Comparing two correlated C indices with right‐censored survival outcome: a one‐shot nonparametric approach

The area under the receiver operating characteristic curve is often used as a summary index of the diagnostic ability in evaluating biomarkers when the clinical outcome (truth) is binary. When the clinical outcome is right‐censored survival time, the C index, motivated as an extension of area under the receiver operating characteristic curve, has been proposed by Harrell as a measure of concordance between a predictive biomarker and the right‐censored survival outcome. In this work, we investigate methods for statistical comparison of two diagnostic or predictive systems, of which they could either be two biomarkers or two fixed algorithms, in terms of their C indices. We adopt a U‐statistics‐based C estimator that is asymptotically normal and develop a nonparametric analytical approach to estimate the variance of the C estimator and the covariance of two C estimators. A z‐score test is then constructed to compare the two C indices. We validate our one‐shot nonparametric method via simulation studies in terms of the type I error rate and power. We also compare our one‐shot method with resampling methods including the jackknife and the bootstrap. Simulation results show that the proposed one‐shot method provides almost unbiased variance estimations and has satisfactory type I error control and power. Finally, we illustrate the use of the proposed method with an example from the Framingham Heart Study. Copyright © 2014 John Wiley & Sons, Ltd.     

Avoiding the high Bonferroni penalty in genome‐wide association studies

A major challenge in genome‐wide association studies (GWASs) is to derive the multiple testing threshold when hypothesis tests are conducted using a large number of single nucleotide polymorphisms. Permutation tests are considered the gold standard in multiple testing adjustment in genetic association studies. However, it is computationally intensive, especially for GWASs, and can be impractical if a large number of random shuffles are used to ensure accuracy. Many researchers have developed approximation algorithms to relieve the computing burden imposed by permutation. One particularly attractive alternative to permutation is to calculate the effective number of independent tests, M_eff, which has been shown to be promising in genetic association studies. In this study, we compare recently developed M_eff methods and validate them by the permutation test with 10,000 random shuffles using two real GWAS data sets: an Illumina 1M BeadChip and an Affymetrix GeneChip^® Human Mapping 500K Array Set. Our results show that the simpleM method produces the best approximation of the permutation threshold, and it does so in the shortest amount of time. We also show that M_eff is indeed valid on a genome‐wide scale in these data sets based on statistical theory and significance tests. The significance thresholds derived can provide practical guidelines for other studies using similar population samples and genotyping platforms. Genet. Epidemiol. 34:100–105, 2010. © 2009 Wiley‐Liss, Inc.     

Kernel machine methods for integrative analysis of genome‐wide methylation and genotyping studies

Many large GWAS consortia are expanding to simultaneously examine the joint role of DNA methylation in addition to genotype in the same subjects. However, integrating information from both data types is challenging. In this paper, we propose a composite kernel machine regression model to test the joint epigenetic and genetic effect. Our approach works at the gene level, which allows for a common unit of analysis across different data types. The model compares the pairwise similarities in the phenotype to the pairwise similarities in the genotype and methylation values; and high correspondence is suggestive of association. A composite kernel is constructed to measure the similarities in the genotype and methylation values between pairs of samples. We demonstrate through simulations and real data applications that the proposed approach can correctly control type I error, and is more robust and powerful than using only the genotype or methylation data in detecting trait‐associated genes. We applied our method to investigate the genetic and epigenetic regulation of gene expression in response to stressful life events using data that are collected from the Grady Trauma Project. Within the kernel machine testing framework, our methods allow for heterogeneity in effect sizes, nonlinear, and interactive effects, as well as rapid P‐value computation.     

