TAM (version 1.6-0)

tam.fit: Item Infit and Outfit Statistic

Description

The item infit and outfit statistic are calculated for objects of classes tam, tam.mml and tam.jml, respectively.

Usage

tam.fit(tamobj, ...)

tam.mml.fit(tamobj, FitMatrix=NULL, Nsimul=NULL,progress=TRUE,
   useRcpp=TRUE, seed=123, fit.facets=TRUE)

tam.jml.fit(tamobj)

## S3 method for class 'tam.fit':
summary(object, \dots)

Arguments

tamobj

An object of class tam, tam.mml or tam.jml

FitMatrix

A fit matrix $F$ for a specific hypothesis of fit of the linear function $F \xi$ (see Simulated Example 3 and Adams & Wu 2007).

Nsimul

Number of simulations used for fit calculation. The default is 100 (less than 400 students), 40 (less than 1000 students), 15 (less than 3000 students) and 5 (more than 3000 students)

progress

An optional logical indicating whether computation progress should be displayed at console.

useRcpp

Optional logical indicating whether Rcpp or pure Rcode should be used for fit calculation. The latter is consistent with TAM (

seed

Fixed simulation seed.

fit.facets

An optional logical indicating whether fit for all facet parameters should be computed.

object

Object of class tam.fit

...

Further arguments to be passed

Value

• In case of tam.mml.fit a data frame as entry itemfit with four columns:
• Outfit
  Item outfit statistic
• Outfit_t
  The $t$ value for the outfit statistic
• Outfit_p
  Significance $p$ value for outfit statistic
• Outfit_pholm
  Significance $p$ value for outfit statistic, adjusted for multiple testing according to the Holm procedure
• Infit
  Item infit statistic
• Infit_t
  The $t$ value for the infit statistic
• Infit_p
  Significance $p$ value for infit statistic
• Infit_pholm
  Significance $p$ value for infit statistic, adjusted for multiple testing according to the Holm procedure

Details

Item fit is automatically calculated in JML estimation using tam.jml.

References

Adams, R. J., & Wu, M. L. (2007). The mixed-coefficients multinomial logit model. A generalized form of the Rasch model. In M. von Davier & C. H. Carstensen (Eds.). Multivariate and mixture distribution Rasch models: Extensions and applications (pp. 55-76). New York: Springer.

See Also

Item fit and person fit based on estimated person parameters can be calculated using the pcm.fit function in the sirt package (see Example 1 and Example 2).

Examples

#############################################################################
# EXAMPLE 1: Dichotomous data sim.rasch
#############################################################################

data(sim.rasch)
# estimate Rasch model
mod1 <- tam.mml(resp=sim.rasch) 
# item fit
fit1 <- tam.fit( mod1 )
summary(fit1)
  ##   > summary(fit1)
  ##      parameter Outfit Outfit_t Outfit_p Infit Infit_t Infit_p
  ##   1         I1  0.966   -0.409    0.171 0.996  -0.087   0.233
  ##   2         I2  1.044    0.599    0.137 1.029   0.798   0.106
  ##   3         I3  1.022    0.330    0.185 1.012   0.366   0.179
  ##   4         I4  1.047    0.720    0.118 1.054   1.650   0.025

#--------
# infit and oufit based on estimated WLEs
library(sirt)

# estimate WLE
wle <- tam.wle(mod1)
# extract item parameters
b1 <- - mod1$AXsi[ , -1 ]
# assess item fit and person fit
fit1a <- sirt::pcm.fit(b=b1, theta=wle$theta , sim.rasch )
fit1a$item       # item fit statistic
fit1a$person     # person fit statistic

#############################################################################
# EXAMPLE 2: Partial credit model data.gpcm
#############################################################################

data( data.gpcm )
dat <- data.gpcm

# estimate partial credit model in ConQuest parametrization 'item+item*step'
mod2 <- tam.mml( resp=dat , irtmodel="PCM2" )
summary(mod2)
# estimate item fit
fit2 <- tam.fit(mod2)
summary(fit2)

#  => The first three rows of the data frame correspond to the fit statistics
#     of first three items Comfort, Work and Benefit.

