estpars ] <- .95
est.d[ d.estpars ] <- d.estpars # different d parameters
mod4 <- smirt(dat,Qmatrix=Q , est.c=est.c , c.init=c.init ,
est.d=est.d , d.init=d.init )
summary(mod4)
#***
# Model 5: Unidimensional 1PL model
Qmatrix <- matrix( 1 , nrow=I , ncol=1 )
mod5 <- smirt( dat , Qmatrix=Qmatrix )
summary(mod5)
#***
# Model 6: Unidimensional 2PL model
mod6 <- smirt( dat , Qmatrix=Qmatrix , est.a="2PL" )
summary(mod6)
#***
# Model 7: Compensatory model with between item dimensionality
# Note that the data is simulated under the noncompensatory condition
# Therefore Model 7 should have a worse model fit than Model 1
Q1 <- Q
Q1[ 6:10 , 1] <- 0
mod7 <- smirt(dat,Qmatrix=Q1 , irtmodel="comp" , maxiter=30)
summary(mod7)
#***
# Model 8: Compensatory model with within item dimensionality
# assuming zero correlation between dimensions
variance.fixed <- as.matrix( cbind( 1,2,0) )
# set the covariance between the first and second dimension to zero
mod8 <- smirt(dat,Qmatrix=Q , irtmodel="comp" , variance.fixed=variance.fixed ,
maxiter=30)
summary(mod8)
#***
# Model 8b: 2PL model with starting values for a and b parameters
b.init <- rep(0,10) # set all item difficulties initially to zero
# b.init <- NULL
a.init <- Q # initialize a.init with Q-matrix
# provide starting values for slopes of first three items on Dimension 1
a.init[1:3,1] <- c( .55 , .32 , 1.3)
mod8b <- smirt(dat,Qmatrix=Q , irtmodel="comp" , variance.fixed=variance.fixed ,
b.init=b.init , a.init=a.init , maxiter=20 , est.a="2PL" )
summary(mod8b)
#***
# Model 9: Unidimensional model with quadratic item response functions
# define theta
theta.k <- seq( - 6 , 6 , len=15 )
theta.k <- as.matrix( theta.k , ncol=1 )
# define design matrix
theta.kDES <- cbind( theta.k[,1] , theta.k[,1]^2 )
# define Q-matrix
Qmatrix <- matrix( 0 , I , 2 )
Qmatrix[,1] <- 1
Qmatrix[ c(3,6,7) , 2 ] <- 1
colnames(Qmatrix) <- c("F1" , "F1sq" )
# estimate model
mod9 <- smirt(dat,Qmatrix=Qmatrix , maxiter=50 , irtmodel="comp" ,
theta.k=theta.k , theta.kDES=theta.kDES , est.a="2PL" )
summary(mod9)
#***
# Model 10: Two-dimensional item response model with latent interaction
# between dimensions
theta.k <- seq( - 6 , 6 , len=15 )
theta.k <- expand.grid( theta.k , theta.k ) # expand theta to 2 dimensions
# define design matrix
theta.kDES <- cbind( theta.k , theta.k[,1]*theta.k[,2] )
# define Q-matrix
Qmatrix <- matrix( 0 , I , 3 )
Qmatrix[,1] <- 1
Qmatrix[ 6:10 , c(2,3) ] <- 1
colnames(Qmatrix) <- c("F1" , "F2" , "F1iF2" )
# estimate model
mod10 <- smirt(dat,Qmatrix=Qmatrix ,irtmodel="comp" , theta.k=theta.k ,
theta.kDES= theta.kDES , est.a="2PL" )
summary(mod10)
#****
# Model 11: Example Quasi Monte Carlo integration
Qmatrix <- matrix( 1 , I , 1 )
mod11 <- smirt( dat , irtmodel="comp" , Qmatrix=Qmatrix , qmcnodes=1000 )
summary(mod11)
#############################################################################
## EXAMPLE 2: Dataset Reading data.read
## Multidimensional models for dichotomous data
#############################################################################
