linking.robust: Robust Linking of Item Intercepts

Description

This function implements a robust alternative of mean-mean linking which employs trimmed means instead of means. The linking constant is calculated for varying trimming parameters \(k\). The treatment of differential item functioning as outliers and application of robust statistics is discussed in Magis and De Boeck (2011, 2012).

Usage

linking.robust(itempars)
# S3 method for linking.robust
summary(object,...)
# S3 method for linking.robust
plot(x, ...)

Value

A list with following entries

ind.kopt: Index for optimal scale parameter
kopt: Optimal scale parameter
meanpars.kopt: Linking constant for optimal scale parameter
se.kopt: Standard error for linking constant obtained with optimal scale parameter
meanpars: Linking constant dependent on the scale parameter
se: Standard error of the linking constant dependent on the scale parameter
sd: DIF standard deviation (non-robust estimate)
mad: DIF standard deviation (robust estimate using the MAD measure)
pars: Original item parameters
k.robust: Used vector of scale parameters
I: Number of items
itempars: Used data frame of item parameters

Arguments

itempars: Data frame of item parameters (item intercepts). The first column contains the item label, the 2nd and 3rd columns item parameters of two studies.
object: Object of class linking.robust
x: Object of class linking.robust
...: Further arguments to be passed

References

Magis, D., & De Boeck, P. (2011). Identification of differential item functioning in multiple-group settings: A multivariate outlier detection approach. Multivariate Behavioral Research, 46(5), 733-755. tools:::Rd_expr_doi("10.1080/00273171.2011.606757")

Magis, D., & De Boeck, P. (2012). A robust outlier approach to prevent type I error inflation in differential item functioning. Educational and Psychological Measurement, 72(2), 291-311. tools:::Rd_expr_doi("10.1177/0013164411416975")

Examples

Run this code

#############################################################################
# EXAMPLE 1: Linking data.si03
#############################################################################

data(data.si03)
res1 <- sirt::linking.robust( itempars=data.si03 )
summary(res1)
  ##   Number of items=27
  ##   Optimal trimming parameter k=8 |  non-robust parameter k=0
  ##   Linking constant=-0.0345 |  non-robust estimate=-0.056
  ##   Standard error=0.0186 |  non-robust estimate=0.027
  ##   DIF SD: MAD=0.0771 (robust) | SD=0.1405 (non-robust)
plot(res1)

if (FALSE) {
#############################################################################
# EXAMPLE 2: Linking PISA item parameters data.pisaPars
#############################################################################

data(data.pisaPars)

# Linking with items
res2 <- sirt::linking.robust( data.pisaPars[, c(1,3,4)] )
summary(res2)
  ##   Optimal trimming parameter k=0 |  non-robust parameter k=0
  ##   Linking constant=-0.0883 |  non-robust estimate=-0.0883
  ##   Standard error=0.0297 |  non-robust estimate=0.0297
  ##   DIF SD: MAD=0.1824 (robust) | SD=0.1487 (non-robust)
##  -> no trimming is necessary for reducing the standard error
plot(res2)

#############################################################################
# EXAMPLE 3: Linking with simulated item parameters containing outliers
#############################################################################

# simulate some parameters
I <- 38
set.seed(18785)
itempars <- data.frame("item"=paste0("I",1:I) )
itempars$study1 <- stats::rnorm( I, mean=.3, sd=1.4 )
# simulate DIF effects plus some outliers
bdif <- stats::rnorm(I,mean=.4,sd=.09)+( stats::runif(I)>.9 )* rep( 1*c(-1,1)+.4, each=I/2 )
itempars$study2 <- itempars$study1 + bdif

# robust linking
res <- sirt::linking.robust( itempars )
summary(res)
  ##   Number of items=38
  ##   Optimal trimming parameter k=12 |  non-robust parameter k=0
  ##   Linking constant=-0.4285 |  non-robust estimate=-0.5727
  ##   Standard error=0.0218 |  non-robust estimate=0.0913
  ##   DIF SD: MAD=0.1186 (robust) | SD=0.5628 (non-robust)
## -> substantial differences of estimated linking constants in this case of
##    deviations from normality of item parameters
plot(res)
}