THE ROBUST BEAUTY OF EQUAL WEIGHTS--OR, IT DON'T MAKE NO NEVERMIND

Some might argue that one aspect of the grade for this course is more valid than others--perhaps because it requires more work, or perhaps for other reasons--and that the more valid aspect should receive a greater weighting than the rest. This page explains why each aspect of the grade for this course is weighted equally.

Psychologists have shown that, under very general conditions, a linear model containing equally weighted predictors is just as good at predicting real-world critera (such as graduate school success, or relationship satisfaction) in a validating sample as is a model containing predictors whose weights are more precisely estimated from an original (non-validating) sample (Dawes, 1979). In addition, equal weights have a number of attractive side benefits. They are easy to estimate; they do not "use up" any degrees of freedom, making models that contain them very parsimonious; they are insensitive to outliers; and they are insensitive to non-normality in the original sample. Equal weights are also very robust--they often yield a higher correlation with criteria in a validating sample than weights estimated from an original sample, and hardly ever yield a drastically lower correlation with criteria (Wainer, 1976).

Equal weights work well under conditions that are almost always true when the predictors are intercorrelated. First, the direction of the relationship between each preditor and the criterion must be known. If one doesn't know whether the relation is positive or negative, then the predictor should not be used. Second, the weights must be between .25 and .75. If the relative influence of a predictor is less than .25, then it should not be used; if it is greater than .75, then it should probably be used to the exclusion of the other predictors (Wainer, 1976).

What if the predictors are not intercorrelated. For example, under the topic of intelligence two models are discussed, one in which the smaller, non-general factors ("specific intelligences" such as verbal and spatial intelligence) are uncorrelated, called the bifactor model, and one in which the non-general factors are correlated, called the hierarchical model. It is noted that one would prefer to include the uncorrelated predictors in a linear regression equation, rather than the correlated predictors.

Wainer's (1976) point is that if the predictors are intercorrelated, then variability in the weights makes little difference, and hence equal (unit) weighting is fine. He does not argue in favor of correlated predictors but rather points out that if they are correlated, then unit weighting is fine. This is called the principle of "it don't make no nevermind." Where the various components of grades for this course are concerned, one can safely assume that they are all intercorrelated; thus, unit weighting is fine. In terms of optimal prediction, however, uncorrelated predictors that are optimally weighted are preferred to unit weighted predictors.

References

Dawes, R. M. (1979). The robust beauty of improper linear models in decision making. American Psychologist, 34, 571-582.

Wainer, H. (1976). Estimating coefficients in linear models: It don't make no nevermind. Psychological Bulletin, 83, 312-317.

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