a maximum likelihood method for studying gene environment interactions under conditional independence of genotype and exposure given the biomedical interest in gene environment interactions along with the difficulties inherent in gathering genetic data from controls epidemiologists need methodologies that can increase precision of estimating interactions while minimizing the genotyping of controls to achieve this purpose many epidemiologists suggested that one can use case only design in this paper we present a maximum likelihood method for making inference about gene environment interactions using case only data the probability of disease development is described by a logistic risk model thus the interactions are model parameters measuring the departure of joint effects of exposure and genotype from multiplicative odds ratios we extend the typical inference method derived under the assumption of independence between genotype and exposure to that under a more general assumption of conditional independence our maximum likelihood method can be applied to analyse both categorical and continuous environmental factors and generalized to make inference about gene gene environment interactions moreover the application of this method can be reduced to simply fitting a multinomial logistic model when we have case only data as a consequence the maximum likelihood estimates of interactions and likelihood ratio tests for hypotheses concerning interactions can be easily computed the methodology is illustrated through an example based on a study about the joint effects of xrcc1 polymorphisms and smoking on bladder cancer we also give two simulation studies to show that the proposed method is reliable in finite sample situation

fitting regression models to case control data by maximum likelihood summary we consider fitting categorical regression models to data obtained by either stratified or nonstratified case control or response selective sampling from a finite population with known population totals in each response category with certain models such as the logistic with appropriate constant terms a method variously known as conditional maximum likelihood breslow cain 1988 or pseudo conditional likelihood wild 1991 which involves the prospective fitting of a pseudo model results in maximum likelihood estimates of case control data we extend these results by showing the maximum likelihood estimates for any model can be found by iterating this process with a simple updating of offset parameters attention is also paid to estimation of the asymptotic covariance matrix one benefit of the results of this paper is the ability to obtain maximum likelihood estimates of the parameters of logistic models for stratified case control studies compare breslow cain 1988 scott wild 1991 using an ordinary logistic regression program even when the stratum constants are modelled

semiparametric maximum likelihood estimation exploiting gene environment independence in case control studies we consider the problem of maximum likelihood estimation in case control studies of gene environment associations with disease when genetic and environmental exposures can be assumed to be independent in the underlying population traditional logistic regression analysis may not be efficient in this setting we study the semiparametric maximum likelihood estimates of logistic regression parameters that exploit the gene environment independence assumption and leave the distribution of the environmental exposures to be nonparametric we use a profile likelihood technique to derive a simple algorithm for obtaining the estimator and we study the asymptotic theory the results are extended to situations where genetic and environmental factors are independent conditional on some other factors simulation studies investigate small sample properties the method is illustrated using data from a case control study designed to investigate the interplay of brca1 2 mutations and oral contraceptive use in the aetiology of ovarian cancer

a note on efficient estimation from case control data summary it is well known that it is valid to treat case control data as if they had been obtained by random sampling when estimating the odds ratio parameters of logistic regression models the result has been generalised to estimation of the regression parameters in multiplicative intercept models and it has further been pointed out that the approach may be applied to non multiplicative intercept models if the parameter space is first enlarged to include a multiplicative intercept term this note is concerned with the efficiency of the approach in epidemiological case control studies it is common to model the influence of predictors on the risk of disease through a finite dimensional regression parameter while treating the marginal distribution of the predictors as an infinite dimensional nuisance parameter see for example breslow day 1980 it is well known that case control data may be treated as if they had been obtained by random sampling when estimating the regression parameters in a logistic regression model in a multiplicative intercept model the log odds ratio of the risk of disease is modelled as a known function of the regression parameters and the predictors plus a variation independent intercept parameter it is also known that the result in breslow day 1980 holds more generally for estimating other than the intercept parameter in any multiplicative intercept model see for example prentice pyke 1979 hsieh manski mcfadden 1985 anderson 1972 wacholder weinberg 1993 robins rotnitzky zhao 1994 or scott wild 1989 with non multiplicative intercept models the data may be treated as if they had been obtained by random sampling if the model for the risk of disease is first replaced in the likelihood by a larger model that has the multiplicative intercept property the larger model is defined implicitly by treating the logit of the risk of disease as if it were equal to the logit of the risk of disease in the original model plus an unspecified intercept term see for example scott wild 1986 robins blevins 1987 breslow storer 1985 or storer wacholder breslow 1983 the results of cosslett 1981 imply that joint maximisation of the likelihood for the marginal distribution of the predictors and the regression parameters results in efficient estimates of the regression parameters in the asymptotic setting where the numbers of cases and controls tend to infinity together scott wild 1986 note that with randomly sampled data in multiplicative intercept models the case control indicators are ancillary for the regression coefficients thus joint maximisation of the semiparametric likelihood for the case control data and maximisation of the likelihood obtained by treating the data as if they were randomly sampled both result in the same estimate of the regression parameters j robins personal communication has pointed out that the argument may be applied to show efficiency of the approach in non multiplicative intercept models as well because the case control likelihood in the larger model and the case control likelihood in the smaller model are equivalent this note presents a direct calculation that demonstrates the efficiency result both for multiplicative and non multiplicative intercept models

