A note on the empirical Bayes heterogeneity variance estimator in meta-analysis

This paper focuses on the empirical Bayes (EB) or Mandel-Paule estimator of the heterogeneity variance in meta-analysis, which was discussed by Morris and proposed in earlier publications by Mandel and Paule in an inter-laboratory context. The relationship of the EB estimator to other heterogeneity variance estimators typically used in meta-analysis is explored, and approximate variance estimators for the EB estimate of the heterogeneity variance are proposed based on the M-estimation method. Statistical inference for the overall treatment effect using the EB estimator and the proposed standard errors is discussed using two example data sets from meta-analysis applications.     

A comparison of heterogeneity variance estimators in combining results of studies

For random effects meta-analysis, seven different estimators of the heterogeneity variance are compared and assessed using a simulation study. The seven estimators are the variance component type estimator (VC), the method of moments estimator (MM), the maximum likelihood estimator (ML), the restricted maximum likelihood estimator (REML), the empirical Bayes estimator (EB), the model error variance type estimator (MV), and a variation of the MV estimator (MVvc). The performance of the estimators is compared in terms of both bias and mean squared error, using Monte Carlo simulation. The results show that the REML and especially the ML and MM estimators are not accurate, having large biases unless the true heterogeneity variance is small. The VC estimator tends to overestimate the heterogeneity variance in general, but is quite accurate when the number of studies is large. The MV estimator is not a good estimator when the heterogeneity variance is small to moderate, but it is reasonably accurate when the heterogeneity variance is large. The MVvc estimator is an improved estimator compared to the MV estimator, especially for small to moderate values of the heterogeneity variance. The two estimators MVvc and EB are found to be the most accurate in general, particularly when the heterogeneity variance is moderate to large.     

Empirical Bayes Gaussian likelihood estimation of exposure distributions from pooled samples in human biomonitoring

Human biomonitoring of exposure to environmental chemicals is important. Individual monitoring is not viable because of low individual exposure level or insufficient volume of materials and the prohibitive cost of taking measurements from many subjects. Pooling of samples is an efficient and cost‐effective way to collect data. Estimation is, however, complicated as individual values within each pool are not observed but are only known up to their average or weighted average. The distribution of such averages is intractable when the individual measurements are lognormally distributed, which is a common assumption. We propose to replace the intractable distribution of the pool averages by a Gaussian likelihood to obtain parameter estimates. If the pool size is large, this method produces statistically efficient estimates, but regardless of pool size, the method yields consistent estimates as the number of pools increases. An empirical Bayes (EB) Gaussian likelihood approach, as well as its Bayesian analog, is developed to pool information from various demographic groups by using a mixed‐effect formulation. We also discuss methods to estimate the underlying mean–variance relationship and to select a good model for the means, which can be incorporated into the proposed EB or Bayes framework. By borrowing strength across groups, the EB estimator is more efficient than the individual group‐specific estimator. Simulation results show that the EB Gaussian likelihood estimates outperform a previous method proposed for the National Health and Nutrition Examination Surveys with much smaller bias and better coverage in interval estimation, especially after correction of bias. Copyright © 2014 John Wiley & Sons, Ltd.     

Joint testing of genotype and ancestry association in admixed families

Current genome-wide association studies (GWAS) often involve populations that have experienced recent genetic admixture. Genotype data generated from these studies can be used to test for association directly, as in a non-admixed population. As an alternative, these data can be used to infer chromosomal ancestry, and thus allow for admixture mapping. We quantify the contribution of allele-based and ancestry-based association testing under a family-design, and demonstrate that the two tests can provide non-redundant information. We propose a joint testing procedure, which efficiently integrates the two sources information. The efficiencies of the allele, ancestry and combined tests are compared in the context of a GWAS. We discuss the impact of population history and provide guidelines for future design and analysis of GWAS in admixed populations.     

