Rater Facets Models with Item/Rater Intercepts and Slopes
This function estimates the unidimensional rater facets model (Lincare, 1994) and an extension to slopes (see Details; Robitzsch & Steinfeld, 2018). The estimation is conducted by an EM algorithm employing marginal maximum likelihood.
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, rater_item_int=FALSE,
est.mean=FALSE, tau.item.fixed=NULL, a.item.fixed=NULL, b.rater.fixed=NULL,
a.rater.fixed=NULL, b.rater.center=2, a.rater.center=2, a.item.center=2, a_lower=.05,
a_upper=10, reference_rater=NULL, max.b.increment=1, numdiff.parm=0.00001,
maxdevchange=0.1, globconv=0.001, maxiter=1000, msteps=4, mstepconv=0.001,
PEM=FALSE, PEM_itermax=maxiter)
## S3 method for class 'rm.facets'
summary(object, file=NULL, ...)
## 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, ...)
## function for processing data
rm_proc_data( dat, pid, rater, rater_item_int=FALSE, reference_rater=NULL )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? |
rater_item_int |
Logical indicating whether rater-item-interactions should be modeled. |
est.mean |
Optional logical indicating whether the mean of the trait distribution should be estimated. |
tau.item.fixed |
Matrix with fixed τ parameters. Non-fixed
parameters must be declared by |
a.item.fixed |
Vector with fixed item discriminations |
b.rater.fixed |
Vector with fixed rater intercept parameters |
a.rater.fixed |
Vector with fixed rater discrimination parameters |
b.rater.center |
Centering method for rater intercept parameters. The
value |
a.rater.center |
Centering method for rater discrimination parameters. The
value |
a.item.center |
Centering method for item discrimination parameters. The
value |
a_lower |
Lower bound for a parameters |
a_upper |
Upper bound for a parameters |
reference_rater |
Identifier for rater as a reference rater for which a fixed rater mean of 0 and a fixed rater slope of 1 is assumed. |
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 |
PEM |
Logical indicating whether the P-EM acceleration should be applied (Berlinet & Roland, 2012). |
PEM_itermax |
Number of iterations in which the P-EM method should be applied. |
object |
Object of class |
file |
Optional file name in which summary should be written. |
type |
Factor score estimation method. Factor score types
|
... |
Further arguments to be passed |
This function models ratings X_{pri} for person p, rater r and item i and category k (see also Robitzsch & Steinfeld, 2018; Uto & Ueno, 2010; Wu, 2017)
P( X_{pri}=k | θ_p ) \propto \exp( a_i a_r q_{ik} θ_p - q_{ik} b_r - τ_{ik} ), θ_p ~ N( 0, σ^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. ∏_i a_i=∏_r a_r=1.
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 |
tau.item |
Item parameters τ_{ik} |
se.tau.item |
Standard error of item parameters τ_{ik} |
a.item |
Item slopes a_i |
se.a.item |
Standard error of item slopes a_i |
delta.item |
Delta item parameter. See
|
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 |
n.ik |
Expected counts |
maxK |
Maximum number of categories |
procdata |
Processed data |
iter |
Number of iterations |
ipars.dat2 |
Item parameters for expanded dataset |
... |
Further values |
If the trait standard deviation sigma strongly
differs from 1, then a user should investigate the sensitivity
of results using different theta integration points theta.k.
Berlinet, A. F., & Roland, C. (2012). Acceleration of the EM algorithm: P-EM versus epsilon algorithm. Computational Statistics & Data Analysis, 56(12), 4122-4137.
Linacre, J. M. (1994). Many-Facet Rasch Measurement. Chicago: MESA Press.
Robitzsch, A., & Steinfeld, J. (2018). Item response models for human ratings: Overview, estimation methods, and implementation in R. Psychological Test and Assessment Modeling, 60(1), 101-139.
Uto, M., & Ueno, M. (2016). Item response theory for peer assessment. IEEE Transactions on Learning Technologies, 9(2), 157-170.
