itemfit.sx2.RdComputes the S-X2 item fit statistic (Orlando & Thissen; 2000, 2003) for dichotomous data. Note that completely observed data is necessary for applying this function.
Object of class din, gdina, gdm,
sirt::rasch.mml,
sirt::smirt or TAM::tam.mml
Object of class din, gdina, gdm,
sirt::rasch.mml,
sirt::smirt or TAM::tam.mml
The minimum expected cell size for merging score groups.
An optional logical indicating whether progress should be displayed.
An optional logical indicating whether every item should be separately displayed.
Further arguments to be passed
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.
A list with following entries
Data frame containing item fit statistics
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.
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.
This function does not work properly for multiple groups.
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
}