itemfit.sx2: S-X2 Item Fit Statistic for Dichotomous Data

Description

Computes the S-X2 item fit statistic (Orlando & Thissen; 2000, 2003) for dichotomous data. Note that completely observed data is necessary for applying this function.

Usage

itemfit.sx2(object, Eik_min=1, progress=TRUE)
# S3 method for itemfit.sx2
summary(object, ...)
# S3 method for itemfit.sx2
plot(x, ask=TRUE, ...)

Value

A list with following entries

itemfit.stat: Data frame containing item fit statistics
itemtable: Data frame with expected and observed proportions for each score group and each item. Beside the ordinary p value, an adjusted p value obtained by correction due to multiple testing is provided (p.holm, see stats::p.adjust.

Arguments

object: Object of class din, gdina, gdm, sirt::rasch.mml, sirt::smirt or TAM::tam.mml
x: Object of class din, gdina, gdm, sirt::rasch.mml, sirt::smirt or TAM::tam.mml
Eik_min: The minimum expected cell size for merging score groups.
progress: An optional logical indicating whether progress should be displayed.
ask: An optional logical indicating whether every item should be separately displayed.
...: Further arguments to be passed

Author

Alexander Robitzsch

Details

The S-X2 item fit statistic compares observed and expected proportions $O_{jk}$ and $E_{jk}$ for item $j$ and each score group $k$ and forms a chi-square distributed statistic $$S-X_j^2=\sum_{k=1}^{J-1} N_k \frac{ ( O_{jk} - E_{jk} )^2 } { E_{jk} ( 1 - E_{jk} ) } $$ The degrees of freedom are $J-1-P_j$ where $P_j$ denotes the number of estimated item parameters.

References

Li, Y., & Rupp, A. A. (2011). Performance of the S-X2 statistic for full-information bifactor models. Educational and Psychological Measurement, 71, 986-1005.

Orlando, M., & Thissen, D. (2000). Likelihood-based item-fit indices for dichotomous item response theory models. Applied Psychological Measurement, 24, 50-64.

Orlando, M., & Thissen, D. (2003). Further investigation of the performance of S-X2: An item fit index for use with dichotomous item response theory models. Applied Psychological Measurement, 27, 289-298.

Zhang, B., & Stone, C. A. (2008). Evaluating item fit for multidimensional item response models. Educational and Psychological Measurement, 68, 181-196.

Examples

Run this code

if (FALSE) {
#############################################################################
# EXAMPLE 1: Items with unequal item slopes
#############################################################################

# simulate data
set.seed(9871)
I <- 11
b <- seq( -1.5, 1.5, length=I)
a <- rep(1,I)
a[4] <- .4
N <- 1000
library(sirt)
dat <- sirt::sim.raschtype( theta=stats::rnorm(N), b=b, fixed.a=a)

#*** 1PL model estimated with gdm
mod1 <- CDM::gdm( dat, theta.k=seq(-6,6,len=21), irtmodel="1PL" )
summary(mod1)
# estimate item fit statistic
fitmod1 <- CDM::itemfit.sx2(mod1)
summary(fitmod1)
  ##       item itemindex   S-X2 df     p S-X2_df RMSEA Nscgr Npars p.holm
  ##   1  I0001         1  4.173  9 0.900   0.464 0.000    10     1  1.000
  ##   2  I0002         2 12.365  9 0.193   1.374 0.019    10     1  1.000
  ##   3  I0003         3  6.158  9 0.724   0.684 0.000    10     1  1.000
  ##   4  I0004         4 37.759  9 0.000   4.195 0.057    10     1  0.000
  ##   5  I0005         5 12.307  9 0.197   1.367 0.019    10     1  1.000
  ##   6  I0006         6 19.358  9 0.022   2.151 0.034    10     1  0.223
  ##   7  I0007         7 14.610  9 0.102   1.623 0.025    10     1  0.818
  ##   8  I0008         8 15.568  9 0.076   1.730 0.027    10     1  0.688
  ##   9  I0009         9  8.471  9 0.487   0.941 0.000    10     1  1.000
  ##   10 I0010        10  8.330  9 0.501   0.926 0.000    10     1  1.000
  ##   11 I0011        11 12.351  9 0.194   1.372 0.019    10     1  1.000
  ##
  ##   -- Average Item Fit Statistics --
  ##   S-X2=13.768 | S-X2_df=1.53
# -> 4th item does not fit to the 1PL model

