immer_cml | R Documentation |
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).
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 class 'immer_cml'
summary(object, digits=3, file=NULL, ...)
## S3 method for class 'immer_cml'
logLik(object,...)
## S3 method for class 'immer_cml'
anova(object,...)
## S3 method for class 'immer_cml'
coef(object,...)
## S3 method for class 'immer_cml'
vcov(object,...)
dat |
Data frame with item responses |
weights |
Optional vector of sample weights |
W |
Design matrix |
b_const |
Optional vector of parameter constants |
par_init |
Optional vector of initial parameter estimates |
a |
Optional vector of integer item discriminations |
irtmodel |
Type of item response model. |
normalization |
The type of normalization in partial credit models. Can be |
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 |
diff |
Logical indicating whether the difference algorithm should be used. See
|
use_rcpp |
Logical indicating whether Rcpp package should be used for computation. |
... |
Further arguments to be passed to |
object |
Object of class |
digits |
Number of digits after decimal to be rounded. |
file |
Name of a file in which the output should be sunk |
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
).
List with following entries:
item |
Data frame with item-category parameters |
b |
Item-category parameters |
coefficients |
Estimated basis parameters |
vcov |
Covariance matrix of basis parameters |
par_summary |
Summary for basis parameters |
loglike |
Value of conditional log-likelihood |
deviance |
Deviance |
result_optim |
Result from optimization in |
W |
Used design matrix |
b_const |
Used constant vector |
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 |
parm_index |
Parameter indices |
item_index |
Item indices |
score |
Raw score for each person |
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.
For CML estimation see also the eRm and psychotools packages and the
functions eRm::RM
and
psychotools::raschmodel
for the Rasch model
and eRm::PCM
and
psychotools::pcmodel
for the partial
credit model.
See eRm::LLTM
for the linear logistic test model
and eRm::LPCM
for the linear logistic partial
credit model in the eRm package for CML implementations.
The immer_cml
function makes use of
psychotools::elementary_symmetric_functions
.
For CML estimation with sample weights see also the RM.weights package.
#############################################################################
# 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
## Not run:
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)
## End(Not run)
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