sirt (version 1.14-0)

rasch.pml3: Pairwise Marginal Likelihood Estimation for the Probit Rasch Model

Description

This function estimates unidimensional 1PL and 2PL models with the probit link using pairwise marginal maximum likelihood estimation (PMML; Renard, Molenberghs & Geys, 2004). Item pairs within an itemcluster can be excluded from the pairwise likelihood (argument itemcluster). The other alternative is to model a residual error structure with itemclusters (argument error.corr).

Usage

rasch.pml3(dat, est.b = seq(1, ncol(dat)), est.a=rep(0,ncol(dat)) , est.sigma = TRUE, itemcluster = NULL,  weight = rep(1, nrow(dat)), numdiff.parm = 0.001,  b.init = NULL, a.init=NULL ,  sigma.init = NULL,  error.corr = 0*diag( 1 , ncol(dat) ) , err.constraintM=NULL , err.constraintV=NULL , glob.conv = 10^(-6), conv1 = 10^(-4), pmliter = 300, progress = TRUE, use.maxincrement=TRUE )   
"summary"(object,...)

Arguments

dat

An $N \times I$ data frame of dichotomous item responses

est.b

Vector of integers of length $I$. Same integers mean that the corresponding items do have the same item difficulty b. Entries of 0 mean fixing item parameters to values specified in b.init.

est.a

Vector of integers of length $I$. Same integers mean that the corresponding items do have the same item slope a. Entries of 0 mean fixing item parameters to values specified in a.init.

est.sigma

Should sigma (the trait standard deviation) be estimated? The default is TRUE.

itemcluster

Optional vector of length $I$ of integers which indicates itemclusters. Same integers correspond to the same itemcluster. An entry of 0 correspond to an item which is not included in any itemcluster.

weight

Optional vector of person weights

numdiff.parm

Step parameter for numerical differentiation

b.init

Initial or fixed item difficulty

a.init

Initial or fixed item slopes

sigma.init

Initial or fixed trait standard deviation

error.corr

An optional $I \times I$ integer matrix which defines the estimation of residual correlations. Entries of zero indicate that the corresponding residual correlation should not be estimated. Integers which differ from zero indicate correlations to be estimated. All entries with an equal integer are estimated by the same residual correlation. The default of error.corr is a diagonal matrix which means that no residual correlation is estimated. If error.corr deviates from this default, then the argument itemcluster is set to NULL. If some error correlations are estimated, then no itempairs in itemcluster can be excluded from the pairwise modeling.

err.constraintM

An optional $P \times L$ matrix where $P$ denotes the number of item pairs in pseudolikelihood estimation and $L$ is the number of linear constraints for residual correlations (see Details).

err.constraintV

An optional $L \times 1$ matrix with specified values for linear constraints on residual correlations (see Details).

glob.conv

Global convergence criterion

conv1

Convergence criterion for model parameters

pmliter

Maximum number of iterations

progress

Display progress?

use.maxincrement

Optional logical whether increments in slope parameters should be controlled in size in iterations. The default is TRUE.

object

Object of class rasch.pml

...

Further arguments to be passed

Value

A list with following entries: A list with following entries:

Details

The probit item response model can be estimated with this function: $$P(X_{pi}=1|\theta_p ) = \Phi( a_i \theta_p - b_i ) \quad , \quad \theta_p \sim N ( 0 , \sigma^2 )$$ where $\Phi$ denotes the normal distribution function. This model can also be expressed as a latent variable model which assumes a latent response tendency $X_pi*$ which is equal to 1 if $X_{pi}> - b_i$ and otherwise zero. If $\epsilon_{pi}$ is standard normally distributed, then $$X_{pi}^{\ast} = a_i \theta_p - b_i + \epsilon_{pi} $$ An arbitrary pattern of residual correlations between $\epsilon_{pi}$ and $\epsilon_{pj}$ for item pairs $i$ and $j$ can be imposed using the error.corr argument.

