Conditional Maximum Likelihood Estimation for the Linear Logistic Partial Credit Model

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, use_rcpp=FALSE, ...)

# S3 method for immer_cml
summary(object, digits=3, file=NULL, ...)

# S3 method for immer_cml
logLik(object,...)

# S3 method for immer_cml
anova(object,...)

# S3 method for immer_cml
coef(object,...)

# S3 method for immer_cml
vcov(object,...)

Arguments

dat: Data frame with item responses
weights: Optional vector of sample weights
W: 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
a: 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 details.
use_rcpp: Logical indicating whether Rcpp package should be used for computation.
...: Further arguments to be passed to stats::optim.
object: Object of class immer_cml
digits: Number of digits after decimal to be rounded.
file: Name of a file in which the output should be sunk

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).

Value

List with following entries:

item: Data frame with item-category parameters
b: Item-category parameters $b_{ih}$
coefficients: Estimated basis parameters $\beta_{v}$
vcov: Covariance matrix of basis parameters $\beta_{v}$
par_summary: Summary for basis parameters
loglike: Value of conditional log-likelihood
deviance: Deviance
result_optim: Result from optimization in stats::optim
W: Used design matrix $\bold{W}$
b_const: Used constant vector $b_{0ih}$
par_init: Used initial parameters
suffstat: Sufficient statistics
score_freq: Score frequencies
dat: Used dataset
used_persons: Used persons
NP: Number of missing data patterns
N: Number of persons
I: Number of items
maxK: Maximum number of categories per item
K: Maximum score of all items
npars: Number of estimated parameters
pars_info: Information of definition of item-category parameters $b_{ih}$
parm_index: Parameter indices
item_index: Item indices
score: Raw score for each person

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

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

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

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

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

if (FALSE) {
library(eRm)

# 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::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::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::immer_opcat( a, hmean=2, min=1, max=3 )
# estimate one-parameter model with fixed integer item discriminations
mod4 <- immer::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::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::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::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::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::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::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::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::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::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)
}