# plot item fit
plot(fitmod1)

#*** 2PL model estimated with gdm
mod2 <- CDM::gdm( dat, theta.k=seq(-6,6,len=21), irtmodel="2PL", maxiter=100 )
summary(mod2)
# estimate item fit statistic
fitmod2 <- CDM::itemfit.sx2(mod2)
summary(fitmod2)
  ##       item itemindex   S-X2 df     p S-X2_df RMSEA Nscgr Npars p.holm
  ##   1  I0001         1  4.083  8 0.850   0.510 0.000    10     2  1.000
  ##   2  I0002         2 13.580  8 0.093   1.697 0.026    10     2  0.747
  ##   3  I0003         3  6.236  8 0.621   0.780 0.000    10     2  1.000
  ##   4  I0004         4  6.049  8 0.642   0.756 0.000    10     2  1.000
  ##   5  I0005         5 12.792  8 0.119   1.599 0.024    10     2  0.834
  ##   6  I0006         6 14.397  8 0.072   1.800 0.028    10     2  0.648
  ##   7  I0007         7 15.046  8 0.058   1.881 0.030    10     2  0.639
  ##   [...]
  ##
  ##   -- Average Item Fit Statistics --
  ##   S-X2=10.22 | S-X2_df=1.277

#*** 1PL model estimation in smirt (sirt package)
Qmatrix <- matrix(1, nrow=I, ncol=1 )
mod1a <- sirt::smirt( dat, Qmatrix=Qmatrix )
summary(mod1a)
# item fit statistic
fitmod1a <- CDM::itemfit.sx2(mod1a)
summary(fitmod1a)

#*** 2PL model estimation in smirt (sirt package)
mod2a <- sirt::smirt( dat, Qmatrix=Qmatrix, est.a="2PL")
summary(mod2a)
# item fit statistic
fitmod2a <- CDM::itemfit.sx2(mod2a)
summary(fitmod2a)

#*** 1PL model estimated with rasch.mml2 (in sirt)
mod1b <- sirt::rasch.mml2(dat)
summary(mod1b)
# estimate item fit statistic
fitmod1b <- CDM::itemfit.sx2(mod1b)
summary(fitmod1b)

#*** 1PL estimated in TAM
library(TAM)
mod1c <- TAM::tam.mml( resp=dat )
summary(mod1c)
# item fit
summary( CDM::itemfit.sx2( mod1c) )
# conversion to mirt object
library(sirt)
library(mirt)
cmod1c <- sirt::tam2mirt( mod1c )
# item fit in mirt
mirt::itemfit( cmod1c$mirt )

#*** 2PL estimated in TAM
mod2c <- TAM::tam.mml.2pl( resp=dat )
summary(mod2c)
# item fit
summary( CDM::itemfit.sx2( mod2c) )
# conversion to mirt object and item fit in mirt
cmod2c <- sirt::tam2mirt( mod2c )
mirt::itemfit( cmod2c$mirt )

# estimation in mirt
mod1d <- mirt::mirt( dat, 1, itemtype="Rasch" )
mirt::itemfit( mod1d )    # compute item fit

#############################################################################
# EXAMPLE 2: Item fit statistics sim.dina dataset
#############################################################################

data(sim.dina, package="CDM")
data(sim.qmatrix, package="CDM")

#*** Model 1: DINA model (correctly specified model)
mod1 <- CDM::din( data=sim.dina, q.matrix=sim.qmatrix )
summary(mod1)
# item fit statistic
summary( CDM::itemfit.sx2( mod1 ) )
  ##   -- Average Item Fit Statistics --
  ##   S-X2=7.397 | S-X2_df=1.233

#*** Model 2: Mixed DINA/DINO model
#***  1th item is misspecified according to DINO rule
I <- ncol(CDM::sim.dina)
rule <- rep("DINA", I )
rule[1] <- "DINO"
mod2 <- CDM::din( data=CDM::sim.dina, q.matrix=CDM::sim.qmatrix, rule=rule)
summary(mod2)
# item fit statistic
summary( CDM::itemfit.sx2( mod2 ) )
  ##   -- Average Item Fit Statistics --
  ##   S-X2=9.925 | S-X2_df=1.654

#*** Model 3: Additive GDINA model
mod3 <- CDM::gdina( data=CDM::sim.dina, q.matrix=CDM::sim.qmatrix, rule="ACDM")
summary(mod3)
# item fit statistic
summary( CDM::itemfit.sx2( mod3 ) )
  ##   -- Average Item Fit Statistics --
  ##   S-X2=8.416 | S-X2_df=1.678
}

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