GEE‐Based SNP Set Association Test for Continuous and Discrete Traits in Family‐Based Association Studies

Family‐based genetic association studies of related individuals provide opportunities to detect genetic variants that complement studies of unrelated individuals. Most statistical methods for family association studies for common variants are single marker based, which test one SNP a time. In this paper, we consider testing the effect of an SNP set, e.g., SNPs in a gene, in family studies, for both continuous and discrete traits. Specifically, we propose a generalized estimating equations (GEEs) based kernel association test, a variance component based testing method, to test for the association between a phenotype and multiple variants in an SNP set jointly using family samples. The proposed approach allows for both continuous and discrete traits, where the correlation among family members is taken into account through the use of an empirical covariance estimator. We derive the theoretical distribution of the proposed statistic under the null and develop analytical methods to calculate the P‐values. We also propose an efficient resampling method for correcting for small sample size bias in family studies. The proposed method allows for easily incorporating covariates and SNP‐SNP interactions. Simulation studies show that the proposed method properly controls for type I error rates under both random and ascertained sampling schemes in family studies. We demonstrate through simulation studies that our approach has superior performance for association mapping compared to the single marker based minimum P‐value GEE test for an SNP‐set effect over a range of scenarios. We illustrate the application of the proposed method using data from the Cleveland Family GWAS Study.     

Finding Novel Genes by Testing G × E Interactions in a Genome‐Wide Association Study

In a genome‐wide association study (GWAS), investigators typically focus their primary analysis on the direct (marginal) associations of each single nucleotide polymorphism (SNP) with the trait. Some SNPs that are truly associated with the trait may not be identified in this scan if they have a weak marginal effect and thus low power to be detected. However, these SNPs may be quite important in subgroups of the population defined by an environmental or personal factor, and may be detectable if such a factor is carefully considered in a gene–environment (G × E) interaction analysis. We address the question “Using a genome wide interaction scan (GWIS), can we find new genes that were not found in the primary GWAS scan?” We review commonly used approaches for conducting a GWIS in case‐control studies, and propose a new two‐step screening and testing method (EDG×E) that is optimized to find genes with a weak marginal effect. We simulate several scenarios in which our two‐step method provides 70–80% power to detect a disease locus while a marginal scan provides less than 5% power. We also provide simulations demonstrating that the EDG×E method outperforms other GWIS approaches (including case only and previously proposed two‐step methods) for finding genes with a weak marginal effect. Application of this method to a G × Sex scan for childhood asthma reveals two potentially interesting SNPs that were not identified in the marginal‐association scan. We distribute a new software program (G×Escan, available at http://biostats.usc.edu/software ) that implements this new method as well as several other GWIS approaches.     

Case‐only genome‐wide interaction study of disease risk,prognosis and treatment

Case‐control genome‐wide association (GWA) studies have facilitated the identification of susceptibility loci for many complex diseases; however, these studies are often not adequately powered to detect gene‐environment (G×E) and gene‐gene (G×G) interactions. Case‐only studies are more efficient than case‐control studies for detecting interactions and require no data on control subjects. In this article, we discuss the concept and utility of the case‐only genome‐wide interaction (COGWI) study, in which common genetic variants, measured genome‐wide, are screened for association with environmental exposures or genetic variants of interest. An observed G‐E (or G‐G) association, as measured by the case‐only odds ratio (OR), suggests interaction, but only if the interacting factors are unassociated in the population from which the cases were drawn. The case‐only OR is equivalent to the interaction risk ratio. In addition to risk‐related interactions, we discuss how the COGWI design can be used to efficiently detect G×G, G×E and pharmacogenetic interactions related to disease outcomes in the context of observational clinical studies or randomized clinical trials. Such studies can be conducted using only data on individuals experiencing an outcome of interest or individuals not experiencing the outcome of interest. Sharing data among GWA and COGWI studies of disease risk and outcome can further enhance efficiency. Sample size requirements for COGWI studies, as compared to case‐control GWA studies, are provided. In the current era of genome‐wide analyses, the COGWI design is an efficient and straightforward method for detecting G×G, G×E and pharmacogenetic interactions related to disease risk, prognosis and treatment response. Genet. Epidemiol. 34:7–15, 2010. © 2009 Wiley‐Liss, Inc.     

