rasch.pml3 | R Documentation |
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
).
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 class 'rasch.pml'
summary(object,...)
dat |
An |
est.b |
Vector of integers of length |
est.a |
Vector of integers of length |
est.sigma |
Should sigma (the trait standard deviation) be estimated?
The default is |
itemcluster |
Optional vector of length |
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 |
err.constraintM |
An optional |
err.constraintV |
An optional |
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 |
object |
Object of class |
... |
Further arguments to be passed |
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).
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 |
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. |
... |
This function needs the combinat library.
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.
Get a summary of rasch.pml3
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
.
#############################################################################
# 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)
## 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 <- 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)
## End(Not run)
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