Linear constraints $Me=v$ on residual correlations $e=Cov( \epsilon_{pi} , \epsilon_{pj})_{ij}$ (in a vectorized form) can be specified using the arguments err.constraintM (matrix $M$) and err.constraintV (vector $v$). The estimation is described in Neuhaus (1996).

For the pseudo likelihood information criterion (PLIC) see Stanford and Raftery (2002).

References

Green, S. B., & Yang, Y. (2009). Reliability of summed item scores using structural equation modeling: An alternative to coefficient alpha. Psychometrika, 74, 155-167. Neuhaus, W. (1996). Optimal estimation under linear constraints. Astin Bulletin, 26, 233-245.

Renard, D., Molenberghs, G., & Geys, H. (2004). A pairwise likelihood approach to estimation in multilevel probit models. Computational Statistics & Data Analysis, 44, 649-667.

Stanford, D. C., & Raftery, A. E. (2002). Approximate Bayes factors for image segmentation: The pseudolikelihood information criterion (PLIC). IEEE Transactions on Pattern Analysis and Machine Intelligence, 24, 1517-1520.

See Also

Get a summary of rasch.pml2 with summary.rasch.pml.

For simulation of locally dependent items see sim.rasch.dep.

For pairwise conditional likelihood estimation see rasch.pairwise or rasch.pairwise.itemcluster.

For an assessment of global model fit see modelfit.sirt.

Examples

#############################################################################
# EXAMPLE 1: Reading data set
#############################################################################

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

#******
# Model 1: Rasch model with PML estimation
mod1 <- rasch.pml3( dat )
summary(mod1)

#******
# Model 2: Excluding item pairs with local dependence 
#          from bivariate composite likelihood
itemcluster <- rep( 1:3 , each=4)
mod2 <- rasch.pml3( dat , itemcluster = itemcluster )
summary(mod2)

