TAM (version 1.6-0)

tam.jml: Joint Maximum Likelihood Estimation

Description

This function estimate unidimensional item response models with joint maximum likelihood (JML, see e.g. Linacre, 1994).

Usage

tam.jml(resp, group = NULL, adj = .3, disattenuate = FALSE, bias = TRUE, 
    xsi.fixed = NULL, xsi.inits = NULL, theta.fixed=NULL, A = NULL, B = NULL, Q = NULL, 
    ndim = 1, pweights = NULL, control = list())

tam.jml2(resp, group = NULL, adj = .3 , disattenuate = FALSE, bias = TRUE, 
    xsi.fixed = NULL, xsi.inits = NULL, A = NULL, B = NULL, Q = NULL, 
    ndim = 1, pweights = NULL, control = list())

## S3 method for class 'tam.jml':
summary(object,file=NULL,\dots)

Arguments

resp

A matrix of item responses. Missing responses must be declared as NA.

group

An optional vector of group identifier

disattenuate

An optional logical indicating whether the person parameters should be disattenuated due to unreliability? The disattenuation is conducted by applying the Kelley formula.

adj

Adjustment constant which is subtracted or added to extreme scores (score of zero or maximum score. The default is a value of 0.3

bias

A logical which indicates if JML bias should be reduced by multiplying item parameters by the adjustment factor of $(I-1)/I$

xsi.fixed

An optional matrix with two columns for fixing some of the basis parameters $\xi$ of item intercepts. 1st column: Index of $\xi$ parameter, 2nd column: Fixed value of $\xi$ parameter

xsi.inits

An optional vector of initial $\xi$ parameters. Note that all parameters must be specified and the vector is not of the same format as xsi.fixed.

theta.fixed

Matrix for fixed person parameters $\theta$. The first column includes the index whereas the second column includes the fixed value. This argument only applies to tam.jml.

A

A design array $A$ for item category intercepts. For item $i$ and category $k$, the threshold is specified as $\sum _j a_{ikj} \xi_j$.

B

A design array for scoring item category responses. Entries in $B$ represent item loadings on abilities $\theta$.

Q

A Q-matrix which defines loadings of items on dimensions.

ndim

Number of dimensions in the model. The default is 1.

pweights

An optional vector of person weights.

control

A list of control arguments. See tam.mml for more details.

object

Object of class tam.jml (only for summary.tam function)

file

A file name in which the summary output will be written (only for summary.tam.jml function)

...

Further arguments to be passed

Value

• A list with following entries
• item
  Data frame with item parameters
• xsi
  Vector of item parameters $\xi$
• errorP
  Standard error of item parameters $\xi$
• theta
  MLE in final step
• errorWLE
  Standard error of WLE
• WLE
  WLE in last iteration
• WLEreliability
  WLE reliability
• PersonScores
  Scores for each person (sufficient statistic)
• ItemScore
  Sufficient statistic for each item parameter
• PersonMax
  Maximum person score
• ItemMax
  Maximum item score
• deviance
  Deviance
• deviance.history
  Deviance history in iterations
• resp
  Original data frame
• resp.ind
  Response indicator matrix
• group
  Vector of group identifiers (if provided as an argument)
• pweights
  Vector of person weights
• A
  Design matrix $A$ of item intercepts
• B
  Loading (or scoring) matrix $B$
• nitems
  Number of items
• maxK
  Maximum number of categories
• nstud
  Number of persons in resp
• resp.ind.list
  Like resp.ind, only in the format of a list
• xsi.fixed
  Fixed $\xi$ item parameters
• control
  Control list
• ...

Details

The function tam.jml2 is just a bit more efficient (for most applications) implementation than tam.jml.

References

Linacre, J. M. (1994). Many-Facet Rasch Measurement. Chicago: MESA Press.

See Also

For estimating the same class of models with marginal maximum likelihood estimation see tam.mml.

Examples

#############################################################################
# EXAMPLE 1: Dichotomous data
#############################################################################
data(sim.rasch)
resp <- sim.rasch[1:700 , seq( 1 , 40 , len=10)  ]  # subsample
# estimate the Rasch model with JML (function 'tam.jml')
mod1a <- tam.jml(resp=resp) 
summary(mod1a)
itemfit <- tam.fit(mod1a)$fit.item

# the same model but using function 'tam.jml2'
mod1b <- tam.jml2(resp=resp)
summary(mod1b)
itemfit <- tam.fit(mod1b)$fit.item

# compare results with Rasch model estimated by MML
mod1c <- tam.mml(resp=resp )
# plot estimated parameters
plot( mod1b$xsi , mod1c$xsi$xsi , pch=16 ,
    xlab= expression( paste( xi[i] , "(JML)" )) ,
    ylab= expression( paste( xi[i] , "(MML)" )) ,
    main="Item Parameter Estimate Comparison")
lines( c(-5,5) , c(-5,5) , col="gray" )

# Now, the adjustment pf .05 instead of the default .3 is used.
mod1d <- tam.jml2(resp=resp , adj=.05)
# compare item parameters
round( rbind( mod1b$xsi , mod1d$xsi ) , 3 )
  ##          [,1]   [,2]   [,3]   [,4]   [,5]  [,6]  [,7]  [,8]  [,9] [,10]
  ##   [1,] -2.076 -1.743 -1.217 -0.733 -0.338 0.147 0.593 1.158 1.570 2.091
  ##   [2,] -2.105 -1.766 -1.233 -0.746 -0.349 0.139 0.587 1.156 1.574 2.108

