SEAsic (version 0.1)

emaxd: Expected Maximum Difference

Description

The expected maximum difference index, $EMAXD$, locates the maximum absolute difference between subpopulation equated scores, $y_j(x)$, and the equated scores based on the overall population, $y(x)$, at each score level $x$, and then takes the expectation of those maximum scores across score levels. Formally, $$EMAXD=\frac{\sum_{x}P_x\{\mbox{max}\lbrack\mid y_j(x)-y(x)\mid\rbrack\}}{\sigma_x},$$ where $P$ represents a proportion of examinees based on the population distribution specified in argument $f$, and $s$ is the standard deviation of $x$ scores in the (sub)population of interest. It is considered an omnibus, unconditional index that was originally presented by Dorans and Holland (2000). It provides practitioners with a summary of the maximum differences found between subpopulation and overall equated scores.

Usage

emaxd(x, o, g, n, f, s)

Arguments

x

a column vector of scores on which the rsd is conditioned

o

a column vector of equated scores based on the overall population (aligned with elements in x)

g

column vectors of equated scores based on various subpopulations (aligned with elements in x)

n

a scalar indicating the number of groups

f

a column vector of relative frequency associated with each raw score (can be based on either overall population or a subpopulation) (aligned with elements in x)

s

a scalar representing the standard deviation of x for any (sub)population of interest (e.g., synthetic population) (default is 1, which leads to calculation of the unstandardized emaxd)

Value

expected maximum difference

References

• Dorans, N.J., & Holland, P.W. (2000). Population invariance and the equitability of tests: Theory and the linear case. Journal of Educational Measurement, 37, 281-306.

See Also

maxd

Examples

#Unstandardized EMAXD across subpopulation 1 and subpopulation 2 in the example data set, ex.data
emaxd(x=ex.data[,1],o=ex.data[,2],
g=c(ex.data[,3],ex.data[,4]),
n=2,f=ex.data[,8])

#Unstandardized EMAXD across subpopulations 1 thru 5 in the example data set, ex.data
emaxd(x=ex.data[,1],o=ex.data[,2],
g=c(ex.data[,3],ex.data[,4],ex.data[,5],ex.data[,6],ex.data[,7]),
n=5,f=ex.data[,8])

#Standardized EMAXD across subpopulations 1 thru 5 in the example data set, ex.data
emaxd(x=ex.data[,1],o=ex.data[,2],
g=c(ex.data[,3],ex.data[,4],ex.data[,5],ex.data[,6],ex.data[,7]),
n=5,f=ex.data[,8],s=4.2)