sirt (version 1.5-0)

rm.facets: Rater Facets Models with Item/Rater Intercepts and Slopes

Description

This function estimates the unidimensional rater facets model (Lincare, 1994) and an extension to slopes (see Details). The estimation is conducted by an EM algorithm employing marginal maximum likelihood.

Usage

rm.facets(dat, pid=NULL, rater=NULL, Qmatrix=NULL, theta.k=seq(-9, 9, len=30), 
    est.b.rater=TRUE, est.a.item=FALSE, est.a.rater=FALSE, est.mean=FALSE ,
    tau.item.fixed=NULL , a.item.fixed=NULL , b.rater.fixed=NULL , a.rater.fixed=NULL ,
    max.b.increment=1, numdiff.parm=0.00001, maxdevchange=0.1, globconv=0.001, 
    maxiter=1000, msteps=4, mstepconv=0.001)

## S3 method for class 'rm.facets':
summary(object,...)	  

## S3 method for class 'rm.facets':
anova(object,...)

## S3 method for class 'rm.facets':
logLik(object,...)

## S3 method for class 'rm.facets':
IRT.irfprob(object,...)

## S3 method for class 'rm.facets':
IRT.factor.scores(object, type="EAP", ...)

## S3 method for class 'rm.facets':
IRT.likelihood(object,...)

## S3 method for class 'rm.facets':
IRT.posterior(object,...)

## S3 method for class 'rm.facets':
IRT.modelfit(object,...)

## S3 method for class 'IRT.modelfit.rm.facets':
summary(object,...)

Arguments

dat

Original data frame. Ratings on variables must be in rows, i.e. every row corresponds to a person-rater combination.

pid

Person identifier.

rater

Rater identifier

Qmatrix

An optional Q-matrix. If this matrix is not provided, then by default the ordinary scoring of categories (from 0 to the maximum score of $K$) is used.

theta.k

A grid of theta values for the ability distribution.

est.b.rater

Should the rater severities $b_r$ be estimated?

est.a.item

Should the item slopes $a_i$ be estimated?

est.a.rater

Should the rater slopes $a_r$ be estimated?

est.mean

Optional logical indicating whether the mean of the trait distribution should be estimated.

tau.item.fixed

Matrix with fixed $\tau$ parameters. Non-fixed parameters must be declared by NA values.

a.item.fixed

Vector with fixed item discriminations

b.rater.fixed

Vector with fixed rater intercept parameters

a.rater.fixed

Vector with fixed rater discrmination parameters

max.b.increment

Maximum increment of item parameters during estimation

numdiff.parm

Numerical differentiation step width

maxdevchange

Maximum relative deviance change as a convergence criterion

globconv

Maximum parameter change

maxiter

Maximum number of iterations

msteps

Maximum number of iterations during an M step

mstepconv

Convergence criterion in an M step

object

Object of class rm.facets

type

Factor score estimation method. Factor score types "EAP", "MLE" and "WLE" are supported.

...

Further arguments to be passed

Value

• A list with following entries:
• deviance
  Deviance
• ic
  Information criteria and number of parameters
• item
  Data frame with item parameters
• rater
  Data frame with rater parameters
• person
  Data frame with person parameters: EAP and corresponding standard errors
• EAP.rel
  EAP reliability
• mu
  Mean of the trait distribution
• sigma
  Standard deviation of the trait distribution
• theta.k
  Grid of theta values
• pi.k
  Fitted distribution at theta.k values
• tau.item
  Item parameters $\tau_{ik}$
• se.tau.item
  Standard error of item parameters $\tau_{ik}$
• a.item
  Item slopes $a_i$
• se.a.item
  Standard error of item slopes $a_i$
• delta.item
  Delta item parameter. See pcm.conversion.
• b.rater
  Rater severity parameter $b_r$
• se.b.rater
  Standard error of rater severity parameter $b_r$
• a.rater
  Rater slope parameter $a_r$
• se.a.rater
  Standard error of rater slope parameter $a_r$
• f.yi.qk
  Individual likelihood
• f.qk.yi
  Individual posterior distribution
• probs
  Item probabilities at grid theta.k
• n.ik
  Expected counts
• maxK
  Maximum number of categories
• procdata
  Processed data
• iter
  Number of iterations
• ipars.dat2
  Item parameters for expanded dataset dat2
• ...
  Further values

Details

This function models ratings $X_{pri}$ for person $p$, rater $r$ and item $i$ and category $k$ $$P( X_{pri} = k | \theta_p ) \propto exp( a_i a_r q_{ik} \theta_p - q_{ik} b_r - \tau_{ik} ) \quad , \quad \theta_p \sim N( 0 , \sigma^2 )$$ By default, the scores in the $Q$ matrix are $q_{ik}=k$. Item slopes $a_i$ and rater slopes $a_r$ are standardized such that their product equals one, i.e. $\prod_i a_i = \prod_r a_r = 1$.

References

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

See Also

See also the TAM package for the estimation of more complicated facet models. See rm.sdt for estimating a hierarchical rater model.

