Multidimensional IRT Copula Model
This function handles local dependence by specifying copulas for residuals in multidimensional item response models for dichotomous item responses (Braeken, 2011; Braeken, Tuerlinckx & de Boeck, 2007; Schroeders, Robitzsch & Schipolowski, 2014). Estimation is allowed for item difficulties, item slopes and a generalized logistic link function (Stukel, 1988).
The function rasch.copula3 allows the estimation of multidimensional
models while rasch.copula2 only handles unidimensional models.
rasch.copula2(dat, itemcluster, weights=NULL, copula.type="bound.mixt",
progress=TRUE, mmliter=1000, delta=NULL,
theta.k=seq(-4, 4, len=21), alpha1=0, alpha2=0,
numdiff.parm=1e-06, est.b=seq(1, ncol(dat)),
est.a=rep(1, ncol(dat)), est.delta=NULL, b.init=NULL, a.init=NULL,
est.alpha=FALSE, glob.conv=0.0001, alpha.conv=1e-04, conv1=0.001,
dev.crit=.2, increment.factor=1.01)
rasch.copula3(dat, itemcluster, dims=NULL, copula.type="bound.mixt",
progress=TRUE, mmliter=1000, delta=NULL,
theta.k=seq(-4, 4, len=21), alpha1=0, alpha2=0,
numdiff.parm=1e-06, est.b=seq(1, ncol(dat)),
est.a=rep(1, ncol(dat)), est.delta=NULL, b.init=NULL, a.init=NULL,
est.alpha=FALSE, glob.conv=0.0001, alpha.conv=1e-04, conv1=0.001,
dev.crit=.2, rho.init=.5, increment.factor=1.01)
## S3 method for class 'rasch.copula2'
summary(object, file=NULL, digits=3, ...)
## S3 method for class 'rasch.copula3'
summary(object, file=NULL, digits=3, ...)
## S3 method for class 'rasch.copula2'
anova(object,...)
## S3 method for class 'rasch.copula3'
anova(object,...)
## S3 method for class 'rasch.copula2'
logLik(object,...)
## S3 method for class 'rasch.copula3'
logLik(object,...)
## S3 method for class 'rasch.copula2'
IRT.likelihood(object,...)
## S3 method for class 'rasch.copula3'
IRT.likelihood(object,...)
## S3 method for class 'rasch.copula2'
IRT.posterior(object,...)
## S3 method for class 'rasch.copula3'
IRT.posterior(object,...)dat |
An N \times I data frame. Cases with only missing responses are removed from the analysis. |
itemcluster |
An integer vector of length I (number of items). Items with the same integers define a joint item cluster of (positively) locally dependent items. Values of zero indicate that the corresponding item is not included in any item cluster of dependent responses. |
weights |
Optional vector of sampling weights |
dims |
A vector indicating to which dimension an item is allocated. The default is that all items load on the first dimension. |
copula.type |
A character or a vector containing one of the following copula
types: |
progress |
Print progress? Default is |
mmliter |
Maximum number of iterations. |
delta |
An optional vector of starting values for the dependency parameter |
theta.k |
Discretized trait distribution |
alpha1 |
|
alpha2 |
|
numdiff.parm |
Parameter for numerical differentiation |
est.b |
Integer vector of item difficulties to be estimated |
est.a |
Integer vector of item discriminations to be estimated |
est.delta |
Integer vector of length |
b.init |
Initial b parameters |
a.init |
Initial a parameters |
est.alpha |
Should both alpha parameters be estimated? Default is |
glob.conv |
Convergence criterion for all parameters |
alpha.conv |
Maximal change in alpha parameters for convergence |
conv1 |
Maximal change in item parameters for convergence |
dev.crit |
Maximal change in the deviance. Default is |
rho.init |
Initial value for off-diagonal elements in correlation matrix |
increment.factor |
A numeric value larger than one which controls the size of increments in iterations. To stabilize convergence, choose values 1.05 or 1.1 in some situations. |
object |
Object of class |
file |
Optional file name for |
digits |
Number of digits after decimal in |
... |
Further arguments to be passed |
A list with following entries
N.itemclusters |
Number of item clusters |
item |
Estimated item parameters |
iter |
Number of iterations |
dev |
Deviance |
delta |
Estimated dependency parameters δ |
b |
Estimated item difficulties |
a |
Estimated item slopes |
mu |
Mean |
sigma |
Standard deviation |
alpha1 |
Parameter α_1 in the generalized item response model |
alpha2 |
Parameter α_2 in the generalized item response model |
ic |
Information criteria |
theta.k |
Discretized ability distribution |
pi.k |
Fixed θ distribution |
deviance |
Deviance |
pattern |
Item response patterns with frequencies and posterior distribution |
person |
Data frame with person parameters |
datalist |
List of generated data frames during estimation |
EAP.rel |
Reliability of the EAP |
copula.type |
Type of copula |
summary.delta |
Summary for estimated δ parameters |
f.qk.yi |
Individual posterior |
f.yi.qk |
Individual likelihood |
... |
Further values |
Braeken, J. (2011). A boundary mixture approach to violations of conditional independence. Psychometrika, 76(1), 57-76. doi: 10.1007/s11336-010-9190-4
Braeken, J., Kuppens, P., De Boeck, P., & Tuerlinckx, F. (2013). Contextualized personality questionnaires: A case for copulas in structural equation models for categorical data. Multivariate Behavioral Research, 48(6), 845-870. doi: 10.1080/00273171.2013.827965
Braeken, J., & Tuerlinckx, F. (2009). Investigating latent constructs with item response models: A MATLAB IRTm toolbox. Behavior Research Methods, 41(4), 1127-1137.