Estimating treatment effect via simple cross design synthesis

Randomized controlled trials (RCTs) are the traditional gold standard evidence for medical decision-making. However, protocols that limit enrollment eligibility introduce selection error that severely limits a RCT's applicability to a wide range of patients. Conversely, high quality observational data can be representative of entire populations, but freedom to choose treatment can bias estimators based on this data. Cross design synthesis (CDS) is an approach to combining both RCT and observational data in a single analysis that capitalizes on the RCT's strong internal validity and the observational study's strong external validity. We proposed and assessed a simple estimator of effect size based on the CDS approach. We evaluated its properties within a formal framework of causal estimation and compared our estimator with more traditional estimators based on single sources of evidence. We show that under ideal conditions the simple CDS estimator is unbiased whenever the observational data-based estimators' treatment selection error is constant across those who are and are not eligible for RCT participation. Whereas this assumption may not often hold in practice, assumptions required for the unbiasedness of usual single-source estimators may also be implausible. We show that, under some reasonable data assumptions, our simple CDS estimator has smaller bias and better coverage than commonly used estimates based on randomized or observational studies alone.     

Smooth bootstrap methods for analysis of longitudinal data

In analysis of longitudinal data, the variance matrix of the parameter estimates is usually estimated by the 'sandwich' method, in which the variance for each subject is estimated by its residual products. We propose smooth bootstrap methods by perturbing the estimating functions to obtain 'bootstrapped' realizations of the parameter estimates for statistical inference. Our extensive simulation studies indicate that the variance estimators by our proposed methods can not only correct the bias of the sandwich estimator but also improve the confidence interval coverage. We applied the proposed method to a data set from a clinical trial of antibiotics for leprosy.     

A flexible genome-wide bootstrap method that accounts for ranking and threshold-selection bias in GWAS interpretation and replication study design

The phenomenon known as the winner's curse is a form of selection bias that affects estimates of genetic association. In genome-wide association studies (GWAS) the bias is exacerbated by the use of stringent selection thresholds and ranking over hundreds of thousands of single nucleotide polymorphisms (SNPs). We develop an improved multi-locus bootstrap point estimate and confidence interval, which accounts for both ranking- and threshold-selection bias in the presence of genome-wide SNP linkage disequilibrium structure. The bootstrap method easily adapts to various study designs and alternative test statistics as well as complex SNP selection criteria. The latter is demonstrated by our application to the Wellcome Trust Case Control Consortium findings, in which the selection criterion was the minimum of the p-values for the additive and genotypic genetic effect models. In contrast, existing likelihood-based bias-reduced estimators account for the selection criterion applied to an SNP as if it were the only one tested, and so are more simple computationally, but do not address ranking across SNPs. Our simulation studies show that the bootstrap bias-reduced estimates are usually closer to the true genetic effect than the likelihood estimates and are less variable with a narrower confidence interval. Replication study sample size requirements computed from the bootstrap bias-reduced estimates are adequate 75-90 per cent of the time compared to 53-60 per cent of the time for the likelihood method. The bootstrap methods are implemented in a user-friendly package able to provide point and interval estimation for both binary and quantitative phenotypes in large-scale GWAS.     

Accounting for Linkage Disequilibrium in Association Analysis of Diverse Populations

The National Human Genome Research Institute's catalog of published genome‐wide association studies (GWAS) lists over 10,000 genetic variants collectively associated with over 800 human diseases or traits. Most of these GWAS have been conducted in European‐ancestry populations. Findings gleaned from these studies have led to identification of disease‐associated loci and biologic pathways involved in disease etiology. In multiple instances, these genomic findings have led to the development of novel medical therapies or evidence for prescribing a given drug as the appropriate treatment for a given individual beyond phenotypic appearances or socially defined constructs of race or ethnicity. Such findings have implications for populations throughout the globe and GWAS are increasingly being conducted in more diverse populations. A major challenge for investigators seeking to follow up genomic findings between diverse populations is discordant patterns of linkage disequilibrium (LD). We provide an overview of common measures of LD and opportunities for their use in novel methods designed to address challenges associated with following up GWAS conducted in European‐ancestry populations in African‐ancestry populations or, more generally, between populations with discordant LD patterns. We detail the strengths and weaknesses associated with different approaches. We also describe application of these strategies in follow‐up studies of populations with concordant LD patterns (replication) or discordant LD patterns (transferability) as well as fine‐mapping studies. We review application of these methods to a variety of traits and diseases.     