Examples

#############################################################################
# EXAMPLE 1: Partial Credit Model and Generalized partial credit model 
#                   5 items and 1 rater
#############################################################################
data(data.ratings1)
dat <- data.ratings1

# select rater db01
dat <- dat[ paste(dat$rater) == "db01" , ]

# Model 1: Partial Credit Model
mod1 <- rm.facets( dat[ , paste0( "k",1:5) ] , pid=dat$idstud , maxiter=15)

# Model 2: Generalized Partial Credit Model
mod2 <- rm.facets( dat[ , paste0( "k",1:5) ] ,  pid=dat$idstud  , 
           est.a.item=TRUE , maxiter=15)

summary(mod1)
summary(mod2)

#############################################################################
# EXAMPLE 2: Facets Model: 5 items, 7 raters
#############################################################################

data(data.ratings1)
dat <- data.ratings1
maxit <- 15    # maximum number of iterations, increase it in applications!

# Model 1: Partial Credit Model: no rater effects
mod1 <- rm.facets( dat[ , paste0( "k",1:5) ] , rater=dat$rater , 
             pid=dat$idstud  , est.b.rater=FALSE  , maxiter=maxit)

# Model 2: Partial Credit Model: intercept rater effects
mod2 <- rm.facets( dat[ , paste0( "k",1:5) ] , rater=dat$rater  , 
             pid=dat$idstud  , maxiter=maxit)

# extract individual likelihood
lmod1 <- IRT.likelihood(mod1)   
str(lmod1)
# likelihood value
logLik(mod1)
# extract item response functions
pmod1 <- IRT.irfprob(mod1)
str(pmod1)
# model comparison
anova(mod1,mod2)             
# absolute and relative model fit
smod1 <- IRT.modelfit(mod1)
summary(smod1)
smod2 <- IRT.modelfit(mod2)
summary(smod2)
IRT.compareModels( smod1 , smod2 )
# extract factor scores (EAP is the default)
IRT.factor.scores(mod2)
# extract WLEs
IRT.factor.scores(mod2 , type="WLE")

# Model 2a: compare results with TAM package
#   Results should be similar to Model 2
library(TAM)
mod2a <- TAM::tam.mml.mfr( resp= dat[ , paste0( "k",1:5) ] , 
             facets= dat[ , "rater" , drop=FALSE] ,
             pid= dat$pid , formulaA = ~ item*step + rater )

# Model 2b: Partial Credit Model: some fixed parameters
# fix rater parameters for raters 1, 4 and 5
b.rater.fixed <- rep(NA,7)
b.rater.fixed[ c(1,4,5) ] <- c(1,-.8,0)  # fixed parameters
# fix item parameters of first and second item
tau.item.fixed <- round( mod2$tau.item , 1 )    # use parameters from mod2
tau.item.fixed[ 3:5 , ] <- NA    # free item parameters of items 3, 4 and 5
mod2b <- rm.facets( dat[ , paste0( "k",1:5) ] , rater=dat$rater  , 
             b.rater.fixed=b.rater.fixed , tau.item.fixed=tau.item.fixed , 
             est.mean = TRUE , pid=dat$idstud  , maxiter=maxit)
summary(mod2b)             
             
# Model 3: estimated rater slopes
mod3 <- rm.facets( dat[ , paste0( "k",1:5) ] , rater=dat$rater , 
            est.a.rater=TRUE  , maxiter=maxit)

# Model 4: estimated item slopes
mod4 <- rm.facets( dat[ , paste0( "k",1:5) ] , rater=dat$rater , 
             pid=dat$idstud  ,  est.a.item=TRUE , maxiter=maxit)

# Model 5: estimated rater and item slopes
mod5 <- rm.facets( dat[ , paste0( "k",1:5) ] , rater=dat$rater , 
             pid=dat$idstud  , est.a.rater=TRUE , est.a.item=TRUE , maxiter=maxit)
summary(mod1)
summary(mod2)
summary(mod2a)
summary(mod3)
summary(mod4)
summary(mod5)

# Model 5a: Some fixed parameters in Model 5
# fix rater b parameters for raters 1, 4 and 5
b.rater.fixed <- rep(NA,7)
b.rater.fixed[ c(1,4,5) ] <- c(1,-.8,0)
# fix rater a parameters for first four raters
a.rater.fixed <- rep(NA,7)
a.rater.fixed[ c(1,2,3,4) ] <- c(1.1,0.9,.85,1)
# fix item b parameters of first item
tau.item.fixed <- matrix( NA , nrow=5 , ncol=3 )
tau.item.fixed[ 1 , ] <- c(-2,-1.5 , 1 )   
# fix item a parameters
a.item.fixed <- rep(NA,5)
a.item.fixed[ 1:4 ] <- 1
# estimate model
mod5a <- rm.facets( dat[ , paste0( "k",1:5) ] , rater=dat$rater , 
             pid=dat$idstud  , est.a.rater=TRUE , est.a.item=TRUE , 
             tau.item.fixed=tau.item.fixed , b.rater.fixed=b.rater.fixed , 
             a.rater.fixed=a.rater.fixed , a.item.fixed=a.item.fixed , 
             est.mean=TRUE , maxiter=maxit)
summary(mod5a)