## Not run: 
# #*****
# # Model 3: Modelling error correlations: 
# #          joint residual correlations for each itemcluster
# error.corr <- diag(1,ncol(dat))
# for ( ii in 1:3){
#     ind.ii <- which( itemcluster == ii )
#     error.corr[ ind.ii , ind.ii ] <- ii
#         }
# # estimate the model with error correlations
# mod3 <- rasch.pml3( dat , error.corr = error.corr )
# summary(mod3)
# 
# #****
# # Model 4: model separate residual correlations        
# I <- ncol(error.corr)
# error.corr1 <- matrix( 1:(I*I) , ncol= I )
# error.corr <- error.corr1 * ( error.corr > 0 )  
# # estimate the model with error correlations
# mod4 <- rasch.pml3( dat , error.corr = error.corr )
# summary(mod4)
# 
# #****
# # Model 5:  assume equal item difficulties:
# # b_1 = b_7 and b_2 = b_12
# # fix item difficulty of the 6th item to .1
# est.b <- 1:I
# est.b[7] <- 1; est.b[12] <- 2 ; est.b[6] <- 0
# b.init <- rep( 0, I ) ; b.init[6] <- .1
# mod5 <- rasch.pml3( dat , est.b =est.b , b.init=b.init)
# summary(mod5)
# 
# #****
# # Model 6: estimate three item slope groups
# est.a <- rep(1:3 , each=4 )
# mod6 <- rasch.pml3( dat , est.a =est.a , est.sigma=0)
# summary(mod6)
# 
# #############################################################################
# # EXAMPLE 2: PISA reading
# #############################################################################
# 
# data(data.pisaRead)
# dat <- data.pisaRead$data
# 
# # select items
# dat <- dat[ , substring(colnames(dat),1,1)=="R" ]
# 
# #******
# # Model 1: Rasch model with PML estimation
# mod1 <- rasch.pml3( as.matrix(dat) )
#   ## Trait SD (Logit Link) : 1.419 
# 
# #******
# # Model 2: Model correlations within testlets
# error.corr <- diag(1,ncol(dat))
# testlets <- paste( data.pisaRead$item$testlet )
# itemcluster <- match( testlets , unique(testlets ) ) 
# for ( ii in 1:(length(unique(testlets))) ){
#     ind.ii <- which( itemcluster == ii )
#     error.corr[ ind.ii , ind.ii ] <- ii
#         }
# # estimate the model with error correlations
# mod2 <- rasch.pml3( dat , error.corr = error.corr )
#   ## Trait SD (Logit Link) : 1.384
# 
# #****
# # Model 3: model separate residual correlations        
# I <- ncol(error.corr)
# error.corr1 <- matrix( 1:(I*I) , ncol= I )
# error.corr <- error.corr1 * ( error.corr > 0 )  
# # estimate the model with error correlations
# mod3 <- rasch.pml3( dat , error.corr = error.corr )
#   ## Trait SD (Logit Link) : 1.384 
# 
# #############################################################################
# # EXAMPLE 3: 10 locally independent items
# #############################################################################
# 
# #**********
# # simulate some data
# set.seed(554)
# N <- 500	# persons
# I <- 10		# items
# theta <- stats::rnorm(N,sd=1.3 )	# trait SD of 1.3
# b <- seq(-2 , 2 , length=I) # item difficulties
# 
# # simulate data from the Rasch model
# dat <- sim.raschtype( theta = theta , b = b )
# 
# # estimation with rasch.pml and probit link
# mod1 <- rasch.pml3( dat )
# summary(mod1)
# 
# # estimation with rasch.mml2 function
# mod2 <- rasch.mml2( dat )
# 
# # estimate item parameters for groups with five item parameters each
# est.b <- rep( 1:(I/2) , each=2 )
# mod3 <- rasch.pml3( dat , est.b=est.b )
# summary(mod3)
# 
# # compare parameter estimates
# summary(mod1)
# summary(mod2)
# summary(mod3)
# 
# #############################################################################
# # EXAMPLE 4: 11 items and 2 item clusters with 2 and 3 items
# #############################################################################
# 
# set.seed(5698)
# I <- 11                             # number of items
# n <- 5000                           # number of persons
# b <- seq(-2,2, len=I)               # item difficulties
# theta <- stats::rnorm( n , sd = 1 ) # person abilities
# # itemcluster
# itemcluster <- rep(0,I)
# itemcluster[c(3,5)] <- 1
# itemcluster[c(2,4,9)] <- 2
# # residual correlations
# rho <- c( .7 , .5 )
# 
# # simulate data (under the logit link)
# dat <- sim.rasch.dep( theta , b , itemcluster , rho )
# colnames(dat) <- paste("I" , seq(1,ncol(dat)) , sep="")
# 
# #***
# # Model 1: estimation using the Rasch model (with