tam.wle: Weighted Likelihood Estimation and Maximum Likelihood Estimation of Person Parameters

Description

Compute the weighted likelihood estimator (Warm, 1989) for objects of classes tam, tam.mml and tam.jml, respectively.

Usage

tam.wle(tamobj, ...)

tam.mml.wle(tamobj, score.resp=NULL, WLE=TRUE, adj=0.3, Msteps=20, convM=1e-04)

tam.mml.wle2(tamobj, score.resp=NULL, WLE=TRUE, adj=0.3, Msteps=20, convM=1e-04)

tam.jml.WLE(tamobj, resp, resp.ind, A, B, nstud, nitems, maxK, convM, 
    PersonScores, theta, xsi, Msteps, WLE=FALSE, theta.fixed = NULL)

Arguments

tamobj

An object generated by tam.mml or tam.jml

score.resp

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 t

WLE

A logical indicating whether the weighted likelihood estimate (WLE, WLE=TRUE) or the maximum likelihood estimate (MLE, WLE=FALSE) should be used.

adj

Adjustment in WLE estimation for extreme scores (i.e. all or none items were correctly solved)

Msteps

Maximum number of iterations

convM

Convergence criterion

resp

Data frame with item responses (only for tam.jml.WLE)

resp.ind

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)

nstud

Number of persons (applies only to tam.jml.WLE)

nitems

Number of items (applies only to tam.jml.WLE)

maxK

Maximum item score (applies only to tam.jml.WLE)

PersonScores

A vector containing the sufficient statistics for the person parameters (applies only to tam.jml.WLE)

theta

Initial $\theta$ estimate (applies only to tam.jml.WLE)

xsi

Parameter vector $\xi$ (applies only to tam.jml.WLE)

theta.fixed

Matrix for fixed person parameters $\theta$. The first column includes the index whereas the second column includes the fixed value.

...

Further arguments to be passed

Value

For tam.wle.mml and tam.wle.mml2, it is a data frame with following columns:
pidPerson identifier
PersonScoresScore of each person
PersonMaxMaximum score of each person
thetaWeighted likelihood estimate (WLE) or MLE
errorStandard error of the WLE or MLE
WLE.relWLE reliability (same value for all persons)
For tam.jml.WLE, it is a list with following entries:
thetaWeighted likelihood estimate (WLE) or MLE
errorWLEStandard error of the WLE or MLE
meanChangeWLEMean change between updated and previous ability estimates from last iteration

References

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.

Examples

Run this code

#############################################################################
# EXAMPLE 1: 1PL model, sim.rasch data
#############################################################################
data(sim.rasch)
# estimate Rasch model
mod1 <- tam.mml(resp=sim.rasch) 
# WLE estimation
wle1 <- tam.wle( mod1 )
  ## ----
  ## WLE Reliability = 0.894 

# scoring for a different dataset containing same items (first 10 persons in 
#  sim.rasch)
wle2 <- tam.wle( mod1 , score.resp=sim.rasch[1:10,])

# compare results to earlier wle-implementation
wle3 <- tam.mml.wle( mod1 )
stopifnot( all(wle3==wle1) )

#############################################################################
# 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.mml( resp=resp , Q=Q , control=list(snodes=1000,maxiter=50) )
summary(mod)

# WLE estimates
wmod <- 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.mml( dat )
summary(mod1)

#-- WLE estimation in TAM
tamw1 <- 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.mml.2pl( dat , irtmodel="GPCM" )
summary(mod2)

#-- WLE estimation in TAM
tamw2 <- 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