Testing for genetic association in the presence of population stratification in genome‐wide association studies

Genome‐wide case‐control association study is gaining popularity, thanks to the rapid development of modern genotyping technology. In such studies, population stratification is a potential concern especially when the number of study subjects is large as it can lead to seriously inflated false‐positive rates. Current methods addressing this issue are still not completely immune to excess false positives. A simple method that corrects for population stratification is proposed. This method modifies a test statistic such as the Armitage trend test by using an additive constant that measures the variation of the effect size confounded by population stratification across genomic control (GC) markers. As a result, the original statistic is deflated by a multiplying factor that is specific to the marker being tested for association. This deflating multiplying factor is guaranteed to be larger than 1. These properties are in contrast to the conventional GC method where the original statistic is deflated by a common factor regardless of the marker being tested and the deflation factor may turn out to be less than 1. The new method is introduced first for regular case‐control design and then for other situations such as quantitative traits and the presence of covariates. Extensive simulation study indicates that this new method provides an appealing alternative for genetic association analysis in the presence of population stratification. Genet. Epidemiol. 33:637–645, 2009. © 2009 Wiley‐Liss, Inc.     

Quality control and quality assurance in genotypic data for genome‐wide association studies

Genome‐wide scans of nucleotide variation in human subjects are providing an increasing number of replicated associations with complex disease traits. Most of the variants detected have small effects and, collectively, they account for a small fraction of the total genetic variance. Very large sample sizes are required to identify and validate findings. In this situation, even small sources of systematic or random error can cause spurious results or obscure real effects. The need for careful attention to data quality has been appreciated for some time in this field, and a number of strategies for quality control and quality assurance (QC/QA) have been developed. Here we extend these methods and describe a system of QC/QA for genotypic data in genome‐wide association studies (GWAS). This system includes some new approaches that (1) combine analysis of allelic probe intensities and called genotypes to distinguish gender misidentification from sex chromosome aberrations, (2) detect autosomal chromosome aberrations that may affect genotype calling accuracy, (3) infer DNA sample quality from relatedness and allelic intensities, (4) use duplicate concordance to infer SNP quality, (5) detect genotyping artifacts from dependence of Hardy‐Weinberg equilibrium test P‐values on allelic frequency, and (6) demonstrate sensitivity of principal components analysis to SNP selection. The methods are illustrated with examples from the “Gene Environment Association Studies” (GENEVA) program. The results suggest several recommendations for QC/QA in the design and execution of GWAS. Genet. Epidemiol. 34: 591–602, 2010. © 2010 Wiley‐Liss, Inc.     

Using genome‐wide pathway analysis to unravel the etiology of complex diseases

Several genome‐wide association studies (GWAS) have been published on various complex diseases. Although, new loci are found to be associated with these diseases, still only very little of the genetic risk for these diseases can be explained. As GWAS are still underpowered to find small main effects, and gene‐gene interactions are likely to play a role, the data might currently not be analyzed to its full potential. In this study, we evaluated alternative methods to study GWAS data. Instead of focusing on the single nucleotide polymorphisms (SNPs) with the highest statistical significance, we took advantage of prior biological information and tried to detect overrepresented pathways in the GWAS data. We evaluated whether pathway classification analysis can help prioritize the biological pathways most likely to be involved in the disease etiology. In this study, we present the various benefits and limitations of pathway‐classification tools in analyzing GWAS data. We show multiple differences in outcome between pathway tools analyzing the same dataset. Furthermore, analyzing randomly selected SNPs always results in significantly overrepresented pathways, large pathways have a higher chance of becoming statistically significant and the bioinformatics tools used in this study are biased toward detecting well‐defined pathways. As an example, we analyzed data from two GWAS on type 2 diabetes (T2D): the Diabetes Genetics Initiative (DGI) and the Wellcome Trust Case Control Consortium (WTCCC). Occasionally the results from the DGI and the WTCCC GWAS showed concordance in overrepresented pathways, but discordance in the corresponding genes. Thus, incorporating gene networks and pathway classification tools into the analysis can point toward significantly overrepresented molecular pathways, which cannot be picked up using traditional single‐locus analyses. However, the limitations discussed in this study, need to be addressed before these methods can be widely used. Genet. Epidemiol. 33:419–431, 2009. © 2009 Wiley‐Liss, Inc.     

