tam.wle.RdCompute the weighted likelihood estimator (Warm, 1989)
for objects of classes tam, tam.mml and tam.jml,
respectively.
tam.wle(tamobj, ...)
tam.mml.wle( tamobj, score.resp=NULL, WLE=TRUE, adj=.3, Msteps=20,
convM=.0001, progress=TRUE, output.prob=FALSE )
tam.mml.wle2(tamobj, score.resp=NULL, WLE=TRUE, adj=0.3, Msteps=20, convM=1e-04,
progress=TRUE, output.prob=FALSE, pid=NULL, theta_init=NULL )
tam_jml_wle(tamobj, resp, resp.ind, A, B, nstud, nitems, maxK, convM,
PersonScores, theta, xsi, Msteps, WLE=FALSE, theta.fixed=NULL, progress=FALSE,
output.prob=TRUE, damp=0, version=2, theta_proc=NULL)
# S3 method for tam.wle
summary(object, file=NULL, digits=3, ...)
# S3 method for tam.wle
print(x, digits=3, ...)An object generated by tam.mml or tam.jml. The object can also
be a list containing (at least the) inputs AXsi, B and
resp and therefore allows WLE estimation without fitting models
in TAM.
An optional data frame for which WLEs or MLEs
should be calculated. In case of the default NULL,
resp from tamobj (i.e. tamobj$resp) is chosen.
Note that items in score.resp must be the same (and in the
same order) as in tamobj$resp.
A logical indicating whether the weighted likelihood estimate
(WLE, WLE=TRUE) or the maximum likelihood estimate (MLE, WLE=FALSE)
should be used.
Adjustment in MLE estimation for extreme scores (i.e. all or none
items were correctly solved). This argument is not used if
WLE=TRUE.
Maximum number of iterations
Convergence criterion
Logical indicating whether progress should be displayed.
Logical indicating whether evaluated probabilities should be included in the list of outputs.
Optional vector of person identifiers
Initial theta values
Data frame with item responses (only for tam.jml.WLE)
Data frame with response indicators (only for tam.jml.WLE)
Design matrix \(A\) (applies only to tam.jml.WLE)
Design matrix \(B\) (applies only to tam.jml.WLE)
Number of persons (applies only to tam.jml.WLE)
Number of items (applies only to tam.jml.WLE)
Maximum item score (applies only to tam.jml.WLE)
A vector containing the sufficient statistics for the
person parameters (applies only to tam.jml.WLE)
Initial \(\theta\) estimate (applies only to tam.jml.WLE)
Parameter vector \(\xi\) (applies only to tam.jml.WLE)
Matrix for fixed person parameters \(\theta\). The first column includes the index whereas the second column includes the fixed value.
Numeric value between 0 and 1 indicating amount of dampening increments in \(\theta\) estimates during iterations
Integer with possible values 2 or 3. In case of missing item responses,
version=3 will typically be more efficient.
Function for processing theta within iterations.
Further arguments to be passed
Object of class tam.wle
Object of class tam.wle
Optional file name in which the object summary should be written.
Number of digits for rounding
For tam.wle.mml and tam.wle.mml2, it is a data frame with following
columns:
Person identifier
Score of each person
Maximum score of each person
Weighted likelihood estimate (WLE) or MLE
Standard error of the WLE or MLE
WLE reliability (same value for all persons)
For tam.jml.WLE, it is a list with following entries:
Weighted likelihood estimate (WLE) or MLE
Standard error of the WLE or MLE
Mean change between updated and previous ability estimates from last iteration
Penfield, R. D., & Bergeron, J. M. (2005). Applying a weighted maximum likelihood latent trait estimator to the generalized partial credit model. Applied Psychological Measurement, 29, 218-233.
Warm, T. A. (1989). Weighted likelihood estimation of ability in item response theory. Psychometrika, 54, 427-450. doi:10.1007/BF02294627
See the PP::PP_gpcm function
in the PP package for more person
parameter estimators for the partial credit model (Penfield & Bergeron, 2005).
See the S3 method IRT.factor.scores.tam.