A comparison of strategies for analyzing dichotomous outcomes in genome-wide association studies with general pedigrees

Genome-wide association studies (GWAS) have been frequently conducted on general or isolated populations with related individuals. However, there is a lack of consensus on which strategy is most appropriate for analyzing dichotomous phenotypes in general pedigrees. Using simulation studies, we compared several strategies including generalized estimating equations (GEE) strategies with various working correlation structures, generalized linear mixed model (GLMM), and a variance component strategy (denoted LMEBIN) that treats dichotomous outcomes as continuous with special attentions to their performance with rare variants, rare diseases, and small sample sizes. In our simulations, when the sample size is not small, for type I error, only GEE and LMEBIN maintain nominal type I error in most cases with exceptions for GEE with very rare disease and genetic variants. GEE and LMEBIN have similar statistical power and slightly outperform GLMM when the prevalence is low. In terms of computational efficiency, GEE with sandwich variance estimator outperforms GLMM and LMEBIN. We apply the strategies to GWAS of gout in the Framingham Heart Study. Based on our results, we would recommend using GEE ind-san in the GWAS for common variants and GEE ind-fij or LMEBIN for rare variants for GWAS of dichotomous outcomes with general pedigrees.     

Meta-analysis combining parallel and cross-over clinical trials. III: The issue of carry-over

In meta-analysis combining results from parallel and cross-over trials, there is a risk of bias originating from the carry-over effect in cross-over trials. When pooling treatment effects estimated from parallel trials and two-period two-treatment cross-over trials, meta-analytic estimators of treatment effect can be obtained from the combination of parallel trial results either with cross-over trial results based on data of the first period only or with cross-over trial results analysed with data from both periods. Taking data from the first cross-over period protects against carry-over but gives less efficient treatment estimators and may lead to selection bias. This study evaluates in terms of variance reduction and mean square error the cost of calculating meta-analysis estimates with data from the first period instead of data from the two cross-over periods. If the information on cross-over sequence is available, we recommend performing two combined design meta-analyses, one using the first cross-over period data and one based on data from both cross-over periods. To investigate simultaneously the statistical significance of these two estimators as well as the carry-over at meta-analysis level, a method based on a multivariate analysis of the meta-analytic treatment effect and carry-over estimates is proposed.     

Modified robust variance estimator for generalized estimating equations with improved small-sample performance

Generalized estimating equations (GEE (Biometrika 1986; 73(1):13-22) is a general statistical method to fit marginal models for correlated or clustered responses, and it uses a robust sandwich estimator to estimate the variance-covariance matrix of the regression coefficient estimates. While this sandwich estimator is robust to the misspecification of the correlation structure of the responses, its finite sample performance deteriorates as the number of clusters or observations per cluster decreases. To address this limitation, Pan (Biometrika 2001; 88(3):901-906) and Mancl and DeRouen (Biometrics 2001; 57(1):126-134) investigated two modifications to the original sandwich variance estimator. Motivated by the ideas underlying these two modifications, we propose a novel robust variance estimator that combines the strengths of these estimators. Our theoretical and numerical results show that the proposed estimator attains better efficiency and achieves better finite sample performance compared with existing estimators. In particular, when the sample size or cluster size is small, our proposed estimator exhibits lower bias and the resulting confidence intervals for GEE estimates achieve better coverage rates performance. We illustrate the proposed method using data from a dental study.     

Transethnic meta-analysis of metabolic syndrome in a multiethnic study

Genome-wide association studies (GWAS) have been used to establish thousands of genetic associations across numerous phenotypes. To improve the power of GWAS and generalize associations across ethnic groups, transethnic meta-analysis methods are used to combine the results of several GWAS from diverse ancestries. The goal of this study is to identify genetic associations for eight quantitative metabolic syndrome (MetS) traits through a meta-analysis across four ethnic groups. Traits were measured in the GENetics of Noninsulin dependent Diabetes Mellitus (GENNID) Study which consists of African-American (families = 73, individuals = 288), European-American (families = 79, individuals = 519), Japanese-American (families = 17, individuals = 132), and Mexican-American (families = 113, individuals = 610) samples. Genome-wide association results from these four ethnic groups were combined using four meta-analysis methods: fixed effects, random effects, TransMeta, and MR-MEGA. We provide an empirical comparison of the four meta-analysis methods from the GENNID results, discuss which types of loci (characterized by allelic heterogeneity) appear to be better detected by each of the four meta-analysis methods in the GENNID Study, and validate our results using previous genetic discoveries. We specifically compare the two transethnic methods, TransMeta and MR-MEGA, and discuss how each transethnic method's framework relates to the types of loci best detected by each method.     