Evaluation of multi‐outcome longitudinal studies

Evaluation of intervention effects on multiple outcomes is a common scenario in clinical studies. In longitudinal studies, such evaluation is a challenge if one wishes to adequately capture simultaneous data behavior. In this situation, a common approach is to analyze each outcome separately. As a result, multiple statistical statements describing the intervention effect need to be reported and an adjustment for multiple testing is necessary. This is typically done by means of the Bonferroni procedure, which does not take into account the correlation between outcomes, thus resulting in overly conservative conclusions. We propose an alternative approach for multiplicity adjustment that incorporates dependence between outcomes, resulting in an appreciably less conservative evaluation. The ability of the proposed method to control the familywise error rate is evaluated in a simulation study, and the applicability of the method is demonstrated in two examples from the literature. Copyright © 2015 John Wiley & Sons, Ltd.     

Estimation of a significance threshold for epigenome‐wide association studies

Epigenome‐wide association studies (EWAS) are designed to characterise population‐level epigenetic differences across the genome and link them to disease. Most commonly, they assess DNA‐methylation status at cytosine‐guanine dinucleotide (CpG) sites, using platforms such as the Illumina 450k array that profile a subset of CpGs genome wide. An important challenge in the context of EWAS is determining a significance threshold for declaring a CpG site as differentially methylated, taking multiple testing into account. We used a permutation method to estimate a significance threshold specifically for the 450k array and a simulation extrapolation approach to estimate a genome‐wide threshold. These methods were applied to five different EWAS datasets derived from a variety of populations and tissue types. We obtained an estimate of for the 450k array, and a genome‐wide estimate of . We further demonstrate the importance of these results by showing that previously recommended sample sizes for EWAS should be adjusted upwards, requiring samples between ～10% and ～20% larger in order to maintain type‐1 errors at the desired level.     

Integrative modeling of multi‐platform genomic data under the framework of mediation analysis

Given the availability of genomic data, there have been emerging interests in integrating multi‐platform data. Here, we propose to model genetics (single nucleotide polymorphism (SNP)), epigenetics (DNA methylation), and gene expression data as a biological process to delineate phenotypic traits under the framework of causal mediation modeling. We propose a regression model for the joint effect of SNPs, methylation, gene expression, and their nonlinear interactions on the outcome and develop a variance component score test for any arbitrary set of regression coefficients. The test statistic under the null follows a mixture of chi‐square distributions, which can be approximated using a characteristic function inversion method or a perturbation procedure. We construct tests for candidate models determined by different combinations of SNPs, DNA methylation, gene expression, and interactions and further propose an omnibus test to accommodate different models. We then study three path‐specific effects: the direct effect of SNPs on the outcome, the effect mediated through expression, and the effect through methylation. We characterize correspondences between the three path‐specific effects and coefficients in the regression model, which are influenced by causal relations among SNPs, DNA methylation, and gene expression. We illustrate the utility of our method in two genomic studies and numerical simulation studies. Copyright © 2014 John Wiley & Sons, Ltd.     

How to link call rate and p‐values for Hardy–Weinberg equilibrium as measures of genome‐wide SNP data quality

We study the link between two quality measures of SNP (single nucleotide polymorphism) data in genome‐wide association (GWA) studies, that is, per SNP call rates (CR) and p‐values for testing Hardy–Weinberg equilibrium (HWE). The aim is to improve these measures by applying methods based on realized randomized p‐values, the false discovery rate and estimates for the proportion of false hypotheses. While exact non‐randomized conditional p‐values for testing HWE cannot be recommended for estimating the proportion of false hypotheses, their realized randomized counterparts should be used. P‐values corresponding to the asymptotic unconditional chi‐square test lead to reasonable estimates only if SNPs with low minor allele frequency are excluded. We provide an algorithm to compute the probability that SNPs violate HWE given the observed CR, which yields an improved measure of data quality. The proposed methods are applied to SNP data from the KORA (Cooperative Health Research in the Region of Augsburg, Southern Germany) 500 K project, a GWA study in a population‐based sample genotyped by Affymetrix GeneChip 500 K arrays using the calling algorithm BRLMM 1.4.0. We show that all SNPs with CR = 100 per cent are nearly in perfect HWE which militates in favor of the population to meet the conditions required for HWE at least for these SNPs. Moreover, we show that the proportion of SNPs not being in HWE increases with decreasing CR. We conclude that using a single threshold for judging HWE p‐values without taking the CR into account is problematic. Instead we recommend a stratified analysis with respect to CR. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

