Statistical Properties of Biostatistical Methods for Correlated Processes with Application to Data Arising in the Legal Settings Open Access
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Statistical methods are widely used in equal employment cases to analyze data submitted as evidence of discrimination. In this dissertation, statistical procedures are developed to resolve issues arising in actual discrimination cases. The first problem arose in the Alexander v. Milwaukee case, where white male police lieutenants filed a reverse discrimination claim that they were treated unfairly in the promotion to captain. At the same time, individuals whose opportunities for promotion were limited by discrimination might decide to retire earlier than otherwise. Two Cox proportional hazards (PH) models are employed to describe the promotion and retirement processes, respectively. However, the two processes are likely to be negatively related and data on a relevant factor might not have been available reflecting the possible effect of the omitted variable. Therefore, frailty terms are introduced in the Cox PH models. The joint Cox PH models with frailty terms are fitted using the Monte-Carlo expectation and maximization (MCEM) algorithm. Model selection and diagnosis in the presence of frailty are challenging as the frailty terms are unobserved. An inappropriate choice of the functional form of the frailty term or its distribution may influence the ultimate inference about the main effects. If one includes a frailty term when it is not needed, the estimates of the main effects remain reasonably accurate. On the other hand, ignoring a frailty term, i.e., only fitting two marginal models can lead to a noticeable bias in the estimates of the main effects. Th effect of assuming an incorrect distribution for the frailty term is also studied. The Gamma frailty model is seen to yield robust results over the family of distributions (Lognormal and Inverse Gaussian distributions) considered. The effect of the choice of functional relationship of the frailty term to the outcome of interest when the frailty term follows the commonly Gamma distribution is explored. A relative simple model performs well in the simulation study. The proposed joint models are applied to the Alexander v. Milwaukee data. The original finding of liability is shown to be robust for several models, over the choice of several possible frailty distributions, implying that unadjusted frailty is very unlikely to alter the legal conclusion. Although our focus is on negatively correlated processes, a small study examining the robustness of frailty models for positively correlated processes, which are common in biostatistical applications, is carried out. The second problem arose from the situation where both the hiring and promotion processes of an employer may be affected by discriminatory practices. An under-appreciated issue in promotion cases is that the standard Fisher's exact test may have low power of finding a statistically significant disparity in promotion rates if the number of minorities previously hired was small due to unfairness in the hiring process. In this situation, one should check whether the hiring is fair, even if discrimination in promotion is the main focus. Depending on the result of the statistical test on hiring, an appropriate test for fairness in promotions is specified. A second approach analyzes both the hiring and promotion data to select the appropriate procedure. The statistical properties of the proposed methods are presented and the procedures are applied to four data sets from actual cases. Because the choice of test statistic at the second stage (promotion) depends on the results of the first stage (hiring) analysis, the results indicate that in cases concerning promotion, courts should allow plaintiffs to have access to data concerning hiring or lower level promotions. Thus, the courts should take a broad view of potentially relevant data at the discovery stage.