sirt (version 1.14-0)

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)
"summary"(object,...)	  
"anova"(object,...)
"logLik"(object,...)
"IRT.irfprob"(object,...)
"IRT.factor.scores"(object, type="EAP", ...)
"IRT.likelihood"(object,...)
"IRT.posterior"(object,...)
"IRT.modelfit"(object,...)
"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: A list with following entries:

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)

## Not run: 
# #############################################################################
# # 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)
# ## End(Not run)