semiparametric estimation in the secondary analysis of case control studies we study the regression relationship among covariates in case control data an area known as the secondary analysis of case control studies the context is such that only the form of the regression mean is specified so that we allow an arbitrary regression error distribution which can depend on the covariates and thus can be heteroscedastic under mild regularity conditions we establish the theoretical identifiability of such models previous work in this context has either a specified a fully parametric distribution for the regression errors b specified a homoscedastic distribution for the regression errors c has specified the rate of disease in the population we refer this as true population or d has made a rare disease approximation we construct a class of semiparametric estimation procedures that rely on none of these the estimators differ from the usual semiparametric ones in that they draw conclusions about the true population while technically operating in a hypothetic superpopulation we also construct estimators with a unique feature in that they are robust against the misspecification of the regression error distribution in terms of variance structure while all other nonparametric effects are estimated despite of the biased samples we establish the asymptotic properties of the estimators and illustrate their finite sample performance through simulation studies as well as through an empirical example on the relation between red meat consumption and heterocyclic amines our analysis verified the positive relationship between red meat consumption and two forms of hca indicating that increased red meat consumption leads to increased levels of meiqa and phip both being risk factors for colorectal cancer computer software as well as data to illustrate the methodology are available at http wileyonlinelibrary com journal rss datasets

two stage testing procedures with independent filtering for genome wide gene environment interaction several two stage multiple testing procedures have been proposed to detect gene environment interaction in genome wide association studies in this article we elucidate general conditions that are required for validity and power of these procedures and we propose extensions of two stage procedures using the case only estimator of gene treatment interaction in randomized clinical trials we develop a unified estimating equation approach to proving asymptotic independence between a filtering statistic and an interaction test statistic in a range of situations including marginal association and interaction in a generalized linear model with a canonical link we assess the performance of various two stage procedures in simulations and in genetic studies from women s health initiative clinical trials

exploiting gene environment independence for analysis of case control studies an empirical bayes type shrinkage estimator to trade off between bias and efficiency summary standard prospective logistic regression analysis of case control data often leads to very imprecise estimates of gene environment interactions due to small numbers of cases or controls in cells of crossing genotype and exposure in contrast under the assumption of gene environment independence modern retrospective methods including the case only approach can estimate the interaction parameters much more precisely but they can be seriously biased when the underlying assumption of gene environment independence is violated in this article we propose a novel empirical bayes type shrinkage estimator to analyze case control data that can relax the gene environment independence assumption in a data adaptive fashion in the special case involving a binary gene and a binary exposure the method leads to an estimator of the interaction log odds ratio parameter in a simple closed form that corresponds to an weighted average of the standard case only and case control estimators we also describe a general approach for deriving the new shrinkage estimator and its variance within the retrospective maximum likelihood framework developed by chatterjee and carroll 2005 biometrika92 399 418 both simulated and real data examples suggest that the proposed estimator strikes a balance between bias and efficiency depending on the true nature of the gene environment association and the sample size for a given study

unified analysis of secondary traits in case control association studies it has been repeatedly shown that in case control association studies analysis of a secondary trait that ignores the original sampling scheme can produce highly biased risk estimates although a number of approaches have been proposed to properly analyze secondary traits most approaches fail to reproduce the marginal logistic model assumed for the original case control trait and or do not allow for interaction between secondary trait and genotype marker on primary disease risk in addition the flexible handling of covariates remains challenging we present a general retrospective likelihood framework to perform association testing for both binary and continuous secondary traits which respects marginal models and incorporates the interaction term we provide a computational algorithm based on a reparameterized approximate profile likelihood for obtaining the maximum likelihood ml estimate and its standard error for the genetic effect on secondary traits in the presence of covariates for completen