Complete Effect‐Profile Assessment in Association Studies With Multiple Genetic and Multiple Environmental Factors

Studying complex diseases in the post genome‐wide association studies (GWAS) era has led to developing methods that consider factor‐sets rather than individual genetic/environmental factors (i.e., Multi‐G‐Multi‐E studies), and mining for potential gene‐environment (G×E) interactions has proven to be an invaluable aid in both discovery and deciphering underlying biological mechanisms. Current approaches for examining effect profiles in Multi‐G‐Multi‐E analyses are either underpowered due to large degrees of freedom, ill‐suited for detecting G×E interactions due to imprecise modeling of the G and E effects, or lack of capacity for modeling interactions between two factor‐sets (e.g., existing methods focus primarily on a single E factor). In this work, we illustrate the issues encountered in constructing kernels for investigating interactions between two factor‐sets, and propose a simple yet intuitive solution to construct the G×E kernel that retains the ease‐of‐interpretation of classic regression. We also construct a series of kernel machine (KM) score tests to evaluate the complete effect profile (i.e., the G, E, and G×E effects individually or in combination). We show, via simulations and a data application, that the proposed KM methods outperform the classic and PC regressions across a range of scenarios, including varying effect size, effect structure, and interaction complexity. The largest power gain was observed when the underlying effect structure involved complex G×E interactions; however, the proposed methods have consistent, powerful performance when the effect profile is simple or complex, suggesting that the proposed method could be a useful tool for exploratory or confirmatory G×E analysis.     

Efficient Two‐Step Testing of Gene‐Gene Interactions in Genome‐Wide Association Studies

Exhaustive testing of all possible SNP pairs in a genome‐wide association study (GWAS) generally yields low power to detect gene‐gene (G × G) interactions because of small effect sizes and stringent requirements for multiple‐testing correction. We introduce a new two‐step procedure for testing G × G interactions in case‐control GWAS to detect interacting single nucleotide polymorphisms (SNPs) regardless of their marginal effects. In an initial screening step, all SNP pairs are tested for gene‐gene association in the combined sample of cases and controls. In the second step, the pairs that pass the screening are followed up with a traditional test for G × G interaction. We show that the two‐step method is substantially more powerful to detect G × G interactions than the exhaustive testing approach. For example, with 2,000 cases and 2,000 controls, the two‐step method can have more than 90% power to detect an interaction odds ratio of 2.0 compared to less than 50% power for the exhaustive testing approach. Moreover, we show that a hybrid two‐step approach that combines our newly proposed two‐step test and the two‐step test that screens for marginal effects retains the best power properties of both. The two‐step procedures we introduce have the potential to uncover genetic signals that have not been previously identified in an initial single‐SNP GWAS. We demonstrate the computational feasibility of the two‐step G × G procedure by performing a G × G scan in the asthma GWAS of the University of Southern California Children's Health Study.     

A Robust Method for Genome‐Wide Association Meta‐Analysis With the Application to Circulating Insulin‐Like Growth Factor I Concentrations

Genome‐wide association studies (GWAS) offer an excellent opportunity to identify the genetic variants underlying complex human diseases. Successful utilization of this approach requires a large sample size to identify single nucleotide polymorphisms (SNPs) with subtle effects. Meta‐analysis is a cost‐efficient means to achieve large sample size by combining data from multiple independent GWAS; however, results from studies performed on different populations can be variable due to various reasons, including varied linkage equilibrium structures as well as gene‐gene and gene‐environment interactions. Nevertheless, one should expect effects of the SNP are more similar between similar populations than those between populations with quite different genetic and environmental backgrounds. Prior information on populations of GWAS is often not considered in current meta‐analysis methods, rendering such analyses less optimal for the detecting association. This article describes a test that improves meta‐analysis to incorporate variable heterogeneity among populations. The proposed method is remarkably simple in computation and hence can be performed in a rapid fashion in the setting of GWAS. Simulation results demonstrate the validity and higher power of the proposed method over conventional methods in the presence of heterogeneity. As a demonstration, we applied the test to real GWAS data to identify SNPs associated with circulating insulin‐like growth factor I concentrations.     