# person parameters for persons with a score 0, 5 and 10
pers1 <- data.frame( "score_adj0.3"= mod1b$PersonScore , "theta_adj0.3"= mod1b$theta , 
           "score_adj0.05"= mod1d$PersonScore , "theta_adj0.05"= mod1d$theta  )
round( pers1[ c(698, 683, 608) , ] ,3  )
  ##       score_adj0.3 theta_adj0.3 score_adj0.05 theta_adj0.05
  ##   698          0.3       -4.404          0.05        -6.283
  ##   683          5.0       -0.070          5.00        -0.081
  ##   608          9.7        4.315          9.95         6.179

#*** Models in which some item parameters are fixed
xsi.fixed <- cbind( c(1,3,9,10) , c(-2 , -1.2 , 1.6 , 2 ) )
mod1e <- tam.jml2( resp=resp , xsi.fixed = xsi.fixed )
summary(mod1e)
# the same fixing in tam.jml
mod1f <- tam.jml( resp=resp , xsi.fixed = xsi.fixed )
summary(mod1f)

#*** Model in which also some person parameters theta are fixed
# fix theta parameters of persons 2, 3, 4 and 33 to values -2.9, ...
theta.fixed <- cbind( c(2,3,4,33) , c( -2.9 , 4 , -2.9 , -2.9 ) )
mod1g <- tam.jml( resp=resp , xsi.fixed = xsi.fixed , theta.fixed=theta.fixed )
# look at estimated results
ind.person <- c( 1:5 , 30:33 )
cbind( mod1g$WLE , mod1g$errorWLE )[ind.person,]

#############################################################################
# EXAMPLE 2: Partial credit model
#############################################################################

data(data.gpcm)
# the same model but using function 'tam.jml2'
mod2 <- tam.jml2(resp=data.gpcm)
mod2$xsi     # extract item parameters
summary(mod2)
tam.fit(mod2)	# item and person infit/outfit statistic

#############################################################################
# EXAMPLE 3: Facet model estimation using joint maximum likelihood
#            data.ex10; see also Example 10 in tam.mml
#############################################################################

data(data.ex10)
dat <- data.ex10
  ## > head(dat)
  ##  pid rater I0001 I0002 I0003 I0004 I0005
  ##    1     1     0     1     1     0     0
  ##    1     2     1     1     1     1     0
  ##    1     3     1     1     1     0     1
  ##    2     2     1     1     1     0     1
  ##    2     3     1     1     0     1     1

facets <- dat[ , "rater" , drop=FALSE ] # define facet (rater)
pid <- dat$pid      # define person identifier (a person occurs multiple times)
resp <- dat[ , -c(1:2) ]        # item response data
formulaA <- ~ item * rater      # formula

# use MML function only to restructure data and input obtained design matrices
# and processed response data to tam.jml (-> therefore use only 2 iterations) 
mod3a <- tam.mml.mfr( resp=resp , facets=facets , formulaA = formulaA , 
             pid=dat$pid ,  control=list(maxiter=2) )

# use modified response data mod3a$resp and design matrix mod3a$A
resp1 <- mod3a$resp
# JML with 'tam.jml'
mod3b <- tam.jml( resp=resp1 , A=mod3a$A , control=list(maxiter=200) )
# 'tam.jml2'
mod3c <- tam.jml2( resp=resp1 , A=mod3a$A , control=list(maxiter=200) )	

#############################################################################
# EXAMPLE 4: Multi faceted model with some anchored item and person parameters
#############################################################################

data(data.exJ03)
resp <- data.exJ03$resp
X <- data.exJ03$X

#*** (0) preprocess data with tam.mml.facets
mod0 <- tam.mml.mfr( resp=resp , facets = X , pid = X$rater , 
                formulaA = ~ leader + item + step ,
                control=list(maxiter=2) )
summary(mod0)

#*** (1) estimation with tam.jml (no parameter fixings)

# extract processed data and design matrix from tam.mml.mfr
resp1 <- mod0$resp
A1 <- mod0$A
# estimate model with tam.jml
mod1 <- tam.jml( resp=resp1 , A=A1 , control=list( Msteps=4 , maxiter=100 ) )
summary(mod1)

#*** (2) fix some parameters (persons and items)

# look at indices in mod1$xsi
mod1$xsi
# fix step parameters
xsi.index1 <- cbind( 21:25 , c( -2.44 , 0.01 , -0.15 , 0.01 ,  1.55 ) )
# fix some item parameters of items 1,2,3,6 and 13
xsi.index2 <- cbind( c(1,2,3,6,13) , c(-2,-1,-1,-1.32 , -1 ) )
xsi.index <- rbind( xsi.index1 , xsi.index2 )
# fix some theta parameters of persons 1, 15 and 20
theta.fixed <- cbind(  c(1,15,20) , c(0.4 , 1 , 0 ) )
# estimate model
mod2 <- tam.jml( resp=resp1 , A=A1 , xsi.fixed=xsi.fixed , theta.fixed=theta.fixed , 
            control=list( Msteps=4 , maxiter=100 ) )                                
summary(mod2)
cbind( mod2$WLE , mod2$errorWLE )