Braeken, J., Tuerlinckx, F., & De Boeck, P. (2007). Copula functions for residual dependency. Psychometrika, 72(3), 393-411. doi: 10.1007/s11336-007-9005-4
Schroeders, U., Robitzsch, A., & Schipolowski, S. (2014). A comparison of different psychometric approaches to modeling testlet structures: An example with C-tests. Journal of Educational Measurement, 51(4), 400-418. doi: 10.1111/jedm.12054
Stukel, T. A. (1988). Generalized logistic models. Journal of the American Statistical Association, 83(402), 426-431. doi: 10.1080/01621459.1988.10478613
For a summary see summary.rasch.copula2.
For simulating locally dependent item responses see sim.rasch.dep.
Person parameters estimates are obtained by person.parameter.rasch.copula.
See rasch.mml2 for the generalized logistic link function.
See also Braeken and Tuerlinckx (2009) for alternative (and more expanded) copula models implemented in the MATLAB software. See https://ppw.kuleuven.be/okp/software/irtm/.
See Braeken, Kuppens, De Boeck and Tuerlinckx (2013) for an extension of the copula modeling approach to polytomous data.
#############################################################################
# EXAMPLE 1: Reading Data
#############################################################################
data(data.read)
dat <- data.read
# define item clusters
itemcluster <- rep( 1:3, each=4 )
# estimate Copula model
mod1 <- sirt::rasch.copula2( dat=dat, itemcluster=itemcluster)
## Not run:
# estimate Rasch model
mod2 <- sirt::rasch.copula2( dat=dat, itemcluster=itemcluster,
delta=rep(0,3), est.delta=rep(0,3) )
summary(mod1)
summary(mod2)
# estimate copula 2PL model
I <- ncol(dat)
mod3 <- sirt::rasch.copula2( dat=dat, itemcluster=itemcluster, est.a=1:I,
increment.factor=1.05)
summary(mod3)
#############################################################################
# EXAMPLE 2: 11 items nested within 2 item clusters (testlets)
# with 2 resp. 3 dependent and 6 independent items
#############################################################################
set.seed(5698)
I <- 11 # number of items
n <- 3000 # number of persons
b <- seq(-2,2, len=I) # item difficulties
theta <- stats::rnorm( n, sd=1 ) # person abilities
# define item clusters
itemcluster <- rep(0,I)
itemcluster[ c(3,5 )] <- 1
itemcluster[c(2,4,9)] <- 2
# residual correlations
rho <- c( .7, .5 )
# simulate data
dat <- sirt::sim.rasch.dep( theta, b, itemcluster, rho )
colnames(dat) <- paste("I", seq(1,ncol(dat)), sep="")
# estimate Rasch copula model
mod1 <- sirt::rasch.copula2( dat, itemcluster=itemcluster )
summary(mod1)
# both item clusters have Cook-Johnson copula as dependency
mod1c <- sirt::rasch.copula2( dat, itemcluster=itemcluster,
copula.type="cook.johnson")
summary(mod1c)
# first item boundary mixture and second item Cook-Johnson copula
mod1d <- sirt::rasch.copula2( dat, itemcluster=itemcluster,
copula.type=c( "bound.mixt", "cook.johnson" ) )
summary(mod1d)
# compare result with Rasch model estimation in rasch.copula2
# delta must be set to zero
mod2 <- sirt::rasch.copula2( dat, itemcluster=itemcluster, delta=c(0,0),
est.delta=c(0,0) )
summary(mod2)
#############################################################################