A Genome‐Wide Association Study for Serum Bilirubin Levels and Gene‐Environment Interaction in a Chinese Population

Bilirubin is an effective antioxidant and is influenced by both genetic and environmental factors. Recent genome‐wide association studies (GWAS) have identified multiple loci affecting serum total bilirubin levels. However, most of the studies were conducted in European populations and little attention has been devoted either to genetic variants associated with direct and indirect bilirubin levels or to the gene‐environment interactions on bilirubin levels. In this study, a two‐stage GWAS was performed to identify genetic variants associated with all types of bilirubin levels in 10,282 Han Chinese individuals. Gene‐environment interactions were further examined. Briefly, two previously reported loci, UGT1A1 on 2q37 (rs6742078 and rs4148323, combined P = 1.44 × 10^?89 and P = 5.05 × 10^?69, respectively) and SLCO1B3 on 12p12 (rs2417940, combined P = 6.93 × 10^?19) were successfully replicated. The two loci explained 9.2% and 0.9% of the total variations of total bilirubin levels, respectively. Ethnic genetic differences were observed between Chinese and European populations. More importantly, a significant interaction was found between rs2417940 in SLCO1B3 gene and smoking on total bilirubin levels (P = 1.99 × 10^?3). Single nucleotide polymorphism (SNP) rs2417940 had stronger effects on total bilirubin levels in nonsmokers than in smokers, suggesting that the effects of SLCO1B3 genotype on bilirubin levels were partly dependent on smoking status. Consistent associations and interactions were observed for serum direct and indirect bilirubin levels.     

Meta-analysis of gene-environment interaction: joint estimation of SNP and SNP × environment regression coefficients

Introduction: Genetic discoveries are validated through the meta‐analysis of genome‐wide association scans in large international consortia. Because environmental variables may interact with genetic factors, investigation of differing genetic effects for distinct levels of an environmental exposure in these large consortia may yield additional susceptibility loci undetected by main effects analysis. We describe a method of joint meta‐analysis (JMA) of SNP and SNP by Environment (SNP × E) regression coefficients for use in gene‐environment interaction studies. Methods: In testing SNP × E interactions, one approach uses a two degree of freedom test to identify genetic variants that influence the trait of interest. This approach detects both main and interaction effects between the trait and the SNP. We propose a method to jointly meta‐analyze the SNP and SNP × E coefficients using multivariate generalized least squares. This approach provides confidence intervals of the two estimates, a joint significance test for SNP and SNP × E terms, and a test of homogeneity across samples. Results: We present a simulation study comparing this method to four other methods of meta‐analysis and demonstrate that the JMA performs better than the others when both main and interaction effects are present. Additionally, we implemented our methods in a meta‐analysis of the association between SNPs from the type 2 diabetes‐associated gene PPARG and log‐transformed fasting insulin levels and interaction by body mass index in a combined sample of 19,466 individuals from five cohorts. Genet. Epidemiol. 35:11–18, 2011. © 2010 Wiley‐Liss, Inc.     

