immer_cml: Conditional Maximum Likelihood Estimation for the Linear Logistic Partial Credit Model

Description

Conditional maximum likelihood estimation for the linear logistic partial credit model (Molenaar, 1995; Andersen, 1995; Fischer, 1995). The immer_cml function allows for known integer discrimination parameters like in the one-parameter logistic model (Verhelst & Glas, 1995).

Usage

immer_cml(dat, weights = NULL, W = NULL, b_const = NULL, par_init = NULL, 
    a = NULL, irtmodel = NULL, normalization = "first", nullcats = "zeroprob", 
    diff = FALSE, ...)

## S3 method for class 'immer_cml':
summary(object,digits=3,...)

## S3 method for class 'immer_cml':
logLik(object,...)

## S3 method for class 'immer_cml':
anova(object,...)

Arguments

dat

Data frame with item responses

weights

Optional vector of sample weights

Design matrix $\bold{W}$ for linear logistic partial credit model. Every row corresponds to a parameter for item $i$ in category $h$

b_const

Optional vector of parameter constants $b_{0ih}$ which can be used for parameter fixings.

par_init

Optional vector of initial parameter estimates

Optional vector of integer item discriminations

irtmodel

Type of item response model. irtmodel="PCM" and irtmodel="PCM2" follow the conventions of the TAM package.

normalization

The type of normalization in partial credit models. Can be "first" for the first item or "sum" for a sum constraint.

nullcats

A string indicating whether categories with zero frequencies should have a probability of zero (by fixing the constant parameter to a large value of 99).

diff

Logical indicating whether the difference algorithm should be used. See psychotools::elementary_symmetric_functions for detail

...

Further arguments to be passed to stats::optim.

object

Object of class immer_cml

digits

Number of digits after decimal to be rounded.

Value

List with following entries:
itemData frame with item-category parameters
bItem-category parameters $b_{ih}$
coefficientsEstimated basis parameters $\beta_{v}$
vcovCovariance matrix of basis parameters $\beta_{v}$
par_summarySummary for basis parameters
loglikeValue of conditional log-likelihood
devianceDeviance
result_optimResult from optimiztation in stats::optim
WUsed design matrix $\bold{W}$
b_constUsed constant vector $b_{0ih}$
par_initUsed initial parameters
suffstatSufficient statistics
score_freqScore frequencies
datUsed dataset
used_personsUsed persons
NPNumber of missing data patterns
NNumber of persons
INumber of items
maxKMaximum number of categories per item
KMaximum score of all items
nparsNumber of estimated parameters
pars_infoInformation of definition of item-category parameters $b_{ih}$
parm_indexParameter indices
item_indexItem indices
scoreRaw score for each person

Details

The partial credit model can be written as $$P(X_{pi} = h ) \propto \exp( a_i h \theta_p - b_{ih})$$ where the item-category parameters $b_{ih}$ are linearly decomposed according to $$b_{ih} = \sum_{v} w_{ihv} \beta_v + b_{0ih}$$ with unknown basis parameters $\beta_v$ and fixed values $w_{ihv}$ of the design matrix $\bold{W}$ (specified in W) and constants $b_{0ih}$ (specified in b_const).

References

Andersen, E. B. (1995). Polytomous Rasch models and their estimation. In G. H. Fischer & I. W. Molenaar (Eds.). Rasch Models (pp. 39--52). New York: Springer. Fischer, G. H. (1995). The linear logistic test model. In G. H. Fischer & I. W. Molenaar (Eds.). Rasch Models (pp. 131--156). New York: Springer. Molenaar, I. W. (1995). Estimation of item parameters. In G. H. Fischer & I. W. Molenaar (Eds.). Rasch Models (pp. 39--52). New York: Springer. Verhelst, N. D. &, Glas, C. A. W. (1995). The one-parameter logistic model. In G. H. Fischer & I. W. Molenaar (Eds.). Rasch Models (pp. 215--238). New York: Springer.

