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.

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

`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) |

root expected mean square difference

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.

Anne Corinne Huggins-Manley

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.

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)
``` |

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