Two‐phase designs for joint quantitative‐trait‐dependent and genotype‐dependent sampling in post‐GWAS regional sequencing

We evaluate two‐phase designs to follow‐up findings from genome‐wide association study (GWAS) when the cost of regional sequencing in the entire cohort is prohibitive. We develop novel expectation‐maximization‐based inference under a semiparametric maximum likelihood formulation tailored for post‐GWAS inference. A GWAS‐SNP (where SNP is single nucleotide polymorphism) serves as a surrogate covariate in inferring association between a sequence variant and a normally distributed quantitative trait (QT). We assess test validity and quantify efficiency and power of joint QT‐SNP‐dependent sampling and analysis under alternative sample allocations by simulations. Joint allocation balanced on SNP genotype and extreme‐QT strata yields significant power improvements compared to marginal QT‐ or SNP‐based allocations. We illustrate the proposed method and evaluate the sensitivity of sample allocation to sampling variation using data from a sequencing study of systolic blood pressure.     

A genetic stochastic process model for genome‐wide joint analysis of biomarker dynamics and disease susceptibility with longitudinal data

Unraveling the underlying biological mechanisms or pathways behind the effects of genetic variations on complex diseases remains one of the major challenges in the post‐GWAS (where GWAS is genome‐wide association study) era. To further explore the relationship between genetic variations, biomarkers, and diseases for elucidating underlying pathological mechanism, a huge effort has been placed on examining pleiotropic and gene‐environmental interaction effects. We propose a novel genetic stochastic process model (GSPM) that can be applied to GWAS and jointly investigate the genetic effects on longitudinally measured biomarkers and risks of diseases. This model is characterized by more profound biological interpretation and takes into account the dynamics of biomarkers during follow‐up when investigating the hazards of a disease. We illustrate the rationale and evaluate the performance of the proposed model through two GWAS. One is to detect single nucleotide polymorphisms (SNPs) having interaction effects on type 2 diabetes (T2D) with body mass index (BMI) and the other is to detect SNPs affecting the optimal BMI level for protecting from T2D. We identified multiple SNPs that showed interaction effects with BMI on T2D, including a novel SNP rs11757677 in the CDKAL1 gene (P = 5.77 × 10^?7). We also found a SNP rs1551133 located on 2q14.2 that reversed the effect of BMI on T2D (P = 6.70 × 10^?7). In conclusion, the proposed GSPM provides a promising and useful tool in GWAS of longitudinal data for interrogating pleiotropic and interaction effects to gain more insights into the relationship between genes, quantitative biomarkers, and risks of complex diseases.     

Impact of repeated measures and sample selection on genome‐wide association studies of fasting glucose

Although genome‐wide association studies (GWAS) have been performed in longitudinal studies, most used only a single trait measure. GWAS of fasting glucose have generally included only normoglycemic individuals. We examined the impact of both repeated measures and sample selection on GWAS in Atherosclerosis Risk In Communities (ARIC), a study which obtained four longitudinal measures of fasting glucose and included both individuals with and without prevalent diabetes. The sample included Caucasians and the Affymetrix 6.0 chip was used for genotyping. Sample sizes for GWAS analyses ranged from 8,372 (first study visit) to 5,782 (average fasting glucose). Candidate SNP analyses with SNPs identified through fasting glucose or diabetes GWAS were conducted in 9,133 individuals, including 761 with prevalent diabetes. For a constant sample size, smaller P‐values were obtained for the average measure of fasting glucose compared to values at any single visit, and two additional significant GWAS signals were detected. For four candidate SNPs (rs780094, rs10830963, rs7903146, and rs4607517), the strength of association between genotype and glucose was significantly (P‐interaction<0.05) different in those with and without prevalent diabetes, and for all five fasting glucose candidate SNPs (rs780094, rs10830963, rs560887, rs4607517, and rs13266634) the association with measured fasting glucose was more significant in the smaller sample without prevalent diabetes than in the larger combined sample of those with and without diabetes. This analysis demonstrates the potential utility of averaging trait values in GWAS studies and explores the advantage of using only individuals without prevalent diabetes in GWAS of fasting glucose. Genet. Epidemiol. 34: 665‐673, 2010. © 2010 Wiley‐Liss, Inc.     