#############################################################################
# EXAMPLE 1: 1PL model, data.sim.rasch
#############################################################################
data(data.sim.rasch)
# estimate Rasch model
mod1 <- TAM::tam.mml(resp=data.sim.rasch)
# WLE estimation
wle1 <- TAM::tam.wle( mod1 )
## WLE Reliability=0.894
print(wle1)
summary(wle1)
# scoring for a different dataset containing same items (first 10 persons in sim.rasch)
wle2 <- TAM::tam.wle( mod1, score.resp=data.sim.rasch[1:10,])
#--- WLE estimation without using a TAM object
#* create an input list
input <- list( resp=data.sim.rasch, AXsi=mod1$AXsi, B=mod1$B )
#* estimation
wle2b <- TAM::tam.mml.wle2( input )
if (FALSE) {
#############################################################################
# EXAMPLE 2: 3-dimensional Rasch model | data.read from sirt package
#############################################################################
data(data.read, package="sirt")
# define Q-matrix
Q <- matrix(0,12,3)
Q[ cbind( 1:12, rep(1:3,each=4) ) ] <- 1
# redefine data: create some missings for first three cases
resp <- data.read
resp[1:2, 5:12] <- NA
resp[3,1:4] <- NA
## > head(resp)
## A1 A2 A3 A4 B1 B2 B3 B4 C1 C2 C3 C4
## 2 1 1 1 1 NA NA NA NA NA NA NA NA
## 22 1 1 0 0 NA NA NA NA NA NA NA NA
## 23 NA NA NA NA 1 0 1 1 1 1 1 1
## 41 1 1 1 1 1 1 1 1 1 1 1 1
## 43 1 0 0 1 0 0 1 1 1 0 1 0
## 63 1 1 0 0 1 0 1 1 1 1 1 1
# estimate 3-dimensional Rasch model
mod <- TAM::tam.mml( resp=resp, Q=Q, control=list(snodes=1000,maxiter=50) )
summary(mod)
# WLE estimates
wmod <- TAM::tam.wle(mod, Msteps=3)
summary(wmod)
## head(round(wmod,2))
## pid N.items PersonScores.Dim01 PersonScores.Dim02 PersonScores.Dim03
## 2 1 4 3.7 0.3 0.3
## 22 2 4 2.0 0.3 0.3
## 23 3 8 0.3 3.0 3.7
## 41 4 12 3.7 3.7 3.7
## 43 5 12 2.0 2.0 2.0
## 63 6 12 2.0 3.0 3.7
## PersonMax.Dim01 PersonMax.Dim02 PersonMax.Dim03 theta.Dim01 theta.Dim02
## 2 4.0 0.6 0.6 1.06 NA
## 22 4.0 0.6 0.6 -0.96 NA
## 23 0.6 4.0 4.0 NA -0.07
## 41 4.0 4.0 4.0 1.06 0.82
## 43 4.0 4.0 4.0 -0.96 -1.11
## 63 4.0 4.0 4.0 -0.96 -0.07
## theta.Dim03 error.Dim01 error.Dim02 error.Dim03 WLE.rel.Dim01
## 2 NA 1.50 NA NA -0.1
## 22 NA 1.11 NA NA -0.1
## 23 0.25 NA 1.17 1.92 -0.1
## 41 0.25 1.50 1.48 1.92 -0.1
## 43 -1.93 1.11 1.10 1.14 -0.1
# (1) Note that estimated WLE reliabilities are not trustworthy in this example.
# (2) If cases do not possess any observations on dimensions, then WLEs
# and their corresponding standard errors are set to NA.
#############################################################################
# EXAMPLE 3: Partial credit model | Comparison WLEs with PP package
#############################################################################
library(PP)
data(data.gpcm)
dat <- data.gpcm
I <- ncol(dat)
#****************************************
#*** Model 1: Partial Credit Model
# estimation in TAM
mod1 <- TAM::tam.mml( dat )
summary(mod1)
#-- WLE estimation in TAM
tamw1 <- TAM::tam.wle( mod1 )
#-- WLE estimation with PP package
# convert AXsi parameters into thres parameters for PP
AXsi0 <- - mod1$AXsi[,-1]
b <- AXsi0
K <- ncol(AXsi0)
for (cc in 2:K){
b[,cc] <- AXsi0[,cc] - AXsi0[,cc-1]
}
# WLE estimation in PP
ppw1 <- PP::PP_gpcm( respm=as.matrix(dat), thres=t(b), slopes=rep(1,I) )
#-- compare results
dfr <- cbind( tamw1[, c("theta","error") ], ppw1$resPP)
head( round(dfr,3))
## theta error resPP.estimate resPP.SE nsteps
## 1 -1.006 0.973 -1.006 0.973 8
## 2 -0.122 0.904 -0.122 0.904 8
## 3 0.640 0.836 0.640 0.836 8
## 4 0.640 0.836 0.640 0.836 8
## 5 0.640 0.836 0.640 0.836 8
## 6 -1.941 1.106 -1.941 1.106 8
plot( dfr$resPP.estimate, dfr$theta, pch=16, xlab="PP", ylab="TAM")
lines( c(-10,10), c(-10,10) )
#****************************************
#*** Model 2: Generalized partial Credit Model
# estimation in TAM
mod2 <- TAM::tam.mml.2pl( dat, irtmodel="GPCM" )
summary(mod2)
#-- WLE estimation in TAM
tamw2 <- TAM::tam.wle( mod2 )
#-- WLE estimation in PP
# convert AXsi parameters into thres and slopes parameters for PP
AXsi0 <- - mod2$AXsi[,-1]
slopes <- mod2$B[,2,1]
K <- ncol(AXsi0)
slopesM <- matrix( slopes, I, ncol=K )
AXsi0 <- AXsi0 / slopesM
b <- AXsi0
for (cc in 2:K){
b[,cc] <- AXsi0[,cc] - AXsi0[,cc-1]
}
# estimation in PP
ppw2 <- PP::PP_gpcm( respm=as.matrix(dat), thres=t(b), slopes=slopes )
#-- compare results
dfr <- cbind( tamw2[, c("theta","error") ], ppw2$resPP)
head( round(dfr,3))
## theta error resPP.estimate resPP.SE nsteps
## 1 -0.476 0.971 -0.476 0.971 13
## 2 -0.090 0.973 -0.090 0.973 13
## 3 0.311 0.960 0.311 0.960 13
## 4 0.311 0.960 0.311 0.960 13
## 5 1.749 0.813 1.749 0.813 13
## 6 -1.513 1.032 -1.513 1.032 13
}