Minority Discrimination Case: An Analysis of Ill-Conditioned Data

The court case of Hogan v. Pierce (Gastwirth, 1984) involved the following promotion data, by race.  

	Whites		Blacks
Date of Promotion	Promoted	Not	Promoted	Not
7/74	4	16	0	7
8/74	4	13	0	7
9/74	2	13	0	8
4/75	1	17	0	8
5/75	1	17	0	8
10/75	1	29	0	10
11/75	2	29	0	10
2/76	1	30	0	10
3/76	1	30	0	10
11/77	1	33	0	13

The most notable feature of these data is that at each promotion opportunity not a single black was promoted, whereas small numbers of whites were promoted. Thus the data support an infinite upper bound on the odds ratio of being promoted for whites relative to blacks. But it is fair to ask how low the odds ratio could be. StatXact produces the following exact and Mantel-Haenszel output.

The Mantel-Haenszel method cannot estimate the lower bound on the odds ratio because the point and variance estimators are undefined when the observed statistic is at its maximum. However, both the exact and mid-p methods can. The mid-p method claims with 95% confidence that the odds of promotion for whites is no less than 2.298 times the odds of promotion for blacks. The more conservative exact method places a lower bound of 1.819 on the odds ratio.

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