In Top-Down Decisions, Weighting Variables does Not Matter: A Consequence of Wilks' Theorem

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Abstract
It is often appropriate to weight variables to form a composite for making decisions. Examples include selection systems, organizational performance criteria, test items, and decision modeling. Frequently, criterion-based regression-weighting is employed, but a sizable literature argues for unit or simple weighting. Wainer demonstrated small loss from equal weights compared to regression weights. Usually, weights are of little importance for rank ordering, echoing Wainer's "it don't make no nevermind." Wilks proved a general theorem, that under common circumstances, almost all weighted composites of a set of variables are highly correlated. That is, if a single set of variables is weighted two different ways to form two composites, the expected correlation for the two composites is very high. The authors demonstrate the effect of Wilks' theorem through illustrative examples. Implications of Wilks' theorem are discussed. When top-down decisions are made, weighting variables does not matter because the rank ordering remains almost constant.

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