Examples

Run this code

library(sirt)
library(psychotools)
library(TAM)
library(CDM)
library(eRm)

#############################################################################
# EXAMPLE 1: Dichotomous data data.read
#############################################################################

data(data.read , package="sirt")
dat <- data.read
I <- ncol(dat)

#----------------------------------------------------------------
#--- Model 1: Rasch model, setting first item difficulty to zero
mod1a <- immer_cml( dat=dat)
summary(mod1a)
logLik(mod1a) # extract log likelihood
coef(mod1a)   # extract coefficients

# estimate model in psychotools package
mod1b <- psychotools::raschmodel(dat)
summary(mod1b)
logLik(mod1b)

# estimate model in eRm package
mod1c <- eRm::RM(dat, sum0=FALSE)
summary(mod1c)
mod1c$etapar

# compare estimates of three packages
cbind( coef(mod1a) , coef(mod1b) , mod1c$etapar )

#----------------------------------------------------------------
#-- Model 2: Rasch model sum normalization
mod2a <- immer_cml( dat=dat , normalization = "sum")
summary(mod2a)

# compare estimation in TAM
mod2b <- tam.mml( dat , constraint="items"  )
summary(mod2b)
mod2b$A[,2,]

#----------------------------------------------------------------
#--- Model 3: some fixed item parameters
# fix item difficulties of items 1,4,8
# define fixed parameters in constant parameter vector
b_const <- rep(0,I)
fix_items <- c(1,4,8)
b_const[ fix_items ] <- c( -2.1 , .195 , -.95 )
# design matrix
W <- matrix( 0 , nrow=12 , ncol=9)
W[ cbind( setdiff( 1:12 , fix_items ) , 1:9 ) ] <- 1
colnames(W) <- colnames(dat)[ - fix_items ]
# estimate model
mod3 <- immer_cml( dat=dat , W=W , b_const=b_const)
summary(mod3)

#----------------------------------------------------------------
#--- Model 4: One parameter logistic model
# estimate non-integer item discriminations with 2PL model
I <- ncol(dat)
mod4a <- sirt::rasch.mml2( dat , est.a = 1:I )
summary(mod4a)
a <- mod4a$item$a     # extract (non-integer) item discriminations 
# estimate integer item discriminations ranging from 1 to 3
a_integer <- immer_opcat( a , hmean = 2 , min = 1 , max = 3 )
# estimate one-parameter model with fixed integer item discriminations
mod4 <- immer_cml( dat=dat , a = a_integer )
summary(mod4)

#----------------------------------------------------------------
#--- Model 5: Linear logistic test model

# define design matrix
W <- matrix( 0 , nrow=12 , ncol=5 )
colnames(W) <- c("B","C", paste0("Pos" , 2:4))
rownames(W) <- colnames(dat)
W[ 5:8 , "B" ] <- 1
W[ 9:12 , "C" ] <- 1
W[ c(2,6,10) , "Pos2" ] <- 1
W[ c(3,7,11) , "Pos3" ] <- 1
W[ c(4,8,12) , "Pos4" ] <- 1

# estimation with immer_cml
mod5a <- immer_cml( dat , W=W )
summary(mod5a)

# estimation in eRm package
mod5b <- eRm::LLTM( dat , W=W )
summary(mod5b)

# compare models 1 and 5 by a likelihood ratio test
anova( mod1a , mod5a )

#############################################################################
# EXAMPLE 2: Polytomous data | data.Students
#############################################################################

data(data.Students,package="CDM")
dat <- data.Students
dat <- dat[ , grep("act" , colnames(dat) ) ]
dat <- dat[1:400,]  # select a subdataset
dat <- dat[ rowSums( 1 - is.na(dat) ) > 1 , ] 
    # remove persons with less than two valid responses

#----------------------------------------------------------------
#--- Model 1: Partial credit model with constraint on first parameter
mod1a <- immer_cml( dat= dat )
summary(mod1a)
# compare pcmodel function from psychotools package
mod1b <- psychotools::pcmodel( dat )
summary(mod1b)
# estimation in eRm package
mod1c <- eRm::PCM( dat  , sum0=FALSE )
  # -> subjects with only one valid response must be removed
summary(mod1c)

#----------------------------------------------------------------
#-- Model 2: Partial credit model with sum constraint on item difficulties
mod2a <- immer_cml( dat=dat , irtmodel="PCM2" , normalization="sum")
summary(mod2a)
# compare with estimation in TAM
mod2b <- TAM::tam.mml( dat , irtmodel="PCM2" , constraint="items")
summary(mod2b)

#----------------------------------------------------------------
#-- Model 3: Partial credit model with fixed integer item discriminations
mod3 <- immer_cml( dat= dat , normalization="first", a=c(2,2,1,3,1) )
summary(mod3)