# EXAMPLE 3: 12 items nested within 3 item clusters (testlets)
# Cluster 1 -> Items 1-4; Cluster 2 -> Items 6-9; Cluster 3 -> Items 10-12
#############################################################################
set.seed(967)
I <- 12 # number of items
n <- 450 # number of persons
b <- seq(-2,2, len=I) # item difficulties
b <- sample(b) # sample item difficulties
theta <- stats::rnorm( n, sd=1 ) # person abilities
# itemcluster
itemcluster <- rep(0,I)
itemcluster[ 1:4 ] <- 1
itemcluster[ 6:9 ] <- 2
itemcluster[ 10:12 ] <- 3
# residual correlations
rho <- c( .35, .25, .30 )
# simulate data
dat <- sirt::sim.rasch.dep( theta, b, itemcluster, rho )
colnames(dat) <- paste("I", seq(1,ncol(dat)), sep="")
# estimate Rasch copula model
mod1 <- sirt::rasch.copula2( dat, itemcluster=itemcluster )
summary(mod1)
# person parameter estimation assuming the Rasch copula model
pmod1 <- sirt::person.parameter.rasch.copula(raschcopula.object=mod1 )
# Rasch model estimation
mod2 <- sirt::rasch.copula2( dat, itemcluster=itemcluster,
delta=rep(0,3), est.delta=rep(0,3) )
summary(mod1)
summary(mod2)
#############################################################################
# EXAMPLE 4: Two-dimensional copula model
#############################################################################
set.seed(5698)
I <- 9
n <- 1500 # number of persons
b <- seq(-2,2, len=I) # item difficulties
theta0 <- stats::rnorm( n, sd=sqrt( .6 ) )
#*** Dimension 1
theta <- theta0 + stats::rnorm( n, sd=sqrt( .4 ) ) # person abilities
# itemcluster
itemcluster <- rep(0,I)
itemcluster[ c(3,5 )] <- 1
itemcluster[c(2,4,9)] <- 2
itemcluster1 <- itemcluster
# residual correlations
rho <- c( .7, .5 )
# simulate data
dat <- sirt::sim.rasch.dep( theta, b, itemcluster, rho )
colnames(dat) <- paste("A", seq(1,ncol(dat)), sep="")
dat1 <- dat
# estimate model of dimension 1
mod0a <- sirt::rasch.copula2( dat1, itemcluster=itemcluster1)
summary(mod0a)
#*** Dimension 2
theta <- theta0 + stats::rnorm( n, sd=sqrt( .8 ) ) # person abilities
# itemcluster
itemcluster <- rep(0,I)
itemcluster[ c(3,7,8 )] <- 1
itemcluster[c(4,6)] <- 2
itemcluster2 <- itemcluster
# residual correlations
rho <- c( .2, .4 )
# simulate data
dat <- sirt::sim.rasch.dep( theta, b, itemcluster, rho )
colnames(dat) <- paste("B", seq(1,ncol(dat)), sep="")
dat2 <- dat
# estimate model of dimension 2
mod0b <- sirt::rasch.copula2( dat2, itemcluster=itemcluster2)
summary(mod0b)
# both dimensions
dat <- cbind( dat1, dat2 )
itemcluster2 <- ifelse( itemcluster2 > 0, itemcluster2 + 2, 0 )
itemcluster <- c( itemcluster1, itemcluster2 )
dims <- rep( 1:2, each=I)
# estimate two-dimensional copula model
mod1 <- sirt::rasch.copula3( dat, itemcluster=itemcluster, dims=dims, est.a=dims,
theta.k=seq(-5,5,len=15) )
summary(mod1)
#############################################################################
# EXAMPLE 5: Subset of data Example 2
#############################################################################
set.seed(5698)
I <- 11 # number of items
n <- 3000 # number of persons
b <- seq(-2,2, len=I) # item difficulties
theta <- stats::rnorm( n, sd=1.3 ) # person abilities
# define item clusters
itemcluster <- rep(0,I)
itemcluster[ c(3,5)] <- 1
itemcluster[c(2,4,9)] <- 2
# residual correlations
rho <- c( .7, .5 )
