rasch.copula2: 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). 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, 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)  
"summary"(object,...)
"summary"(object,...)
"anova"(object,...)
"anova"(object,...)
"logLik"(object,...)
"logLik"(object,...)
"IRT.likelihood"(object,...)
"IRT.likelihood"(object,...)
"IRT.posterior"(object,...)
"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.

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 reponse 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 reponse 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

...

Further arguments to be passed

Value

A list with following entries A list with following entries

References

Braeken, J. (2011). A boundary mixture approach to violations of conditional independence. Psychometrika, 76, 57-76.

Braeken, J., & Tuerlinckx, F. (2009). Investigating latent constructs with item response models: A MATLAB IRTm toolbox. Behavior Research Methods, 41, 1127-1137. Braeken, J., Tuerlinckx, F., & De Boeck, P. (2007). Copulas for residual dependencies. Psychometrika, 72, 393-411.

Stukel, T. A. (1988). Generalized logistic models. Journal of the American Statistical Association, 83, 426-431.

Examples

Run this code

#############################################################################
# EXAMPLE 1: Reading Data
#############################################################################

data(data.read)
dat <- data.read

# define item clusters
itemcluster <- rep( 1:3 , each=4 )

# estimate Copula model
mod1 <- rasch.copula2( dat=dat , itemcluster=itemcluster)

# estimate Rasch model
mod2 <- rasch.copula2( dat=dat , itemcluster=itemcluster , 
        delta=rep(0,3) , est.delta=rep(0,3 ) )
summary(mod1)
summary(mod2)

## Not run: 
# #############################################################################
# # 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 <- sim.rasch.dep( theta , b , itemcluster , rho )
# colnames(dat) <- paste("I" , seq(1,ncol(dat)) , sep="")
# 
# # estimate Rasch copula model
# mod1 <- rasch.copula2( dat , itemcluster = itemcluster )
# summary(mod1)
# 
# # both item clusters have Cook-Johnson copula as dependency
# mod1c <- rasch.copula2( dat , itemcluster = itemcluster ,
# 			copula.type ="cook.johnson")
# summary(mod1c)
# 
# # first item boundary mixture and second item Cook-Johnson copula
# mod1d <- 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 <- 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 <- sim.rasch.dep( theta , b , itemcluster , rho )
# colnames(dat) <- paste("I" , seq(1,ncol(dat)) , sep="")
# 
# # estimate Rasch copula model 
# mod1 <- rasch.copula2( dat , itemcluster = itemcluster )
# summary(mod1)
# 
# # person parameter estimation assuming the Rasch copula model
# pmod1 <- person.parameter.rasch.copula(raschcopula.object = mod1 )
# 
# # Rasch model estimation
# mod2 <- 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 <- sim.rasch.dep( theta , b , itemcluster , rho )
# colnames(dat) <- paste("A" , seq(1,ncol(dat)) , sep="")
# dat1 <- dat
# # estimate model of dimension 1
# mod0a <- 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 <- sim.rasch.dep( theta , b , itemcluster , rho )
# colnames(dat) <- paste("B" , seq(1,ncol(dat)) , sep="")
# dat2 <- dat
# # estimate model of dimension 2
# mod0b <- 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 <- 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 <- 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 <- 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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