Reply: Unquestionably Impossible

It appears that Poole & Greenland disagree with us much more than we disagree with them. In Maloley v. National Revenue Service of Canada (Juriansz, 1987), males passed a test for promotion at twice the rate of females, and Maloley argued that the test was a discriminatory practice that needed to be validated as job related before it could legally be used for promotion decisions. The Revenue Service responded, in part, by reviewing its personnel records and offering as an explanation a 2 x 2 table relating "some college attendance" to gender, finding that more men had attended some college. The Appeals Board accepted the Revenue Service's explanation, and the Revenue Service won the case. The explanation the court accepted is impossiblethere is no way, unquestionably no way, that an adjustment for "some college attendance" as recorded in the 2 x 2 table could reduce the disparity to less than three standard deviations, the most extreme legally acceptable disparity. So far, Poole & Greenland agree with us and we with them. Most fields of application develop conventions and standards for the use of statistical data and analysis. The data arose in a disparate impact case for which there are established conventions and standards (Gastwirth, 1992). In the 1970s, the U.S. Supreme Court in Castenada (1977) and Hazelwood (1977) adopted the .05 or .01 levels of significance (disparities should exceed 2 to 3 standard deviations), common in academic publications. Government guidelines (Uniform Guidelines 1978), which are used to measure the impact of the employment practice at issue, compare the ratio of selection rates for females and males with 4/5. The relative risk measure was preferred by Cornfield, et al. (1959). Often, courts compare the pass rates of the protected minority and majority groups and require statistical significance at the .05 level together with a selection ratio less than 4/5. These conventions have largely been adopted in Canada. We used the Supreme Court standards and conventions of statistical significance in this setting. In contradiction to Poole and Greenland's inference at the beginning of their section 3, we have considered the sensitivity of significance levels, point estimates and confidence intervals in other settings; see Gastwirth (1988, v2, p. 809), Rosenbaum (1987, ?3.3, 1993, 1995a,b). This was an adversary proceeding and the Revenue Service had reason to offer the best explanation it could. It determined how accurately education was measured and had full access to its own personnel records. Had the Revenue Service wished to offer an alternative explanation using other measures of education, it was free to do so. It chose, for whatever reason, to present a 2 x 2 table involving "some college attendance." It could have presented 2 x 2 tables describing "high school degree or not" or "college degree or not," or it could have used several categories or a continuous measure of education, but the Revenue Service chose to present a 2 x 2 table for "some college attendance." This is the explanation offered the court, the explanation considered by the court, and the basis for the court's decision. It is the explanation we examined in our paper. Poole and Greenland consider other explanations. For example, they argue that there might have been less evidence of unlawful discrimination if a continuous measure of education had been used. This is a paradoxical claim. Only the Revenue Service had the data. Indeed, according to Juriansz (1987, p. 691), the Service, "performed a comprehensive analysis of the profiles of the males and females." The Service was strongly motivated to present convincing evidence, and yet the Service chose to present this particular 2 x 2 table, and not the full data. If Poole and Greenland had been experts advising the defense, the Appeals Board would surely have been puzzled had they submitted their hypothetical Table 3 instead of the actual data. On another matter, there are some interesting theorems about the many 2 x 2 tables that can be formed by cutting in different places a continuous variable measured in two groups; see Gail and Green (1976). They show that if one picks the very best place to cut a continuous variable to form a 2 x 2 table, that very best table shows a bigger difference than a representative table, a difference governed by the distribution of the Kolmogorov-Smirnov statistic. Poole and Greenland go further and say that, assuming the unobserved data were configured in precisely the way that is most favorable to the Revenue Service, the difference remains statistically significant, but it becomes plausible that the difference in pass rates for men and women is comprised of 72% lawful distinctions based on qualifications and 28% unjustified distinctions between men and women. Even if one accepted the strained premises of their argument, their conclusion would suggest that women have not insubstantial grievances against the Revenue Service of Canada. In a similar case, Allen v. Seidman (1989), against the Federal Deposit Insurance Corporation, Judge Posner wrote: