Differential Item Functioning using Logistic Regression Analysis
This function assesses differential item functioning using logistic regression analysis (Zumbo, 1999).
dif.logistic.regression(dat, group, score,quant=1.645)
dat |
Data frame with dichotomous item responses |
group |
Group identifier |
score |
Ability estimate, e.g. the WLE. |
quant |
Used quantile of the normal distribution for assessing statistical significance |
Items are classified into A (negligible DIF), B (moderate DIF) and C (large DIF) levels according to the ETS classification system (Longford, Holland & Thayer, 1993, p. 175). See also Monahan, McHorney, Stump and Perkins (2007) for further DIF effect size classifications.
A data frame with following variables:
itemnr |
Numeric index of the item |
sortDIFindex |
Rank of item with respect to the uniform DIF (from negative to positive values) |
item |
Item name |
N |
Sample size per item |
R |
Value of |
F |
Value of |
nR |
Sample size per item in reference group |
nF |
Sample size per item in focal group |
p |
Item p value |
pR |
Item p value in reference group |
pF |
Item p value in focal group |
pdiff |
Item p value differences |
pdiff.adj |
Adjusted p value difference |
uniformDIF |
Uniform DIF estimate |
se.uniformDIF |
Standard error of uniform DIF |
t.uniformDIF |
The t value for uniform DIF |
sig.uniformDIF |
Significance label for uniform DIF |
DIF.ETS |
DIF classification according to the ETS classification system (see Details) |
uniform.EBDIF |
Empirical Bayes estimate of uniform DIF (Longford, Holland & Thayer, 1993) which takes degree of DIF standard error into account |
DIF.SD |
Value of the DIF standard deviation |
nonuniformDIF |
Nonuniform DIF estimate |
se.nonuniformDIF |
Standard error of nonuniform DIF |
t.nonuniformDIF |
The t value for nonuniform DIF |
sig.nonuniformDIF |
Significance label for nonuniform DIF |
Longford, N. T., Holland, P. W., & Thayer, D. T. (1993). Stability of the MH D-DIF statistics across populations. In P. W. Holland & H. Wainer (Eds.). Differential Item Functioning (pp. 171-196). Hillsdale, NJ: Erlbaum.
Magis, D., Beland, S., Tuerlinckx, F., & De Boeck, P. (2010). A general framework and an R package for the detection of dichotomous differential item functioning. Behavior Research Methods, 42(3), 847-862. doi: 10.3758/BRM.42.3.847
Monahan, P. O., McHorney, C. A., Stump, T. E., & Perkins, A. J. (2007). Odds ratio, delta, ETS classification, and standardization measures of DIF magnitude for binary logistic regression. Journal of Educational and Behavioral Statistics, 32(1), 92-109. doi: 10.3102/1076998606298035
Zumbo, B. D. (1999). A handbook on the theory and methods of differential item functioning (DIF): Logistic regression modeling as a unitary framework for binary and Likert-type (ordinal) item scores. Ottawa ON: Directorate of Human Resources Research and Evaluation, Department of National Defense.
For assessing DIF variance see dif.variance and
dif.strata.variance
See also rasch.evm.pcm for assessing differential item
functioning in the partial credit model.
See the difR package for a large collection of DIF detection methods (Magis, Beland, Tuerlinckx, & De Boeck, 2010).
For a download of the free DIF-Pack software (SIBTEST, ...) see http://psychometrictools.measuredprogress.org/home.