Wu, M. (2017). Some IRT-based analyses for interpreting rater effects. Psychological Test and Assessment Modeling, 59(4), 453-470.
See also the TAM package for the estimation of more complicated facet models.
See rm.sdt for estimating a hierarchical rater model.
#############################################################################
# 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$rater)=="db01", ]
#**** Model 1: Partial Credit Model
mod1 <- sirt::rm.facets( dat[, paste0( "k",1:5) ], pid=dat$idstud )
#**** Model 2: Generalized Partial Credit Model
mod2 <- sirt::rm.facets( dat[, paste0( "k",1:5) ], pid=dat$idstud, est.a.item=TRUE)
summary(mod1)
summary(mod2)
## Not run:
#############################################################################
# EXAMPLE 2: Facets Model: 5 items, 7 raters
#############################################################################
data(data.ratings1)
dat <- data.ratings1
#**** Model 1: Partial Credit Model: no rater effects
mod1 <- sirt::rm.facets( dat[, paste0( "k",1:5) ], rater=dat$rater,
pid=dat$idstud, est.b.rater=FALSE )
#**** Model 2: Partial Credit Model: intercept rater effects
mod2 <- sirt::rm.facets( dat[, paste0( "k",1:5) ], rater=dat$rater, pid=dat$idstud)
# extract individual likelihood
lmod1 <- IRT.likelihood(mod1)
str(lmod1)
# likelihood value
logLik(mod1)
# extract item response functions
pmod1 <- IRT.irfprob(mod1)
str(pmod1)
# model comparison
anova(mod1,mod2)
# absolute and relative model fit
smod1 <- IRT.modelfit(mod1)
summary(smod1)
smod2 <- IRT.modelfit(mod2)
summary(smod2)
IRT.compareModels( smod1, smod2 )
# extract factor scores (EAP is the default)
IRT.factor.scores(mod2)
# extract WLEs
IRT.factor.scores(mod2, type="WLE")
#**** Model 2a: compare results with TAM package
# Results should be similar to Model 2
library(TAM)
mod2a <- TAM::tam.mml.mfr( resp=dat[, paste0( "k",1:5) ],
facets=dat[, "rater", drop=FALSE],
pid=dat$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$tau.item, 1 ) # use parameters from mod2
tau.item.fixed[ 3:5, ] <- NA # free item parameters of items 3, 4 and 5
mod2b <- sirt::rm.facets( dat[, paste0( "k",1:5) ], rater=dat$rater,
b.rater.fixed=b.rater.fixed, tau.item.fixed=tau.item.fixed,
est.mean=TRUE, pid=dat$idstud)
summary(mod2b)
#**** Model 3: estimated rater slopes
mod3 <- sirt::rm.facets( dat[, paste0( "k",1:5) ], rater=dat$rater,
est.a.rater=TRUE)
#**** Model 4: estimated item slopes
mod4 <- sirt::rm.facets( dat[, paste0( "k",1:5) ], rater=dat$rater,
pid=dat$idstud, est.a.item=TRUE)
#**** Model 5: estimated rater and item slopes
mod5 <- sirt::rm.facets( dat[, paste0( "k",1:5) ], rater=dat$rater,
pid=dat$idstud, est.a.rater=TRUE, est.a.item=TRUE)
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 <- sirt::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)
summary(mod5a)
#**** Model 6: Estimate rater model with reference rater 'db03'
mod6 <- sirt::rm.facets( dat[, paste0( "k",1:5) ], rater=dat$rater, est.a.item=TRUE,
est.a.rater=TRUE, pid=dat$idstud, reference_rater="db03" )
summary(mod6)
#**** Model 7: Modelling rater-item-interactions
mod7 <- sirt::rm.facets( dat[, paste0( "k",1:5) ], rater=dat$rater, est.a.item=FALSE,
est.a.rater=TRUE, pid=dat$idstud, reference_rater="db03",
rater_item_int=TRUE)
summary(mod7)
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