BAYES' THEOREM

This journal has repeatedly discussed the technical and ethical issues raised by the existence of groups (races, sexes, ethnic groups) that frequently differ in abilities and other job-related characteristics (Eysenck 1991, Jensen, 1992; Levin, 1990, 1991). This paper is meant to add to that discussion by providing mathematical proof that consideration of such groups is, in general, necessary in selecting the best employees or students.

It is almost an article of faith that race, sex, religion, national origin, or similar classifications (which will be referred to here as groups) are irrelevant for hiring, given a goal of selecting the best candidates. The standard wisdom is that those selecting for school admission or employment should devise an unbiased (in the statistical sense) procedure which predicts individual performance, evaluate individuals with this, and then select the highest ranked individuals. However, analysis shows that even with statistically unbiased evaluation procedures, group membership may still be relevant. If the goal is to pick the best individuals for jobs or training, membership in the group with the lower average performance (the disadvantaged group) should properly be held against the individual. In general, not considering group membership and selecting the best candidates are mutually exclusive.

Three definitions will be used:

(1) "Non-discrimination" is selection which does not take into account a particular characteristic of the individual being considered (such as race, sex, age, national origin, etc.).

(2) "Merit Selection" is an endeavor to select the best qualified individual. In the terminology introduced by Hunter and Schmidt (1976), merit selection corresponds to unqualified individualism and non-discrimination to qualified individualism.

(3) "Ability" here refers to the characteristics sought by the selecting employers or schools, or to the characteristics and interests used in advising. It includes not only ability narrowly defined, but also characteristics such as motivation, honesty, etc.

One of the implications of this paper is that common statements taking the form of "Hiring shall be based on ability irrespective of race (or sex, national origin, religion, handicapped status, marital status, sexual preference, etc.)" are at best ambiguous, and at worst illogical. Logically proper statements are, "The best qualified candidates shall be selected without preference for any group (but taking into account group membership to the extent it is relevant)." or "No consideration of group membership shall be permitted (even when it is necessary to select the best candidate)." In practice, antidiscrimination rules appear to have been sold to the public on the basis of the first statement, but administered on the basis of the second statement. Indeed, not only has consideration of group membership been forbidden even when relevant, but there appears to be a tendency to forbid consideration of any characteristics that might be a surrogate for group membership, or even correlated with it (such as test scores). Rational discussion would be greatly facilitated if participants would state which policy they are advocating.

PROOF BY BAYES' THEOREM

The relevance of group membership is clearly shown by an application of Bayes' Theorem. If f(t) is the probability density function for the ability distribution among the candidates, and the distribution of estimated ability (here referred to as e) given the true ability (symbolized by t) is f(e/t), the distribution of ability given the estimated ability f(t/e) is the product of these two probability density functions divided by the density function for the estimated abilities. The proof is by direct substitution into Bayes' theorem (for Bayes' theorem, see Dyckman et al. [1968, pp. 484-489] or any standard statistics text):

EQUATION 1

f(t/e)= f(t) f(e/t)/ f(e)

Note that f(t) enters into this equation. …