傳統的前瞻式 ( prospective ) 邏輯斯迴歸經常被用於病例-對照研究中的相關性分析。然而在病例-對照研究中，資料的取得要先決定生病與健康的人數，再看兩者過去的環境暴露狀況。因此，真正符合病例-對照研究資料的統計模型是回顧式 ( retrospective ) 模型。 在第二章中，我們提出三個重要的回顧式邏輯斯迴歸模型並且得到基因與環境的對數勝算比和交互作用的最大概似估計量以及近似變異數的估計量。我們的模型主要滿足下列兩個條件： (a) 對照組的族群或是子族群滿足哈-溫平衡，(b) 在對照組的族群中，基因與環境因子獨立。由漸近相對效率 ( asymptotic relative efficiency ) 可看出在病例-對照研究中，我們的回顧式分析優於傳統的前瞻式分析，而且我們的估計量計算簡單，方便應用在各種病例-對照研究中。 在基因的相關性分析中， population stratification 和基因型判定錯誤相當常見。由於 population stratification 和基因型判定錯誤會使相關性分析的錯誤率上升，因此修正這兩個問題所造成的偏差成為當今基因統計的重要研究目標。 我們於第三章利用 Genomic Control 的精神建構了一個能修正population stratification 和基因型判定錯誤的穩健型檢定方法 ( robust test )。根據我們的模擬結果，新方法在型一誤差和檢定力的表現非常出色且幾乎不受上述兩個問題的嚴重程度及基因遺傳模式 ( genetic model ) 的影響，明顯優於傳統的概似比檢定。我們的檢定方法簡單易懂，使用上相當便捷。 Association analysis of genetic polymorphisms has been mostly performed in a case-control setting with the traditional logistic regression model. However, in a case-control study, subjects are recruited according to their disease status and their past exposures are determined. Thus the natural model for making inference is the retrospective model. In chapter 2, we present three retrospective logistic models and give maximum likelihood estimators of exposure effects and estimators of asymptotic variances. Two scenarios are considered in this chapter: (a) the control population or its subpopulations are in Hardy-Weinberg equilibrium; and (b) genetic and environmental factors are independent in the control population. According to the concept of asymptotic relative efficiency, we have shown the precision advantages of such retrospective analysis over the traditional prospective analysis. Maximum likelihood estimates and variance estimates under retrospective models are simple in computation and thus can be applied in many practical applications. Population stratification and genotype misclassification occurs frequently in genetic association. Population stratification and genotype misclassification may lead to an increased number of false positive results in genetic association study and so it is im-portant to correct for bias caused by population stratification and genotype misclassification. Using the concept of Genomic Control, we propose a robust test to correct for population stratification and genotyping error in case-control association study in chapter 3. The simulation results given in chapter 3 indicate that this novel test is robust against misspecification of the genetic model and the levels of population stratification and/or genotyping error. The new test obviously surpasses the traditional likelihood ratio test. The test statistic is simple to compute and thus the test is quite easy to apply in applications.