Generalized estimating equations for genome‐wide association studies using longitudinal phenotype data

Many longitudinal cohort studies have both genome‐wide measures of genetic variation and repeated measures of phenotypes and environmental exposures. Genome‐wide association study analyses have typically used only cross‐sectional data to evaluate quantitative phenotypes and binary traits. Incorporation of repeated measures may increase power to detect associations, but also requires specialized analysis methods. Here, we discuss one such method—generalized estimating equations (GEE)—in the contexts of analysis of main effects of rare genetic variants and analysis of gene‐environment interactions. We illustrate the potential for increased power using GEE analyses instead of cross‐sectional analyses. We also address challenges that arise, such as the need for small‐sample corrections when the minor allele frequency of a genetic variant and/or the prevalence of an environmental exposure is low. To illustrate methods for detection of gene‐drug interactions on a genome‐wide scale, using repeated measures data, we conduct single‐study analyses and meta‐analyses across studies in three large cohort studies participating in the Cohorts for Heart and Aging Research in Genomic Epidemiology consortium—the Atherosclerosis Risk in Communities study, the Cardiovascular Health Study, and the Rotterdam Study. Copyright © 2014 John Wiley & Sons, Ltd.     

Smooth‐Threshold Multivariate Genetic Prediction with Unbiased Model Selection

We develop a new genetic prediction method, smooth‐threshold multivariate genetic prediction, using single nucleotide polymorphisms (SNPs) data in genome‐wide association studies (GWASs). Our method consists of two stages. At the first stage, unlike the usual discontinuous SNP screening as used in the gene score method, our method continuously screens SNPs based on the output from standard univariate analysis for marginal association of each SNP. At the second stage, the predictive model is built by a generalized ridge regression simultaneously using the screened SNPs with SNP weight determined by the strength of marginal association. Continuous SNP screening by the smooth thresholding not only makes prediction stable but also leads to a closed form expression of generalized degrees of freedom (GDF). The GDF leads to the Stein's unbiased risk estimation (SURE), which enables data‐dependent choice of optimal SNP screening cutoff without using cross‐validation. Our method is very rapid because computationally expensive genome‐wide scan is required only once in contrast to the penalized regression methods including lasso and elastic net. Simulation studies that mimic real GWAS data with quantitative and binary traits demonstrate that the proposed method outperforms the gene score method and genomic best linear unbiased prediction (GBLUP), and also shows comparable or sometimes improved performance with the lasso and elastic net being known to have good predictive ability but with heavy computational cost. Application to whole‐genome sequencing (WGS) data from the Alzheimer's Disease Neuroimaging Initiative (ADNI) exhibits that the proposed method shows higher predictive power than the gene score and GBLUP methods.     

A Pipeline for Classifying Relationships Using Dense SNP/SNV Data and Putative Pedigree Information

When genome‐wide association studies (GWAS) or sequencing studies are performed on family‐based datasets, the genotype data can be used to check the structure of putative pedigrees. Even in datasets of putatively unrelated people, close relationships can often be detected using dense single‐nucleotide polymorphism/variant (SNP/SNV) data. A number of methods for finding relationships using dense genetic data exist, but they all have certain limitations, including that they typically use average genetic sharing, which is only a subset of the available information. Here, we present a set of approaches for classifying relationships in GWAS datasets or large‐scale sequencing datasets. We first propose an empirical method for detecting identity by descent segments in close relative pairs using un‐phased dense SNP data and demonstrate how that information can assist in building a relationship classifier. We then develop a strategy to take advantage of putative pedigree information to enhance classification accuracy. Our methods are tested and illustrated with two datasets from two distinct populations. Finally, we propose classification pipelines for checking and identifying relationships in datasets containing a large number of small pedigrees.     

