rasch.copula: Multidimensional IRT Copula Model
In alexanderrobitzsch/sirt: Supplementary Item Response Theory Models
rasch.copula2 R Documentation
Multidimensional IRT Copula Model
Description
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.
Usage
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,...)
Arguments
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: bound.mixt (boundary mixture copula),
cook.johnson (Cook-Johnson copula) or frank (Frank copula)
(see Braeken, 2011).
The vector copula.type must match the number of different
itemclusters. For every itemcluster, a different copula type
may be specified (see Examples).
progress
Print progress? Default is TRUE.
mmliter
Maximum number of iterations.
delta
An optional vector of starting values for the dependency parameter delta.
theta.k
Discretized trait distribution
alpha1
alpha1 parameter in the generalized logistic item response model
(Stukel, 1988). The default is 0 which leads together with alpha2=0
to the logistic link function.
alpha2
alpha2 parameter in the generalized logistic item response model
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 length(itemcluster). Nonzero integers
correspond to delta parameters which are estimated.
Equal integers indicate parameter equality constraints.
b.init
Initial b parameters
a.init
Initial a parameters
est.alpha
Should both alpha parameters be estimated? Default is FALSE.
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 .2.
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 rasch.copula2 or rasch.copula3
file
Optional file name for summary output
digits
Number of digits after decimal in summary output
...
Further arguments to be passed
Value
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 \delta
b
Estimated item difficulties
a
Estimated item slopes
mu
Mean
sigma
Standard deviation
alpha1
Parameter \alpha_1 in the generalized item response model
alpha2
Parameter \alpha_2 in the generalized item response model
ic
Information criteria
theta.k
Discretized ability distribution
pi.k
Fixed \theta 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 \delta parameters
f.qk.yi
Individual posterior
f.yi.qk
Individual likelihood
...
Further values
References
Braeken, J. (2011). A boundary mixture approach to violations of conditional
independence. Psychometrika, 76(1), 57-76.
\Sexpr[results=rd]{tools:::Rd_expr_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.
\Sexpr[results=rd]{tools:::Rd_expr_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.
\Sexpr[results=rd]{tools:::Rd_expr_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.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/jedm.12054")}
Stukel, T. A. (1988). Generalized logistic models.
Journal of the American Statistical Association, 83(402), 426-431.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/01621459.1988.10478613")}
See Also
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.
Examples
#############################################################################
# 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
## End(Not run)
alexanderrobitzsch/sirt documentation built on Dec. 1, 2024, 2:18 a.m.
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| rasch.copula2 | R Documentation |
Multidimensional IRT Copula Model
Description
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.
Usage
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,...)
Arguments
dat |
An |
itemcluster |
An integer vector of length |
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 |
a.init |
Initial |
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 |
Value
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 |
b |
Estimated item difficulties |
a |
Estimated item slopes |
mu |
Mean |
sigma |
Standard deviation |
alpha1 |
Parameter |
alpha2 |
Parameter |
ic |
Information criteria |
theta.k |
Discretized ability distribution |
pi.k |
Fixed |
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 |
f.qk.yi |
Individual posterior |
f.yi.qk |
Individual likelihood |
... |
Further values |
References
Braeken, J. (2011). A boundary mixture approach to violations of conditional independence. Psychometrika, 76(1), 57-76. \Sexpr[results=rd]{tools:::Rd_expr_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. \Sexpr[results=rd]{tools:::Rd_expr_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. \Sexpr[results=rd]{tools:::Rd_expr_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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/jedm.12054")}
Stukel, T. A. (1988). Generalized logistic models. Journal of the American Statistical Association, 83(402), 426-431. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/01621459.1988.10478613")}
See Also
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.
Examples
#############################################################################
# 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
## End(Not run)
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