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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=.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 )

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)

## S3 method for class 'tam.wle'
summary(object, file=NULL, digits=3, ...)

## S3 method for class 'tam.wle'
print(x, digits=3, ...)

Arguments

tamobj

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.

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$resp) is chosen. Note that items in score.resp must be the same (and in the same order) as in tamobj$resp.

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 MLE estimation for extreme scores (i.e. all or none items were correctly solved). This argument is not used if WLE=TRUE.

Msteps

Maximum number of iterations

convM

Convergence criterion

progress

Logical indicating whether progress should be displayed.

output.prob

Logical indicating whether evaluated probabilities should be included in the list of outputs.

pid

Optional vector of person identifiers

resp

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

resp.ind

Data frame with response indicators (only for tam.jml.WLE)

A

Design matrix A (applies only to tam.jml.WLE)

B

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 θ estimate (applies only to tam.jml.WLE)

xsi

Parameter vector ξ (applies only to tam.jml.WLE)

theta.fixed

Matrix for fixed person parameters θ. The first column includes the index whereas the second column includes the fixed value.

damp

Numeric value between 0 and 1 indicating amount of dampening increments in θ estimates during iterations

version

Integer with possible values 2 or 3. In case of missing item responses, version=3 will typically be more efficient.

...

Further arguments to be passed

object

Object of class tam.wle

x

Object of class tam.wle

file

Optional file name in which the object summary should be written.

digits

Number of digits for rounding

Value

For tam.wle.mml and tam.wle.mml2, it is a data frame with following columns:

pid

Person identifier

PersonScores

Score of each person

PersonMax

Maximum score of each person

theta

Weighted likelihood estimate (WLE) or MLE

error

Standard error of the WLE or MLE

WLE.rel

WLE reliability (same value for all persons)


For tam.jml.WLE, it is a list with following entries:

theta

Weighted likelihood estimate (WLE) or MLE

errorWLE

Standard error of the WLE or MLE

meanChangeWLE

Mean 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. doi: 10.1007/BF02294627

See Also

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.

Examples

#############################################################################
# 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 )

## Not run: 
#############################################################################
# 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

## End(Not run)

TAM

Test Analysis Modules

v3.6-45
GPL (>= 2)
Authors
Alexander Robitzsch [aut,cre] (<https://orcid.org/0000-0002-8226-3132>), Thomas Kiefer [aut], Margaret Wu [aut]
Initial release
2021-04-22 14:35:52

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