Identifying significant gene‐environment interactions using a combination of screening testing and hierarchical false discovery rate control

Although gene‐environment (G× E) interactions play an important role in many biological systems, detecting these interactions within genome‐wide data can be challenging due to the loss in statistical power incurred by multiple hypothesis correction. To address the challenge of poor power and the limitations of existing multistage methods, we recently developed a screening‐testing approach for G× E interaction detection that combines elastic net penalized regression with joint estimation to support a single omnibus test for the presence of G× E interactions. In our original work on this technique, however, we did not assess type I error control or power and evaluated the method using just a single, small bladder cancer data set. In this paper, we extend the original method in two important directions and provide a more rigorous performance evaluation. First, we introduce a hierarchical false discovery rate approach to formally assess the significance of individual G× E interactions. Second, to support the analysis of truly genome‐wide data sets, we incorporate a score statistic‐based prescreening step to reduce the number of single nucleotide polymorphisms prior to fitting the first stage penalized regression model. To assess the statistical properties of our method, we compare the type I error rate and statistical power of our approach with competing techniques using both simple simulation designs as well as designs based on real disease architectures. Finally, we demonstrate the ability of our approach to identify biologically plausible SNP‐education interactions relative to Alzheimer's disease status using genome‐wide association study data from the Alzheimer's Disease Neuroimaging Initiative (ADNI).     

Detecting Gene–Environment Interactions for a Quantitative Trait in a Genome‐Wide Association Study

A genome‐wide association study (GWAS) typically is focused on detecting marginal genetic effects. However, many complex traits are likely to be the result of the interplay of genes and environmental factors. These SNPs may have a weak marginal effect and thus unlikely to be detected from a scan of marginal effects, but may be detectable in a gene–environment (G × E) interaction analysis. However, a genome‐wide interaction scan (GWIS) using a standard test of G × E interaction is known to have low power, particularly when one corrects for testing multiple SNPs. Two 2‐step methods for GWIS have been previously proposed, aimed at improving efficiency by prioritizing SNPs most likely to be involved in a G × E interaction using a screening step. For a quantitative trait, these include a method that screens on marginal effects [Kooperberg and Leblanc, 2008] and a method that screens on variance heterogeneity by genotype [Paré et al., 2010] In this paper, we show that the Paré et al. approach has an inflated false‐positive rate in the presence of an environmental marginal effect, and we propose an alternative that remains valid. We also propose a novel 2‐step approach that combines the two screening approaches, and provide simulations demonstrating that the new method can outperform other GWIS approaches. Application of this method to a G × Hispanic‐ethnicity scan for childhood lung function reveals a SNP near the MARCO locus that was not identified by previous marginal‐effect scans.     

The Case‐Only Test for Gene–Environment Interaction is Not Uniformly Powerful: An Empirical Example

The case‐only test has been proposed as a more powerful approach to detect gene–environment (G × E) interactions. This approach assumes that the genetic and environmental factors are independent. Although it is well known that Type I error rate will increase if this assumption is violated, it is less widely appreciated that G × E correlation can also lead to power loss. We illustrate this phenomenon by comparing the performance of the case‐only test to other approaches to detect G × E interactions in a genome‐wide association study (GWAS) of esophageal squamous‐cell carcinoma (ESCC) in Chinese populations. Some of these approaches do not use information on the correlation between exposure and genotype (standard logistic regression), whereas others seek to use this information in a robust fashion to boost power without increasing Type I error (two‐step, empirical Bayes, and cocktail methods). G × E interactions were identified involving drinking status and two regions containing genes in the alcohol metabolism pathway, 4q23 and 12q24. Although the case‐only test yielded the most significant tests of G × E interaction in the 4q23 region, the case‐only test failed to identify significant interactions in the 12q24 region which were readily identified using other approaches. The low power of the case‐only test in the 12q24 region is likely due to the strong inverse association between the single nucleotide polymorphism (SNPs) in this region and drinking status. This example underscores the need to consider multiple approaches to detect G × E interactions, as different tests are more or less sensitive to different alternative hypotheses and violations of the G × E independence assumption.     

