Root Expected Mean Square Difference

Description

The root expected mean square difference index (REMSD) is a summary index of the weighted differences between each subpopulation equated score, y_j(x), and the equated score based on the overall population, y(x). Formally,

REMSD=sqrt(sum(P{sum(w_j[y_j(x)-y(x)]^2})))/s,

where w_j is a subpopulation weight, x is a score on the original (i.e., unequated) scale, P is the proportion of examinees scoring at x and s is the standard deviation of x scores in the (sub)population of interest. It is considered an omnibus, unconditional index. It was originally presented by Dorans and Holland (2000). It provides practitioners with a summary of the magnitude of weighted differences between subpopulation equated scores and equated scores based on the overall population.

Usage

1
remsd(x, o, g, f, s, w)

Arguments

x

a column vector of scores on which the rmsd 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)

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 remsd)

w

A row vector of weights for subpopulations 1 thru n (length = number of groups)

Value

root expected mean square difference

Note

The equally weighted version of this index (Kolen & Brennan, 2004) can be obtained by inputting a w vector consisting of identical elements that sum to 1. See example 1 below.

Author(s)

Anne Corinne Huggins-Manley

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

rmsd

Examples

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
#Unstandardized REMSD for subpopulations 1 and 2 in the example data set, ex.data, 
#assuming equal weights for the subpopulations
remsd(x=ex.data[,1],o=ex.data[,2],
g=c(ex.data[,3],ex.data[,4]),f=ex.data[,8],w=c(.5,.5))

#Unstandardized REMSD for all five subpopulations in the example data set, ex.data
remsd(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]),
f=ex.data[,8],w=c(.1,.2,.4,.2,.1))

#Standardized REMSD for all five subpopulations in the example data set, ex.data
remsd(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]),
f=ex.data[,8],w=c(.1,.2,.4,.2,.1),s=4.2)