probit link)
# mod1 <- rasch.pml3( dat )
# #***
# # Model 2: estimation when pairs of locally dependent items are eliminated
# mod2 <- rasch.pml3( dat , itemcluster=itemcluster)
# 
# #***
# # Model 3: Positive correlations within testlets
# est.corrs <- diag( 1 , I )
# est.corrs[ c(3,5) , c(3,5) ] <- 2
# est.corrs[ c(2,4,9) , c(2,4,9) ] <- 3
# mod3 <- rasch.pml3( dat , error.corr=est.corrs )
# 
# #***
# # Model 4: Negative correlations between testlets
# est.corrs <- diag( 1 , I )
# est.corrs[ c(3,5) , c(2,4,9) ] <- 2
# est.corrs[ c(2,4,9) , c(3,5) ] <- 2
# mod4 <- rasch.pml3( dat , error.corr=est.corrs )
# 
# #***
# # Model 5: sum constraint of zero within and between testlets
# est.corrs <- matrix( 1:(I*I) ,  I , I )
# cluster2 <- c(2,4,9)
# est.corrs[ setdiff( 1:I , c(cluster2)) ,  ] <- 0
# est.corrs[ , setdiff( 1:I , c(cluster2))  ] <- 0
# # define an error constraint matrix
# itempairs0 <- mod4$itempairs
# IP <- nrow(itempairs0)
# err.constraint <- matrix( 0 , IP , 1 )
# err.constraint[ ( itempairs0$item1 %in% cluster2 )  
#        & ( itempairs0$item2 %in% cluster2 )  , 1 ] <- 1
# # set sum of error covariances to 1.2
# err.constraintV <- matrix(3*.4,1,1)
# 
# mod5 <- rasch.pml3( dat , error.corr=est.corrs , 
#          err.constraintM=err.constraint, err.constraintV=err.constraintV)
# 
# #****
# # Model 6: Constraint on sum of all correlations
# est.corrs <- matrix( 1:(I*I) ,  I , I )
# # define an error constraint matrix
# itempairs0 <- mod4$itempairs
# IP <- nrow(itempairs0)
# # define two side conditions
# err.constraint <- matrix( 0 , IP , 2 )
# err.constraintV <- matrix( 0 , 2 , 1)
# # sum of all correlations is zero
# err.constraint[  , 1 ] <- 1
# err.constraintV[1,1] <- 0
# # sum of items cluster c(1,2,3) is 0
# cluster2 <- c(1,2,3)
# err.constraint[ ( itempairs0$item1 %in%  cluster2 ) 
#        & ( itempairs0$item2 %in% cluster2 )  , 2 ] <- 1
# err.constraintV[2,1] <- 0
# 
# mod6 <- rasch.pml3( dat , error.corr=est.corrs , 
#     err.constraintM=err.constraint,  err.constraintV=err.constraintV)
# summary(mod6)
# 
# #############################################################################
# # EXAMPLE 5: 10 Items: Cluster 1 -> Items 1,2
# #         Cluster 2 -> Items 3,4,5;   Cluster 3 -> Items 7,8,9
# #############################################################################
# 
# set.seed(7650)
# I <- 10                             # number of items
# n <- 5000                           # number of persons
# b <- seq(-2,2, len=I)               # item difficulties
# bsamp <- b <- sample(b)             # sample item difficulties
# theta <- stats::rnorm( n , sd = 1 ) # person abilities
# # define itemcluster
# itemcluster <- rep(0,I)
# itemcluster[ 1:2 ] <- 1
# itemcluster[ 3:5 ] <- 2
# itemcluster[ 7:9 ] <- 3
# # define residual correlations
# rho <- c( .55 , .35 , .45)
# 
# # simulate data
# dat <- sim.rasch.dep( theta , b , itemcluster , rho )
# colnames(dat) <- paste("I" , seq(1,ncol(dat)) , sep="")
# 
# #***
# # Model 1: residual correlation (equal within item clusters)
# # define a matrix of integers for estimating error correlations
# error.corr <- diag(1,ncol(dat))
# for ( ii in 1:3){
#     ind.ii <- which( itemcluster == ii )
#     error.corr[ ind.ii , ind.ii ] <- ii
#         }
# # estimate the model
# mod1 <- rasch.pml3( dat , error.corr = error.corr )
# 
# #***
# # Model 2: residual correlation (different within item clusters)
# # define again a matrix of integers for estimating error correlations
# error.corr <- diag(1,ncol(dat))
# for ( ii in 1:3){
#     ind.ii <- which( itemcluster == ii )
#     error.corr[ ind.ii , ind.ii ] <- ii
#         }
# I <- ncol(error.corr)
# error.corr1 <- matrix( 1:(I*I) , ncol= I )
# error.corr <- error.corr1 * ( error.corr > 0 )                
# # estimate the model
# mod2 <- rasch.pml3( dat , error.corr = error.corr )
# 
# #***
# # Model 3: eliminate item pairs within itemclusters for PML estimation
# mod3 <- rasch.pml3( dat , itemcluster = itemcluster )
# 
# #***
# # Model 4: Rasch model ignoring dependency
# mod4 <- rasch.pml3( dat )
# 
# # compare different models
# summary(mod1)
# summary(mod2)
# summary(mod3)
# summary(mod4) 
# ## End(Not run)