#--------
# infit and oufit based on estimated WLEs
# compute WLEs
wle <- tam.wle(mod2)
# extract item parameters
b1 <- - mod2$AXsi[ , -1 ]
# assess fit
fit1a <- sirt::pcm.fit(b=b1 , theta=wle$theta , dat)
fit1a$item

#############################################################################
# SIMULATED EXAMPLE 3: Fit statistic testing for local independence
#############################################################################

#generate data with local DEPENDENCE and User-defined fit statistics
set.seed(4888)
I <- 40		# 40 items
N <- 1000       # 1000 persons

delta <- seq(-2,2, len=I)
theta <- rnorm(N, 0, 1)
# simulate data
prob <- plogis(outer(theta, delta, "-"))
rand <- matrix(runif(N*I), nrow=N, ncol=I)
resp <- 1*(rand < prob)
colnames(resp) <- paste("I" , 1:I, sep="")
  
#Introduce some local dependence
for (item in c(10, 20, 30)){ 
 #  20 #are made equal to the previous item 
  row <- round(runif(0.2*N)*N + 0.5)     
  resp[row, item+1] <- resp[row, item]
}
  
#run TAM
mod1 <- tam(resp)
  
#User-defined fit design matrix
F <- array(0, dim=c(dim(mod1$A)[1], dim(mod1$A)[2], 6))
F[,,1] <- mod1$A[,,10] + mod1$A[,,11]
F[,,2] <- mod1$A[,,12] + mod1$A[,,13]
F[,,3] <- mod1$A[,,20] + mod1$A[,,21]
F[,,4] <- mod1$A[,,22] + mod1$A[,,23]
F[,,5] <- mod1$A[,,30] + mod1$A[,,31]
F[,,6] <- mod1$A[,,32] + mod1$A[,,33]
fit <- tam.fit(mod1, FitMatrix=F)
summary(fit)

#############################################################################
# SIMULATED EXAMPLE 4: Fit statistic testing for items with differing slopes
#############################################################################

#*** simulate data
library(sirt)
set.seed(9875)
N <- 2000
I <- 20
b <- sample( seq( -2 , 2 , length=I ) )
a <- rep( 1, I )
# create some misfitting items
a[c(1,3)] <- c(.5 , 1.5 )
# simulate data
dat <- sirt::sim.raschtype( rnorm(N) , b=b , fixed.a=a )
#*** estimate Rasch model
mod1 <- tam.mml(resp=dat) 
#*** assess item fit by infit and outfit statistic
fit1 <- tam.fit( mod1 )$itemfit
round( cbind( "b"=mod1$item$AXsi_.Cat1 , fit1$itemfit[,-1] )[1:7,], 3 )  

#*** compute item fit statistic in mirt package
library(mirt)
library(sirt)
mod1c <- mirt::mirt( dat , model=1 , itemtype="Rasch" , verbose=TRUE)
print(mod1c)                      # model summary
sirt::mirt.wrapper.coef(mod1c)    # estimated parameters
fit1c <- mirt::itemfit(mod1c , method="EAP")    # model fit in mirt package
# compare results of TAM and mirt
dfr <- cbind( "TAM"=fit1 , "mirt"=fit1c[,-c(1:2)] )

# S-X2 item fit statistic (see also the output from mirt) 
library(CDM)
sx2mod1 <- CDM::itemfit.sx2( mod1 )
summary(sx2mod1)

# compare results of CDM and mirt
sx2comp <-  cbind( sx2mod1$itemfit.stat[ , c("S-X2" , "p") ]  , 
                    dfr[  , c("mirt.S_X2" , "mirt.p.S_X2") ] )
round( sx2comp , 3 )