S-X2 Item Fit Statistic for Dichotomous Data
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.
itemfit.sx2(object, Eik_min=1, progress=TRUE) ## S3 method for class 'itemfit.sx2' summary(object, ...) ## S3 method for class 'itemfit.sx2' plot(x, ask=TRUE, ...)
object |
Object of class |
x |
Object of class |
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 |
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=∑_{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
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 ( |
This function does not work properly for multiple groups.
Alexander Robitzsch
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.
#############################################################################
# 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)
## Not run:
#*** 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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