#############################################################################
# EXAMPLE 1: Mathematics data | Gender DIF
#############################################################################
data( data.math )
dat <- data.math$data
items <- grep( "M", colnames(dat))
# estimate item parameters and WLEs
mod <- sirt::rasch.mml2( dat[,items] )
wle <- sirt::wle.rasch( dat[,items], b=mod$item$b )$theta
# assess DIF by logistic regression
mod1 <- sirt::dif.logistic.regression( dat=dat[,items], score=wle, group=dat$female)
# calculate DIF variance
dif1 <- sirt::dif.variance( dif=mod1$uniformDIF, se.dif=mod1$se.uniformDIF )
dif1$unweighted.DIFSD
## > dif1$unweighted.DIFSD
## [1] 0.1963958
# calculate stratified DIF variance
# stratification based on domains
dif2 <- sirt::dif.strata.variance( dif=mod1$uniformDIF, se.dif=mod1$se.uniformDIF,
itemcluster=data.math$item$domain )
## $unweighted.DIFSD
## [1] 0.1455916
## Not run:
#****
# Likelihood ratio test and graphical model test in eRm package
miceadds::library_install("eRm")
# estimate Rasch model
res <- eRm::RM( dat[,items] )
summary(res)
# LR-test with respect to female
lrres <- eRm::LRtest(res, splitcr=dat$female)
summary(lrres)
# graphical model test
eRm::plotGOF(lrres)
#############################################################################
# EXAMPLE 2: Comparison with Mantel-Haenszel test
#############################################################################
library(TAM)
library(difR)
#*** (1) simulate data
set.seed(776)
N <- 1500 # number of persons per group
I <- 12 # number of items
mu2 <- .5 # impact (group difference)
sd2 <- 1.3 # standard deviation group 2
# define item difficulties
b <- seq( -1.5, 1.5, length=I)
# simulate DIF effects
bdif <- scale( stats::rnorm(I, sd=.6 ), scale=FALSE )[,1]
# item difficulties per group
b1 <- b + 1/2 * bdif
b2 <- b - 1/2 * bdif
# simulate item responses
dat1 <- sirt::sim.raschtype( theta=stats::rnorm(N, mean=0, sd=1 ), b=b1 )
dat2 <- sirt::sim.raschtype( theta=stats::rnorm(N, mean=mu2, sd=sd2 ), b=b2 )
dat <- rbind( dat1, dat2 )
group <- rep( c(1,2), each=N ) # define group indicator
#*** (2) scale data
mod <- TAM::tam.mml( dat, group=group )
summary(mod)
#*** (3) extract person parameter estimates
mod_eap <- mod$person$EAP
mod_wle <- tam.wle( mod )$theta
#*********************************
# (4) techniques for assessing differential item functioning
# Model 1: assess DIF by logistic regression and WLEs
dif1 <- sirt::dif.logistic.regression( dat=dat, score=mod_wle, group=group)
# Model 2: assess DIF by logistic regression and EAPs
dif2 <- sirt::dif.logistic.regression( dat=dat, score=mod_eap, group=group)
# Model 3: assess DIF by Mantel-Haenszel statistic
dif3 <- difR::difMH(Data=dat, group=group, focal.name="1", purify=FALSE )
print(dif3)
## Mantel-Haenszel Chi-square statistic:
##
## Stat. P-value
## I0001 14.5655 0.0001 ***
## I0002 300.3225 0.0000 ***
## I0003 2.7160 0.0993 .
## I0004 191.6925 0.0000 ***
## I0005 0.0011 0.9740
## [...]
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Detection threshold: 3.8415 (significance level: 0.05)
##
## Effect size (ETS Delta scale):
##
## Effect size code:
## 'A': negligible effect
## 'B': moderate effect
## 'C': large effect
##
## alphaMH deltaMH
## I0001 1.3908 -0.7752 A
## I0002 0.2339 3.4147 C
## I0003 1.1407 -0.3093 A
## I0004 2.8515 -2.4625 C
## I0005 1.0050 -0.0118 A
## [...]
##
## Effect size codes: 0 'A' 1.0 'B' 1.5 'C'
## (for absolute values of 'deltaMH')
# recompute DIF parameter from alphaMH
uniformDIF3 <- log(dif3$alphaMH)
# compare different DIF statistics
dfr <- data.frame( "bdif"=bdif, "LR_wle"=dif1$uniformDIF,
"LR_eap"=dif2$uniformDIF, "MH"=uniformDIF3 )
round( dfr, 3 )
## bdif LR_wle LR_eap MH
## 1 0.236 0.319 0.278 0.330
## 2 -1.149 -1.473 -1.523 -1.453
## 3 0.140 0.122 0.038 0.132
## 4 0.957 1.048 0.938 1.048
## [...]
colMeans( abs( dfr[,-1] - bdif ))
## LR_wle LR_eap MH
## 0.07759187 0.19085743 0.07501708
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