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.     

A simple hybrid variance estimator for the Kaplan-Meier survival function

In this paper, we propose a hybrid variance estimator for the Kaplan-Meier survival function. This new estimator approximates the true variance by a Binomial variance formula, where the proportion parameter is a piecewise non-increasing function of the Kaplan-Meier survival function and its upper bound, as described below. Also, the effective sample size equals the number of subjects not censored prior to that time. In addition, we consider an adjusted hybrid variance estimator that modifies the regular estimator for small sample sizes. We present a simulation study to compare the performance of the regular and adjusted hybrid variance estimators to the Greenwood and Peto variance estimators for small sample sizes. We show that on average these hybrid variance estimators give closer variance estimates to the true values than the traditional variance estimators, and hence confidence intervals constructed with these hybrid variance estimators have more nominal coverage rates. Indeed, the Greenwood and Peto variance estimators can substantially underestimate the true variance in the left and right tails of the survival distribution, even with moderately censored data. Finally, we illustrate the use of these hybrid and traditional variance estimators on a data set from a leukaemia clinical trial.     

Comparison of case-cohort estimators based on data on premature death of adult adoptees

A case-cohort sample of adoptees was collected to investigate genetic and environmental influences on premature death, which motivated us to supplement existing simulation results to explore the performance of various estimators proposed for case-cohort samples of survival data. We studied six regression coefficients estimators, which differ with regard to the weighting scheme used in a pseudo-likelihood function, and two different estimators of their variances. Compared to earlier simulation studies, we changed the following conditions: type of explanatory variable, the distribution of lifetimes, and the percentage of deaths in the full cohort. The latter condition affected the performance of the estimated variances of the regression coefficients, where we found a systematic bias of the estimator, proposed by Self and Prentice, dependent on the percentages of deaths. This dependence of percentages of death was different for different sizes of case-cohort studies. A robust variance estimator showed a better overall performance. The estimators of regression coefficients compared did not differ much, the estimators proposed by Kalbfleisch and Lawless and by Prentice performing very well. Results of the case-cohort data of adoptees were not in conflict with earlier findings of a moderate genetic influence on premature death in adulthood.     

Quantifying and correcting for the winner's curse in genetic association studies

Genetic association studies are a powerful tool to detect genetic variants that predispose to human disease. Once an associated variant is identified, investigators are also interested in estimating the effect of the identified variant on disease risk. Estimates of the genetic effect based on new association findings tend to be upwardly biased due to a phenomenon known as the “winner's curse.” Overestimation of genetic effect size in initial studies may cause follow‐up studies to be underpowered and so to fail. In this paper, we quantify the impact of the winner's curse on the allele frequency difference and odds ratio estimators for one‐ and two‐stage case‐control association studies. We then propose an ascertainment‐corrected maximum likelihood method to reduce the bias of these estimators. We show that overestimation of the genetic effect by the uncorrected estimator decreases as the power of the association study increases and that the ascertainment‐corrected method reduces absolute bias and mean square error unless power to detect association is high. Genet. Epidemiol. 33:453–462, 2009. © 2009 Wiley‐Liss, Inc.     

Smooth semiparametric receiver operating characteristic curves for continuous diagnostic tests

We propose a semiparametric kernel distribution function estimator, based on which a new smooth semiparametric estimator of the receiver operating characteristic (ROC) curve is constructed. We derive the asymptotic bias and variance of the newly proposed distribution function estimator and show that it is more efficient than the traditional non-parametric kernel distribution estimator. We also derive the asymptotic bias and variance of our new ROC curve estimator and show that it is more efficient than the smooth non-parametric ROC curve estimator proposed by Zou et al. (Stat. Med. 1997; 16:2143-2156) and Lloyd (J. Am. Stat. Assoc. 1998; 93:1356-1364). For our proposed estimators, we derive data-based methods for bandwidth selection. In addition, we present some results on the analysis of two real data sets. Finally, a simulation study is presented to show that our estimators are better than the non-parametric counterparts in terms of bias, standard error, and mean-square error.     