data(data.read)
dat <- data.read
I <- ncol(dat) # number of items
#***
# Model 1: 3-dimensional 2PL model
# define Q-matrix
Qmatrix <- matrix(0,nrow=I,ncol=3)
Qmatrix[1:4,1] <- 1
Qmatrix[5:8,2] <- 1
Qmatrix[9:12,3] <- 1
# estimate model
mod1 <- smirt( dat , Qmatrix=Qmatrix , irtmodel="comp" , est.a="2PL" ,
qmcnodes=1000 , maxiter=20)
summary(mod1)
#***
# Model 2: 3-dimensional Rasch model
mod2 <- smirt( dat , Qmatrix=Qmatrix , irtmodel="comp" ,
qmcnodes=1000 , maxiter=20)
summary(mod2)
#***
# Model 3: 3-dimensional 2PL model with uncorrelated dimensions
# fix entries in variance matrix
variance.fixed <- cbind( c(1,1,2) , c(2,3,3) , 0 )
# set the following covariances to zero: cov[1,2]=cov[1,3]=cov[2,3]=0
# estimate model
mod3 <- smirt( dat , Qmatrix=Qmatrix , irtmodel="comp" , est.a="2PL" ,
variance.fixed=variance.fixed , qmcnodes=1000 , maxiter=20)
summary(mod3)
#***
# Model 4: Bifactor model with one general factor (g) and
# uncorrelated specific factors
# define a new Q-matrix
Qmatrix1 <- cbind( 1 , Qmatrix )
# uncorrelated factors
variance.fixed <- cbind( c(1,1,1,2,2,3) , c(2,3,4,3,4,4) , 0 )
# The first dimension refers to the general factors while the other
# dimensions refer to the specific factors.
# The specification means that:
# Cov[1,2]=Cov[1,3]=Cov[1,4]=Cov[2,3]=Cov[2,4]=Cov[3,4]=0
# estimate model
mod4 <- smirt( dat , Qmatrix=Qmatrix1 , irtmodel="comp" , est.a="2PL" ,
variance.fixed=variance.fixed , qmcnodes=1000 , maxiter=20)
summary(mod4)
#############################################################################
## SIMULATED EXAMPLE 3: Partially compensatory model
#############################################################################
#**** simulate data
set.seed(7656)
I <- 10 # number of items
N <- 2000 # number of subjects
Q <- matrix( 0 , 3*I,2) # Q-matrix
Q[1:I,1] <- 1
Q[1:I + I ,2] <- 1
Q[1:I + 2*I ,1:2] <- 1
b <- matrix( runif( 3*I *2, -2 , 2 ) , nrow=3*I , 2 )
b <- b*Q
b <- round( b , 2 )
mui <- rep(0,3*I)
mui[ seq(2*I+1 , 3*I) ] <- 0.65
# generate data
dat <- matrix( NA , N , 3*I )
colnames(dat) <- paste0("It" , 1:(3*I) )
# simulate item responses
library(MASS)
theta <- MASS::mvrnorm(N , mu=c(0,0) , Sigma = matrix( c( 1.2 , .6,.6,1.6) ,2 , 2 ) )
for (ii in 1:(3*I)){
# define probability
tmp1 <- exp( theta[,1] * Q[ii,1] - b[ii,1] + theta[,2] * Q[ii,2] - b[ii,2] )
# non-compensatory model
nco1 <- ( 1 + exp( theta[,1] * Q[ii,1] - b[ii,1] ) ) *
( 1 + exp( theta[,2] * Q[ii,2] - b[ii,2] ) )
co1 <- ( 1 + tmp1 )
p1 <- tmp1 / ( mui[ii] * nco1 + ( 1 - mui[ii] )*co1 )
dat[,ii] <- 1 * ( runif(N) < p1 )
}
#*** Model 1: Joint mu.i parameter for all items
est.mu.i <- rep(0,3*I)
est.mu.i[ seq(2*I+1,3*I)] <- 1
mod1 <- smirt( dat , Qmatrix = Q , irtmodel = "partcomp" , est.mu.i=est.mu.i)
summary(mod1)
#*** Model 2: Separate mu.i parameter for all items
est.mu.i[ seq(2*I+1,3*I)] <- 1:I
mod2 <- smirt( dat , Qmatrix = Q , irtmodel = "partcomp" , est.mu.i=est.mu.i)
summary(mod2)Run the code above in your browser using DataLab