meta analysis of gene environment interaction exploiting gene environment independence across multiple case control studies multiple papers have studied the use of gene environment g e independence to enhance power for testing gene environment interaction in case control studies however studies that evaluate the role of g e independence in a meta analysis framework are limited in this paper we extend the single study empirical bayes type shrinkage estimators proposed by mukherjee and chatterjee 2008 to a meta analysis setting that adjusts for uncertainty regarding the assumption of g e independence across studies we use the retrospective likelihood framework to derive an adaptive combination of estimators obtained under the constrained model assuming g e independence and unconstrained model without assumptions of g e independence with weights determined by measures of g e association derived from multiple studies our simulation studies indicate that this newly proposed estimator has improved average performance across different simulation scenarios than the standard alternative of using inverse variance covariance weighted estimators that combines study specific constrained unconstrained or empirical bayes estimators the results are illustrated by meta analyzing 6 different studies of type 2 diabetes investigating interactions between genetic markers on the obesity related fto gene and environmental factors body mass index and age

rost nilpotence and higher unramified cohomology abstract we develop an approach to proving the rost nilpotence principle involving higher unramified cohomology along with a decomposition of the diagonal technique we use this to prove the principle for certain threefolds over a perfect field

maximum likelihood inference on a mixed conditionally and marginally specified regression model for genetic epidemiologic studies with two phase sampling summary two phase stratified sampling designs can reduce the cost of genetic epidemiologic studies by limiting expensive ascertainments of genetic and environmental exposure to an efficiently selected subsample phase ii of the main study phase i family history and some covariate information which may be cheaply gathered for all subjects at phase i can be used for sampling of informative subjects at phase ii we develop alternative maximum likelihood methods for analysis of data from such studies by using a novel regression model that permits the estimation of marginal risk parameters that are associated with the genetic and environmental covariates of interest while simultaneously characterizing the conditional risk of the disease associated with family history after adjusting for the other covariates the methods and appropriate asymptotic theories are developed with and without an assumption of gene environment independence allowing the distribution of the environmental factors to remain non parametric the performance of the alternative methods and of sampling strategies is studied by using simulated data involving rare and common genetic variants an application of the methods proposed is illustrated by using a case control study of colorectal adenoma embedded within the prostate lung colorectal and ovarian cancer screening trial

a latent variable approach to study gene environment interactions in the presence of multiple correlated exposures many existing cohort studies initially designed to investigate disease risk as a function of environmental exposures have collected genomic data in recent years with the objective of testing for gene environment interaction g e effects in environmental epidemiology interest in g e arises primarily after a significant effect of the environmental exposure has been documented cohort studies often collect rich exposure data as a result assessing g e effects in the presence of multiple exposure markers further increases the burden of multiple testing an issue already present in both genetic and environment health studies latent variable lv models have been used in environmental epidemiology to reduce dimensionality of the exposure data gain power by reducing multiplicity issues via condensing exposure data and avoid collinearity problems due to presence of multiple correlated exposures we extend the lv framework to characterize gene environment interaction in presence of multiple correlated exposures and genotype categories further similar to what has been done in case control g e studies we use the assumption of gene environment g e independence to boost the power of tests for interaction the consequences of making this assumption or the issue of how to explicitly model g e association has not been previously investigated in lv models we postulate a hierarchy of assumptions about the lv model regarding the different forms of g e dependence and show that making such assumptions may influence inferential results on the g e and g e parameters we implement a class of shrinkage estimators to data adaptively trade off between the most restrictive to most flexible form of g e dependence assumption and note that such class of compromise estimators can serve as a benchmark of model adequacy in lv models we demonstrate the methods with an example from the early life exposures in mexico city to neuro toxicants element study of lead exposure iron metabolism genes and birth weight

likelihood inferences on semiparametric odds ratio model a flexible semiparametric odds ratio model has been proposed to unify and to extend both the log linear model and the joint normal model for data with a mix of discrete and continuous variables the semiparametric odds ratio model is particularly useful for analyzing biased sampling designs however statistical inference of the model has not been systematically studied when more than one nonparametric component is involved in the model in this article we study the maximum semiparametric likelihood approach to estimation and inference of the semiparametric odds ratio model we show that the maximum semiparametric likelihood estimator of the odds ratio parameter is consistent and asymptotically normally distributed we also establish statistical inference under a misspecified semiparametric odds ratio model which is important when handling weak identifiability in conditionally specified models under biased sampling designs we use simulation studies to demonstrate that the proposed approaches have satis

