TAM: tam.wle – R documentation

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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

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