remsd: Root Expected Mean Square Difference In SEAsic: Score Equity Assessment- summary index computation

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.

`rmsd`
 ``` 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) ```