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 )
# S3 method for rasch.pml
summary(object,...)

Value

A list with following entries:

item: Data frame with estimated item parameters
iter: Number of iterations
deviance: Pseudolikelihood multiplied by minus 2
b: Estimated item difficulties
sigma: Estimated standard deviation
dat: Original dataset
ic: Data frame with information criteria (sample size, number of estimated parameters, pseudolikelihood information criterion PLIC)
link: Used link function (only probit is permitted)
itempairs: Estimated statistics of item pairs
error.corr: Estimated error correlation matrix
eps.corr: Vectorized error correlation matrix
omega.rel: Reliability of the sum score according to Green and Yang (2009). If some item pairs are excluded in the estimation, the residual correlation for these item pairs is assumed to be zero.
...

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

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 $X_{pi}^\ast$ 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.

Examples

Run this code

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

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

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

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

if (FALSE) {
#*****
# 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 <- sirt::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 <- sirt::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 <- sirt::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 <- sirt::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 <- sirt::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 <- sirt::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 <- sirt::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 <- sirt::sim.raschtype( theta=theta, b=b )

# estimation with rasch.pml and probit link
mod1 <- sirt::rasch.pml3( dat )
summary(mod1)

# estimation with rasch.mml2 function
mod2 <- sirt::rasch.mml2( dat )

# estimate item parameters for groups with five item parameters each
est.b <- rep( 1:(I/2), each=2 )
mod3 <- sirt::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 <- sirt::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 <- sirt::rasch.pml3( dat )
#***
# Model 2: estimation when pairs of locally dependent items are eliminated
mod2 <- sirt::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 <- sirt::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 <- sirt::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 <- sirt::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 <- sirt::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 <- sirt::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 <- sirt::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 <- sirt::rasch.pml3( dat, error.corr=error.corr )

#***
# Model 3: eliminate item pairs within itemclusters for PML estimation
mod3 <- sirt::rasch.pml3( dat, itemcluster=itemcluster )

#***
# Model 4: Rasch model ignoring dependency
mod4 <- sirt::rasch.pml3( dat )

# compare different models
summary(mod1)
summary(mod2)
summary(mod3)
summary(mod4)
}

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