#############################################################################
# EXAMPLE 3: Polytomous data | Extracting the structure of W matrix
#############################################################################

data(data.mixed1, package="sirt")
dat <- data.mixed1

# use non-exported function "lpcm_data_prep" to extract the meaning
# of the rows in W which are contained in value "pars_info"
res <- immer:::lpcm_data_prep( dat , weights=NULL , a=NULL )
pi2 <- res$pars_info

# create design matrix with some restrictions on item parameters
W <- matrix( 0 , nrow=nrow(pi2) , ncol=2 )
colnames(W) <- c( "P2" , "P3" )
rownames(W) <- res$parnames

# joint item parameter for items I19 and I20 fixed at zero
# item parameter items I21 and I22
W[ 3:10 , 1 ] <- pi2$cat[ 3:10 ]
# item parameters I23, I24 and I25
W[ 11:13 , 2] <- 1

# estimate model with design matrix W
mod <- immer_cml( dat, W=W)
summary(mod)

#############################################################################
# EXAMPLE 4: Partial credit model with raters
#############################################################################

data(data.immer07)
dat <- data.immer07

#*** reshape dataset for one variable
dfr1 <- immer_reshape_wideformat( dat$I1 , rater = dat$rater , pid = dat$pid )

#-- extract structure of design matrix
res <- immer:::lpcm_data_prep( dat=dfr1[,-1] , weights=NULL , a=NULL)
pars_info <- res$pars_info

# specify design matrix for partial credit model and main rater effects
# -> set sum of all rater effects to zero
W <- matrix( 0 , nrow=nrow(pars_info) , ncol=3+2 )
rownames(W) <- rownames(pars_info)
colnames(W) <- c( "Cat1" , "Cat2" , "Cat3" , "R1" , "R2" )
# define item parameters
W[ cbind( pars_info$index , pars_info$cat ) ] <- 1
# define rater parameters
W[ paste(pars_info$item) == "R1" , "R1" ] <- 1
W[ paste(pars_info$item) == "R2" , "R2" ] <- 1
W[ paste(pars_info$item) == "R3" , c("R1","R2") ] <- -1
# set parameter of first category to zero for identification constraints
W <- W[,-1]

# estimate model
mod <- immer_cml( dfr1[,-1] , W=W)
summary(mod)

#############################################################################
# EXAMPLE 5: Multi-faceted Rasch model | Estimation with a design matrix
#############################################################################

data(data.immer07)
dat <- data.immer07

#*** reshape dataset
dfr1 <- immer_reshape_wideformat( dat[, paste0("I",1:4) ] , rater = dat$rater , 
                pid = dat$pid )

#-- structure of design matrix
res <- immer:::lpcm_data_prep( dat=dfr1[,-1] , weights=NULL , a=NULL)
pars_info <- res$pars_info

#--- define design matrix for multi-faceted Rasch model with only main effects
W <- matrix( 0 , nrow=nrow(pars_info) , ncol=3+2+2 )
parnames <- rownames(W) <- rownames(pars_info)
colnames(W) <- c( paste0("I",1:3) , paste0("Cat",1:2) , paste0("R",1:2) )
#+ define item effects
for (ii in c("I1","I2","I3") ){
    ind <- grep( ii , parnames ) 
    W[ ind  , ii ] <- pars_info$cat[ind ]
                }
ind <- grep( "I4" , parnames )
W[ ind  , c("I1","I2","I3") ] <- -pars_info$cat[ind ]               
#+ define step parameters
for (cc in 1:2 ){
    ind <- which( pars_info$cat == cc )
    W[ ind  , paste0("Cat",1:cc) ] <- 1
                }                                         
#+ define rater effects
for (ii in c("R1","R2") ){
    ind <- grep( ii , parnames ) 
    W[ ind  , ii ] <- pars_info$cat[ind ]
                }
ind <- grep( "R3" , parnames )
W[ ind  , c("R1","R2") ] <- -pars_info$cat[ind ]

#--- estimate model with immer_cml
mod1 <- immer_cml( dfr1[,-1] , W=W , par_init = rep(0,ncol(W) ) )
summary(mod1)

#--- comparison with estimation in TAM
resp <- dfr1[,-1]
mod2 <- TAM::tam.mml.mfr( resp=dat[,-c(1:2)] , facets = dat[, "rater" , drop=FALSE ] ,
            pid = dat$pid , formulaA = ~ item + step + rater )
summary(mod2)

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