tam.wle: Weighted Likelihood Estimation and Maximum Likelihood...
In TAM: Test Analysis Modules
tam.wle R Documentation
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, 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)
## 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
theta_init
Initial theta values
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 \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.
damp
Numeric value between 0 and 1 indicating amount of dampening
increments in \theta 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.
\Sexpr[results=rd]{tools:::Rd_expr_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 documentation built on May 29, 2024, 2:20 a.m.
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| tam.wle | R Documentation |
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, 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)
## 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 |
score.resp |
An optional data frame for which WLEs or MLEs
should be calculated. In case of the default |
WLE |
A logical indicating whether the weighted likelihood estimate
(WLE, |
adj |
Adjustment in MLE estimation for extreme scores (i.e. all or none
items were correctly solved). This argument is not used if
|
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 |
theta_init |
Initial theta values |
resp |
Data frame with item responses (only for |
resp.ind |
Data frame with response indicators (only for |
A |
Design matrix |
B |
Design matrix |
nstud |
Number of persons (applies only to |
nitems |
Number of items (applies only to |
maxK |
Maximum item score (applies only to |
PersonScores |
A vector containing the sufficient statistics for the
person parameters (applies only to |
theta |
Initial |
xsi |
Parameter vector |
theta.fixed |
Matrix for fixed person parameters |
damp |
Numeric value between 0 and 1 indicating amount of dampening
increments in |
version |
Integer with possible values 2 or 3. In case of missing item responses,
|
... |
Further arguments to be passed |
object |
Object of class |
x |
Object of class |
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. \Sexpr[results=rd]{tools:::Rd_expr_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)
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