Empirical Hierarchical Bayes Approach to Gene‐Environment Interactions: Development and Application to Genome‐Wide Association Studies of Lung Cancer in TRICL

The analysis of gene‐environment (G × E) interactions remains one of the greatest challenges in the postgenome‐wide association studies (GWASs) era. Recent methods constitute a compromise between the robust but underpowered case‐control and powerful case‐only methods. Inferences of the latter are biased when the assumption of gene‐environment (G‐E) independence in controls fails. We propose a novel empirical hierarchical Bayes approach to G × E interaction (EHB‐GE), which benefits from greater rank power while accounting for population‐based G‐E correlation. Building on Lewinger et al.'s ([2007] Genet Epidemiol 31:871–882) hierarchical Bayes prioritization approach, the method first obtains posterior G‐E correlation estimates in controls for each marker, borrowing strength from G‐E information across the genome. These posterior estimates are then subtracted from the corresponding case‐only G × E estimates. We compared EHB‐GE with rival methods using simulation. EHB‐GE has similar or greater rank power to detect G × E interactions in the presence of large numbers of G‐E correlations with weak to strong effects or only a low number of such correlations with large effect. When there are no or only a few weak G‐E correlations, Murcray et al.'s method ([2009] Am J Epidemiol 169:219–226) identifies markers with low G × E interaction effects better. We applied EHB‐GE and competing methods to four lung cancer case‐control GWAS from the Interdisciplinary Research in Cancer of the Lung/International Lung Cancer Consortium with smoking as environmental factor. A number of genes worth investigating were identified by the EHB‐GE approach.     

The use of imputed values in the meta‐analysis of genome‐wide association studies

In genome‐wide association studies (GWAS), it is a common practice to impute the genotypes of untyped single nucleotide polymorphism (SNP) by exploiting the linkage disequilibrium structure among SNPs. The use of imputed genotypes improves genome coverage and makes it possible to perform meta‐analysis combining results from studies genotyped on different platforms. A popular way of using imputed data is the “expectation‐substitution” method, which treats the imputed dosage as if it were the true genotype. In current practice, the estimates given by the expectation‐substitution method are usually combined using inverse variance weighting (IVM) scheme in meta‐analysis. However, the IVM is not optimal as the estimates given by the expectation‐substitution method are generally biased. The optimal weight is, in fact, proportional to the inverse variance and the expected value of the effect size estimates. We show both theoretically and numerically that the bias of the estimates is very small under practical conditions of low effect sizes in GWAS. This finding validates the use of the expectation‐substitution method, and shows the inverse variance is a good approximation of the optimal weight. Through simulation, we compared the power of the IVM method with several methods including the optimal weight, the regular z‐score meta‐analysis and a recently proposed “imputation aware” meta‐analysis method (Zaitlen and Eskin [2010] Genet Epidemiol 34:537–542). Our results show that the performance of the inverse variance weight is always indistinguishable from the optimal weight and similar to or better than the other two methods. Genet. Epidemiol. 2011. © 2011 Wiley Periodicals, Inc. 35:597‐605, 2011     

On assessing model fit for distribution‐free longitudinal models under missing data

The generalized estimating equation (GEE), a distribution‐free, or semi‐parametric, approach for modeling longitudinal data, is used in a wide range of behavioral, psychotherapy, pharmaceutical drug safety, and healthcare‐related research studies. Most popular methods for assessing model fit are based on the likelihood function for parametric models, rendering them inappropriate for distribution‐free GEE. One rare exception is a score statistic initially proposed by Tsiatis for logistic regression (1980) and later extended by Barnhart and Willamson to GEE (1998). Because GEE only provides valid inference under the missing completely at random assumption and missing values arising in most longitudinal studies do not follow such a restricted mechanism, this GEE‐based score test has very limited applications in practice. We propose extensions of this goodness‐of‐fit test to address missing data under the missing at random assumption, a more realistic model that applies to most studies in practice. We examine the performance of the proposed tests using simulated data and demonstrate the utilities of such tests with data from a real study on geriatric depression and associated medical comorbidities. Copyright © 2013 John Wiley & Sons, Ltd.