Case-control studies of unrelated subjects are now widely used to study the role of genetic susceptibility and gene-environment interactions in the etiology of complex diseases. Exploiting an assumption of gene-environment independence, and treating the distribution of the environmental exposures to be completely nonparametric, Chatterjee and Carroll (2005) recently developed an efficient retrospective maximum-likelihood method for analysis of case-control studies. In this article, we develop an extension of the retrospective maximum-likelihood approach to studies where genetic information may be missing on some study subjects. In particular, special emphasis is given to haplotype-based studies where missing data arises due to linkage-phase ambiguity of genotype data. We use a profile likelihood technique and an appropriate EM algorithm to derive a relatively simple procedure for parameter estimation, with or without a rare disease assumption, and possibly incorporating information on the marginal probability of the disease for the underlying population. We also describe two alternative robust approaches that are less sensitive to the underlying gene-environment independence and Hardy-Weinberg-Equilibrium assumptions. The performance of the proposed methods are studied using simulation studies in the context of haplotype-based studies of gene-environment interaction. An application of the proposed method is illustrated using a case-control study of ovarian cancer designed to study the interaction between BRCA1/2 mutations and reproductive risk factors in the etiology of ovarian cancer.

We have shown that a combination of fish oil (high in n-3 fatty acids) with the butyrate-producing fiber pectin, upregulates apoptosis in colon cells exposed to the carcinogen azoxymethane, protecting against colon tumor development. We now hypothesize that n-3 fatty acids prime the colonocytes such that butyrate can initiate apoptosis. To test this, 30 Sprague-Dawley rats were provided with diets differing in the fatty acid composition (corn oil, fish oil or a purified fatty acid ethyl ester diet). Intact colon crypts were exposed ex vivo to butyrate, and analyzed for reactive oxygen species (ROS), mitochondrial membrane potential (MMP), translocation of cytochrome C to the cytosol, and caspase-3 activity (early events in apoptosis). The fatty acid composition of the three major mitochondrial phospholipids was also determined, and an unsaturation index calculated. The unsaturation index in cardiolipin was correlated with ROS levels (R = 0.99; P = 0.02). When colon crypts from fish oil and FAEE-fed rats were exposed to butyrate, MMP decreased (P = 0.041); and translocation of cytochrome C to the cytosol (P = 0.037) and caspase-3 activation increased (P = 0.032). The data suggest that fish oil may prime the colonocytes for butyrate-induced apoptosis by enhancing the unsaturation of mitochondrial phospholipids, especially cardiolipin, resulting in an increase in ROS and initiating apoptotic cascade.

We showed previously that the dietary combination of fish oil, rich in (n-3) fatty acids, and the fermentable fiber pectin enhances colonocyte apoptosis in a rat model of experimentally induced colon cancer. In this study, we propose that the mechanism by which this dietary combination heightens apoptosis is via modulation of the colonocyte redox environment. Male Sprague-Dawley rats (n = 60) were fed 1 of 2 fats (corn oil or fish oil) and 1 of 2 fibers (cellulose or pectin) for 2 wk before determination of reactive oxygen species (ROS), oxidative DNA damage, antioxidant enzyme activity [superoxide dismutase (SOD), catalase (CAT), glutathione peroxidase (GPx)] and apoptosis in isolated colonocytes. Fish oil enhanced ROS, whereas the combination of fish oil and pectin suppressed SOD and CAT and enhanced the SOD/CAT ratio compared with a corn oil and cellulose diet. Despite this modulation to a seemingly prooxidant environment, oxidative DNA damage was inversely related to ROS in the fish oil and pectin diet, and apoptosis was enhanced relative to other diets. Furthermore, apoptosis increased exponentially as ROS increased. These results suggest that the enhancement of apoptosis associated with fish oil and pectin feeding may be due to a modulation of the redox environment that promotes ROS-mediated apoptosis.

Summary. It is widely believed that risks of many complex diseases are determined by genetic susceptibilities, environmental exposures, and their interaction. Chatterjee and Carroll (2005, Biometrika 92, 399-418) developed an efficient retrospective maximum-likelihood method for analysis of case-control studies that exploits an assumption of gene-environment independence and leaves the distribution of the environmental covariates to be completely nonparametric. Spinka, Carroll, and Chatterjee (2005, Genetic Epidemiology 29, 108-127) extended this approach to studies where certain types of genetic information, such as haplotype phases, may be missing on some subjects. We further extend this approach to situations when some of the environmental exposures are measured with error. Using a polychotomous logistic regression model, we allow disease status to have K + 1 levels. We propose use of a pseudolikelihood and a related EM algorithm for parameter estimation. We prove consistency and derive the resulting asymptotic covariance matrix of parameter estimates when the variance of the measurement error is known and when it is estimated using replications. Inferences with measurement error corrections are complicated by the fact that the Wald test often behaves poorly in the presence of large amounts of measurement error. The likelihoodratio (LR) techniques are known to be a good alternative. However, the LR tests are not technically correct in this setting because the likelihood function is based on an incorrect model, i.e., a prospective model in a retrospective sampling scheme. We corrected standard asymptotic results to account for the fact that the LR test is based on a likelihood-type function. The performance of the proposed method is illustrated using simulation studies emphasizing the case when genetic information is in the form of haplotypes and missing data arises from haplotype-phase ambiguity. An application of our method is illustrated using a population-based case-control study of the association between calcium intake and the risk of colorectal adenoma.

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