testing for gene environment and gene gene interactions under monotonicity constraints recent genome wide association studies gwass designed to detect the main effects of genetic markers have had considerable success with many findings validated by replication studies however relatively few findings of gene gene or gene environment interactions have been successfully reproduced besides the main issues associated with insufficient sample size in current studies a complication is that interactions that rank high based on p values often correspond to extreme forms of joint effects that are biologically less plausible to reduce false positives and to increase power we develop various gene environment gene gene tests based on biologically more plausible constraints using bivariate isotonic regressions for case control data we extend our method to exploit gene environment or gene gene independence information integrating the approach proposed by chatterjee and carroll we propose appropriate nonparametric and parametric permutation procedures for evaluating the significance of the tests

the semiparametric case only estimator we propose a semiparametric case only estimator of multiplicative gene environment or gene gene interactions under the assumption of conditional independence of the two factors given a vector of potential confounding variables our estimator yields valid inferences on the interaction function if either but not necessarily both of two unknown baseline functions of the confounders is correctly modeled furthermore when both models are correct our estimator has the smallest possible asymptotic variance for estimating the interaction parameter in a semiparametric model that assumes that at least one but not necessarily both baseline models are correct

elliptic equations in divergence form with partially bmo coefficients the solvability in sobolev spaces is proved for divergence form second order elliptic equations in the whole space a half space and a bounded lipschitz domain for equations in the whole space or a half space the leading coefficients aij are assumed to be only measurable in one direction and have locally small bmo semi norms in the other directions for equations in a bounded domain additionally we assume that aij have small bmo semi norms in a neighborhood of the boundary of the domain we give a unified approach of both the dirichlet boundary problem and the conormal derivative problem we also investigate elliptic equations in sobolev spaces with mixed norms under the same assumptions on the coefficients

the combination of ecological and case control data ecological studies in which data are available at the level of the group rather than at the level of the individual are susceptible to a range of biases due to their inability to characterize within group variability in exposures and confounders in order to overcome these biases we propose a hybrid design in which ecological data are supplemented with a sample of individual level case control data we develop the likelihood for this design and illustrate its benefits via simulation both in bias reduction when compared to an ecological study and in efficiency gains relative to a conventional case control study an interesting special case of the proposed design is the situation where ecological data are supplemented with case only data the design is illustrated using a dataset of county specific lung cancer mortality rates in the state of ohio from 1988

semiparametric analysis of complex polygenic gene environment interactions in case control studies many methods have recently been proposed for efficient analysis of case control studies of gene environment interactions using a retrospective likelihood framework that exploits the natural assumption of gene environment independence in the underlying population however for polygenic modelling of gene environment interactions which is a topic of increasing scientific interest applications of retrospective methods have been limited due to a requirement in the literature for parametric modelling of the distribution of the genetic factors we propose a general computationally simple semiparametric method for analysis of case control studies that allows exploitation of the assumption of gene environment independence without any further parametric modelling assumptions about the marginal distributions of any of the two sets of factors the method relies on the key observation that an underlying efficient profile likelihood depends on the distribution of genetic factors only through certain expectation terms that can be evaluated empirically we develop asymptotic inferential theory for the estimator and evaluate its numerical performance via simulation studies an application of the method is presented

using covariate specific disease prevalence information to increase the power of case control studies public registration databases and large cohort studies provide vital information on disease prevalence at various levels of a risk factor this auxiliary information can be helpful in conducting statistical inference in a new study we aim to develop a statistical procedure that improves the efficiency of the logistic regression model for a case control study by utilizing auxiliary information on covariate specific disease prevalence via a series of unbiased estimating equations we adopt empirical likelihood for statistical inference and demonstrate its advantages through simulation and an application

