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=NA, fit.facets=TRUE)
tam.jml.fit(tamobj, trim_val=10)
# S3 method for tam.fit
summary(object, file=NULL, ...)

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

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 R code should be used for fit calculation. The latter is consistent with TAM (<=1.1).
seed: Fixed simulation seed.
fit.facets: An optional logical indicating whether fit for all facet parameters should be computed.
trim_val: Optional trimming value. Squared standardized reaisuals larger than trim_val are set to trim_val.
object: Object of class tam.fit
file: Optional file name for summary output
...: Further arguments to be passed

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. tools:::Rd_expr_doi("10.1007/978-0-387-49839-3_4")

Examples

Run this code

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

data(data.sim.rasch)
# estimate Rasch model
mod1 <- TAM::tam.mml(resp=data.sim.rasch)
# item fit
fit1 <- TAM::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

if (FALSE) {

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

# estimate WLE
wle <- TAM::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, data.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::tam.mml( resp=dat, irtmodel="PCM2" )
summary(mod2)
# estimate item fit
fit2 <- TAM::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::tam.wle(mod2)
# extract item parameters
b1 <- - mod2$AXsi[, -1 ]
# assess fit
fit1a <- sirt::pcm.fit(b=b1, theta=wle$theta, dat)
fit1a$item

#############################################################################
# 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 <- stats::rnorm(N, 0, 1)
# simulate data
prob <- stats::plogis(outer(theta, delta, "-"))
rand <- matrix( stats::runif(N*I), nrow=N, ncol=I)
resp <- 1*(rand < prob)
colnames(resp) <- paste("I", 1:I, sep="")

#induce some local dependence
for (item in c(10, 20, 30)){
 #  20
 #are made equal to the previous item
  row <- round( stats::runif(0.2*N)*N + 0.5)
  resp[row, item+1] <- resp[row, item]
}

#run TAM
mod1 <- TAM::tam.mml(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::tam.fit(mod1, FitMatrix=F)
summary(fit)

#############################################################################
# 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::tam.mml(resp=dat)
#*** assess item fit by infit and outfit statistic
fit1 <- TAM::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 )
}