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rasch.copula

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 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

References

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

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)

sirt

Supplementary Item Response Theory Models

v3.10-118
GPL (>= 2)
Authors
Alexander Robitzsch [aut,cre] (<https://orcid.org/0000-0002-8226-3132>)
Initial release
2021-09-22 17:45:34

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