? ? The past decade has witnessed the growing importance of statistical planning and inferential techniques in providing solutions to complex problems in medical and health sciences. Two scientific teams are currently dominating clinical medicine and public health: the molecular biology approach with an emphasis on genetics, and the quantitative approach with an emphasis on epidemiology. The developments in these areas jointly are making fundamental contributions to the study of cancer. This application lies in that new interface of human genetics, epidemiology and statistics in cancer research. Case-control studies are being increasingly used for studying the association between a disease and a candidate gene. However, except for some rare diseases, such as Huntington or Tay Sachs disease which may be the result of a deficiency of a single gene product, most common human diseases like cancer have a multifactorial etiology involving complex interplay of many genetic and environ- mental factors. By identifying and characterizing such complicated gene-environment interactions through clinical and epidemiological studies, one has more opportunities to study etiology, diagnosis, prognosis and treatment of complex diseases. In case-control studies of gene-environment association with disease, when genetic and environmental exposures can be assumed to be independent in the underlying population, one may exploit the independence in order to derive more efficient estimation techniques than the traditional logistic regression analysis. Many of the classical results for case- control analysis, which assume the covariate distribution to be non-parametric, do not hold under a constrained space of exposure distributions. However, the gain in efficiency of modern retrospective methods comes at the cost of lack of robustness, since large biases are introduced in the retrospective estimates under violation of the gene-environment independence assumption. The main objective of this research application is to find a natural analytical tool to solve the model specification dilemma of modern retrospective analysis of studies of gene-environment interaction, under three commonly used epidemiological designs. We posit the problem in a Bayesian framework that incorporates uncertainty regarding the assumed constraint of gene-environment independence in a natural data adaptive way. Preliminary results indicate that the proposed estimator is still able to maintain attractive efficiency properties, without relying on unverifiable model constraints. Epidemiologists have often anguished whether to use the case-control or the case-only estimator of gene-environment interaction for a given study, and the current application tries to resolve the question in a novel Bayesian framework. The methods developed may be routinely applied to various epidemiological studies of gene-environment interaction. ? ? ? ?
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