Comparison of conditional bias‐adjusted estimators for interim analysis in clinical trials with survival data

Group sequential designs are widely used in clinical trials to determine whether a trial should be terminated early. In such trials, maximum likelihood estimates are often used to describe the difference in efficacy between the experimental and reference treatments; however, these are well known for displaying conditional and unconditional biases. Established bias‐adjusted estimators include the conditional mean‐adjusted estimator (CMAE), conditional median unbiased estimator, conditional uniformly minimum variance unbiased estimator (CUMVUE), and weighted estimator. However, their performances have been inadequately investigated. In this study, we review the characteristics of these bias‐adjusted estimators and compare their conditional bias, overall bias, and conditional mean‐squared errors in clinical trials with survival endpoints through simulation studies. The coverage probabilities of the confidence intervals for the four estimators are also evaluated. We find that the CMAE reduced conditional bias and showed relatively small conditional mean‐squared errors when the trials terminated at the interim analysis. The conditional coverage probability of the conditional median unbiased estimator was well below the nominal value. In trials that did not terminate early, the CUMVUE performed with less bias and an acceptable conditional coverage probability than was observed for the other estimators. In conclusion, when planning an interim analysis, we recommend using the CUMVUE for trials that do not terminate early and the CMAE for those that terminate early. Copyright © 2017 John Wiley & Sons, Ltd.     

Analysis of untyped SNPs: maximum likelihood and imputation methods

Analysis of untyped single nucleotide polymorphisms (SNPs) can facilitate the localization of disease-causing variants and permit meta-analysis of association studies with different genotyping platforms. We present two approaches for using the linkage disequilibrium structure of an external reference panel to infer the unknown value of an untyped SNP from the observed genotypes of typed SNPs. The maximum-likelihood approach integrates the prediction of untyped genotypes and estimation of association parameters into a single framework and yields consistent and efficient estimators of genetic effects and gene-environment interactions with proper variance estimators. The imputation approach is a two-stage strategy, which first imputes the untyped genotypes by either the most likely genotypes or the expected genotype counts and then uses the imputed values in a downstream association analysis. The latter approach has proper control of type I error in single-SNP tests with possible covariate adjustments even when the reference panel is misspecified; however, type I error may not be properly controlled in testing multiple-SNP effects or gene-environment interactions. In general, imputation yields biased estimators of genetic effects and gene-environment interactions, and the variances are underestimated. We conduct extensive simulation studies to compare the bias, type I error, power, and confidence interval coverage between the maximum likelihood and imputation approaches in the analysis of single-SNP effects, multiple-SNP effects, and gene-environment interactions under cross-sectional and case-control designs. In addition, we provide an illustration with genome-wide data from the Wellcome Trust Case-Control Consortium (WTCCC) [2007].     

Correcting “winner's curse” in odds ratios from genomewide association findings for major complex human diseases

Genome‐wide association studies (GWAS) provide an important approach for identifying common genetic variants that predispose to human disease. However, odds ratio (OR) estimates for the reported findings from GWAS discovery data are typically affected by a bias away from the null sometimes referred to the “winner's curse”. Also standard confidence intervals (CIs) may have far from the desired coverage rates. We applied a bias reduction method to GWAS findings from several major complex human diseases, including breast cancer, colorectal cancer, lung cancer, prostate cancer, type I diabetes, and type II diabetes. We found the simple bias correction procedure allows one to estimate bias‐adjusted ORs that have substantial consistency with ORs from subsequent replication studies, and that corresponding selection‐adjusted CIs appear to help quantify the uncertainty of the findings. Selection‐adjusted ORs and CIs can provide a reliable summary of GWAS data, and can help to choose single nucleotide polymorphisms for subsequent validation studies. Genet. Epidemiol. 34:78–91, 2010. © 2009 Wiley‐Liss, Inc.