Urn Models for Regression Analysis, with Applications to Employment Discrimination Studies

Abstract

Randomization models can be used to test the statistical significance of differences between two groups, even in observational studies where there is concomitant information, with a meaningful and valid probabilistic interpretation. This article discusses one procedure for incorporating concomitant information into a randomization model, obtaining an urn model for regression analysis. It is motivated in part by the frequency with which standard regression models are used improperly to deal with questions of disparate treatment of sex or race in employment. Although urn models have a very wide applicability, this article presents them specifically for the statistical study of wage disparity between men and women in sex discrimination litigation, an area of application that highlights the problems arising with conventional population sampling models in nonexperimental settings. A characteristic problem is the drawing of inferences in the absence of traditional safeguards found in the natural sciences: experimental control of causal factors, replication of observations, and randomization of experimental units across treatments or conditions. Expert statistical witnesses, who must present their findings without such luxuries and in the face of adversarial criticism, need to base their analyses on a minimum of assumptions in order to support their conclusions. The urn model approach requires much less in the way of assumptions than the population sampling approach and is especially attractive for this reason. The key idea, with which this article begins, is to separate the mean wage disparity between men and women into an explaned portzon (with respect to a specific adjustment procedure) and an unexplaznedportion that reflects any departure from equal earnings for men and women after adjustment for concomitant information. The statistical significance of the unexplained portion is then assessed with a randomization model. The method may be regarded as a generalization of some conventional testing methods, such as the Mantel-Haenszel procedure' for