dif.logistic.regression: Differential Item Functioning using Logistic Regression Analysis

Description

This function estimates differential item functioning using a logistic regression analysis (Zumbo, 1999).

Usage

dif.logistic.regression(dat, group, score,quant=1.645)

Arguments

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

Value

A data frame with following variables:
itemnrNumeric index of the item
sortDIFindexRank of item with respect to the uniform DIF (from negative to positive values)
itemItem name
NSample size per item
RValue of group variable for reference group
FValue of group variable for focal group
nRSample size per item in reference group
nFSample size per item in focal group
pItem $p$ value
pRItem $p$ value in reference group
pFItem $p$ value in focal group
pdiffItem $p$ value differences
pdiff.adjAdjusted $p$ value difference
uniformDIFUniform DIF estimate
se.uniformDIFStandard error of uniform DIF
t.uniformDIFThe $t$ value for uniform DIF
sig.uniformDIFSignificance label for uniform DIF
DIF.ETSDIF classification according to the ETS classification system (see Details)
uniform.EBDIFEmpirical Bayes estimate of uniform DIF (Longford, Holland & Thayer, 1993) which takes degree of DIF standard error into account
DIF.SDValue of the DIF standard deviation
nonuniformDIFNonuniform DIF estimate
se.nonuniformDIFStandard error of nonuniform DIF
t.nonuniformDIFThe $t$ value for nonuniform DIF
sig.nonuniformDIFSignificance label for nonuniform DIF

Details

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 et al. (2007) for further DIF effect size classifications.

References

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. 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, 92-109. 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.

Examples

Run this code

#############################################################################
# EXAMPLE 1: Mathematics data | Gender DIF
#############################################################################

data( data.math )
dat <- data.math$data
items <- grep( "M" , colnames(dat))

# estimate item parameters and WLEs
mod <- rasch.mml2( dat[,items] )
wle <- wle.rasch( dat[,items] , b=mod$item$b )$theta

# assess DIF by logistic regression
mod1 <- dif.logistic.regression( dat=dat[,items] , score=wle , group=dat$female)

# calculate DIF variance
dif1 <- 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 <- dif.strata.variance( dif=mod1$uniformDIF , se.dif = mod1$se.uniformDIF ,
              itemcluster = data.math$item$domain )
  ## $unweighted.DIFSD
  ## [1] 0.1455916

#****
# 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) 

#############################################################################
# SIMULATED 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( 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 <- sim.raschtype( theta = rnorm(N , mean=0 , sd =1 ) , b = b1 )
dat2 <- sim.raschtype( theta = 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 <- dif.logistic.regression( dat=dat , score= mod_wle , group= group)
# Model 2: assess DIF by logistic regression and EAPs
dif2 <- 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

Run the code above in your browser using DataLab