efficient designs of gene environment interaction studies implications of hardy weinberg equilibrium and gene environment independence it is important to investigate whether genetic susceptible variants exercise the same effects in populations that are differentially exposed to environmental risk factors here we assess the power of four two phase case control design strategies for assessing multiplicative gene environment g e interactions or for assessing genetic or environmental effects in the presence of g e interactions with a di allelic snp and a binary e we obtained closed form maximum likelihood estimates of both main effect and interaction odds ratio parameters under the constraints of g e independence and hardy weinberg equilibrium and used the wald statistic for all tests we concluded that i for testing g e interactions or genetic effects in the presence of g e interactions when data for e is fully available it is preferable to ascertain data for g in a subsample of cases with similar numbers of exposed and unexposed and a random subsample of controls and ii for testing g e interactions or environmental effects in the presence of g e interactions when data for g is fully available it is preferable to ascertain data for e in a subsample of cases that has similar numbers for each genotype and a random subsample of controls in addition supplementing external control data to an existing casecontrol sample leads to improved power for assessing effects of g or e in the presence of g e interactions

a unified framework for studying parameter identifiability and estimation in biased sampling designs summary based on the odds ratio representation of a joint density we propose a unified framework to study parameter identifiability in biased sampling designs it is shown that most of these designs encountered in practice can be reformulated within the proposed framework and as a result the question of parameter identifiability can be largely clarified estimation of the identifiable parameters is considered and traditional results on the equivalence of the prospective and retrospective likelihoods are extended information contained in data on certain identifiable parameters is often very limited such parameters can be poorly estimated by the likelihood approach with practically attainable sample sizes which can substantially affect the estimates of parameters of primary interest a partially penalized likelihood approach is proposed to address this simulation results suggest that the proposed approach has good performance

a composite likelihood approach in testing for hardy weinberg equilibrium using family based genetic survey data in population based household surveys for example the national health and nutrition examination survey households are often sampled by stratified multistage cluster sampling and multiple individuals related by blood are often sampled within households therefore genetic data collected from these population based household surveys called national genetic household surveys can be correlated because of two levels of correlation one level of correlation is caused by the multistage geographical cluster sampling and the other is caused by biological inheritance among participants within the same sampled family in this paper we develop an efficient hardy weinberg equilibrium hwe test utilizing pairwise composite likelihood methods that incorporate the sample weighting effect induced by the differential selection probabilities in complex sample designs as well as the two level clustering correlation effects described above monte carlo simulation studies show that the proposed hwe test maintains the nominal levels and is more powerful than existing methods li et al 2011 under various non informative sample designs that depend on genotypes explicitly or implicitly family relationships or both especially when within household sampling depends on the genotypes the developed tests are further evaluated using simulated genetic data based on the hispanic health and nutrition survey copyright 2016 john wiley sons ltd

an easy to implement approach for analyzing case control and case only studies assuming gene environment independence and hardy weinberg equilibrium the case control study is a simple and an useful method to characterize the effect of a gene the effect of an exposure as well as the interaction between the two the control free case only study is yet an even simpler design if interest is centered on gene environment interaction only it requires the sometimes plausible assumption that the gene under study is independent of exposures among the non diseased in the study populations the hardy weinberg equilibrium is also sometimes reasonable to assume this paper presents an easy to implement approach for analyzing case control and case only studies under the above dual assumptions the proposed approach the conditional logistic regression with counterfactuals offers the flexibility for complex modeling yet remains well within the reach to the practicing epidemiologists when the dual assumptions are met the conditional logistic regression with counterfactuals is unbiased and has the correct type i error rates it also results in smaller variances and achieves higher powers as compared with using the conventional analysis unconditional logistic regression

semiparametric estimation exploiting covariate independence in two phase randomized trials recent results for case control sampling suggest when the covariate distribution is constrained by gene environment independence semiparametric estimation exploiting such independence yields a great deal of efficiency gain we consider the efficient estimation of the treatment biomarker interaction in two phase sampling nested within randomized clinical trials incorporating the independence between a randomized treatment and the baseline markers we develop a newton raphson algorithm based on the profile likelihood to compute the semiparametric maximum likelihood estimate spmle our algorithm accommodates both continuous phase one outcomes and continuous phase two biomarkers the profile information matrix is computed explicitly via numerical differentiation in certain situations where computing the spmle is slow we propose a maximum estimated likelihood estimator mele which is also capable of incorporating the covariate independence this estimated likelihood approach uses a one step empirical covariate distribution thus is straightforward to maximize it offers a closed form variance estimate with limited increase in variance relative to the fully efficient spmle our results suggest exploiting the covariate independence in two phase sampling increases the efficiency substantially particularly for estimating treatment biomarker interactions