# simulate data
dat <- sirt::sim.rasch.dep( theta, b, itemcluster, rho )
colnames(dat) <- paste("I", seq(1,ncol(dat)), sep="")
# select subdataset with only one dependent item cluster
item.sel <- scan( what="character", nlines=1 )
I1 I6 I7 I8 I10 I11 I3 I5
dat1 <- dat[,item.sel]
#******************
#*** Model 1a: estimate Copula model in sirt
itemcluster <- rep(0,8)
itemcluster[c(7,8)] <- 1
mod1a <- sirt::rasch.copula2( dat3, itemcluster=itemcluster )
summary(mod1a)
#******************
#*** Model 1b: estimate Copula model in mirt
library(mirt)
#*** redefine dataset for estimation in mirt
dat2 <- dat1[, itemcluster==0 ]
dat2 <- as.data.frame(dat2)
# combine items 3 and 5
dat2$C35 <- dat1[,"I3"] + 2*dat1[,"I5"]
table( dat2$C35, paste0( dat1[,"I3"],dat1[,"I5"]) )
#* define mirt model
mirtmodel <- mirt::mirt.model("
F=1-7
CONSTRAIN=(1-7,a1)
" )
#-- Copula function with two dependent items
# define item category function for pseudo-items like C35
P.copula2 <- function(par,Theta, ncat){
b1 <- par[1]
b2 <- par[2]
a1 <- par[3]
ldelta <- par[4]
P1 <- stats::plogis( a1*(Theta - b1 ) )
P2 <- stats::plogis( a1*(Theta - b2 ) )
Q1 <- 1-P1
Q2 <- 1-P2
# define vector-wise minimum function
minf2 <- function( x1, x2 ){
ifelse( x1 < x2, x1, x2 )
}
# distribution under independence
F00 <- Q1*Q2
F10 <- Q1*Q2 + P1*Q2
F01 <- Q1*Q2 + Q1*P2
F11 <- 1+0*Q1
F_ind <- c(F00,F10,F01,F11)
# distribution under maximal dependence
F00 <- minf2(Q1,Q2)
F10 <- Q2 #=minf2(1,Q2)
F01 <- Q1 #=minf2(Q1,1)
F11 <- 1+0*Q1 #=minf2(1,1)
F_dep <- c(F00,F10,F01,F11)
# compute mixture distribution
delta <- stats::plogis(ldelta)
F_tot <- (1-delta)*F_ind + delta * F_dep
# recalculate probabilities of mixture distribution
L1 <- length(Q1)
v1 <- 1:L1
F00 <- F_tot[v1]
F10 <- F_tot[v1+L1]
F01 <- F_tot[v1+2*L1]
F11 <- F_tot[v1+3*L1]
P00 <- F00
P10 <- F10 - F00
P01 <- F01 - F00
P11 <- 1 - F10 - F01 + F00
prob_tot <- c( P00, P10, P01, P11 )
return(prob_tot)
}
# create item
copula2 <- mirt::createItem(name="copula2", par=c(b1=0, b2=0.2, a1=1, ldelta=0),
est=c(TRUE,TRUE,TRUE,TRUE), P=P.copula2,
lbound=c(-Inf,-Inf,0,-Inf), ubound=c(Inf,Inf,Inf,Inf) )
# define item types
itemtype <- c( rep("2PL",6), "copula2" )
customItems <- list("copula2"=copula2)
# parameter table
mod.pars <- mirt::mirt(dat2, 1, itemtype=itemtype,
customItems=customItems, pars='values')
# estimate model
mod1b <- mirt::mirt(dat2, mirtmodel, itemtype=itemtype, customItems=customItems,
verbose=TRUE, pars=mod.pars,
technical=list(customTheta=as.matrix(seq(-4,4,len=21)) ) )
# estimated coefficients
cmod <- sirt::mirt.wrapper.coef(mod)$coef
# compare common item discrimination
round( c("sirt"=mod1a$item$a[1], "mirt"=cmod$a1[1] ), 4 )
## sirt mirt
## 1.2845 1.2862
# compare delta parameter
round( c("sirt"=mod1a$item$delta[7], "mirt"=stats::plogis( cmod$ldelta[7] ) ), 4 )
## sirt mirt
## 0.6298 0.6297
# compare thresholds a*b
dfr <- cbind( "sirt"=mod1a$item$thresh,
"mirt"=c(- cmod$d[-7],cmod$b1[7]*cmod$a1[1], cmod$b2[7]*cmod$a1[1]))
round(dfr,4)
## sirt mirt
## [1,] -1.9236 -1.9231
## [2,] -0.0565 -0.0562
## [3,] 0.3993 0.3996
## [4,] 0.8058 0.8061
## [5,] 1.5293 1.5295
## [6,] 1.9569 1.9572
## [7,] -1.1414 -1.1404
## [8,] -0.4005 -0.3996
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