A novel haplotype‐sharing approach for genome‐wide case‐control association studies implicates the calpastatin gene in Parkinson's disease

The large number of markers considered in a genome‐wide association study (GWAS) has resulted in a simplification of analyses conducted. Most studies are analyzed one marker at a time using simple tests like the trend test. Methods that account for the special features of genetic association studies, yet remain computationally feasible for genome‐wide analysis, are desirable as they may lead to increased power to detect associations. Haplotype sharing attempts to translate between population genetics and genetic epidemiology. Near a recent mutation that increases disease risk, haplotypes of case participants should be more similar to each other than haplotypes of control participants; conversely, the opposite pattern may be found near a recent mutation that lowers disease risk. We give computationally simple association tests based on haplotype sharing that can be easily applied to GWASs while allowing use of fast (but not likelihood‐based) haplotyping algorithms and properly accounting for the uncertainty introduced by using inferred haplotypes. We also give haplotype‐sharing analyses that adjust for population stratification. Applying our methods to a GWAS of Parkinson's disease, we find a genome‐wide significant signal in the CAST gene that is not found by single‐SNP methods. Further, a missing‐data artifact that causes a spurious single‐SNP association on chromosome 9 does not impact our test. Genet. Epidemiol. 33:657–667, 2009. Published 2009 Wiley‐Liss, Inc.     

Relatedness mapping and tracts of relatedness for genome‐wide data in the presence of linkage disequilibrium

Estimates of relatedness have several applications such as the identification of relatives or in identifying disease related genes through identity by descent (IBD) mapping. Here we present a new method for identifying IBD tracts among individuals from genome‐wide single nucleotide polymorphisms data. We use a continuous time Markov model where the hidden states are the number of alleles shared IBD between pairs of individuals at a given position. In contrast to previous methods, our method accurately accounts for linkage disequilibrium using pairwise haplotype probabilities. The method provides a map of the local relatedness along the genome. We illustrate the potential of the method for mapping disease genes on a real data set, and show that the method has the potential to map causative disease mutations using only a handful of affected individuals. The new IBD mapping method provides considerable improvement in mapping power in natural populations compared to standard association mapping methods. Genet. Epidemiol. 2009. © 2008 Wiley‐Liss, Inc.     

Small sample performance of bias‐corrected sandwich estimators for cluster‐randomized trials with binary outcomes

The sandwich estimator in generalized estimating equations (GEE) approach underestimates the true variance in small samples and consequently results in inflated type I error rates in hypothesis testing. This fact limits the application of the GEE in cluster‐randomized trials (CRTs) with few clusters. Under various CRT scenarios with correlated binary outcomes, we evaluate the small sample properties of the GEE Wald tests using bias‐corrected sandwich estimators. Our results suggest that the GEE Wald z‐test should be avoided in the analyses of CRTs with few clusters even when bias‐corrected sandwich estimators are used. With t‐distribution approximation, the Kauermann and Carroll (KC)‐correction can keep the test size to nominal levels even when the number of clusters is as low as 10 and is robust to the moderate variation of the cluster sizes. However, in cases with large variations in cluster sizes, the Fay and Graubard (FG)‐correction should be used instead. Furthermore, we derive a formula to calculate the power and minimum total number of clusters one needs using the t‐test and KC‐correction for the CRTs with binary outcomes. The power levels as predicted by the proposed formula agree well with the empirical powers from the simulations. The proposed methods are illustrated using real CRT data. We conclude that with appropriate control of type I error rates under small sample sizes, we recommend the use of GEE approach in CRTs with binary outcomes because of fewer assumptions and robustness to the misspecification of the covariance structure. Copyright © 2014 John Wiley & Sons, Ltd.     

Strength of evidence of non‐inferiority trials—The adjustment of the type I error rate in non‐inferiority trials with the synthesis method

In non‐inferiority trials that employ the synthesis method several types of dependencies among test statistics occur due to sharing of the same information from the historical trial. The conditions under which the dependencies appear may be divided into three categories. The first case is when a new drug is approved with single non‐inferiority trial. The second case is when a new drug is approved if two independent non‐inferiority trials show positive results. The third case is when two new different drugs are approved with the same active control. The problem of the dependencies is that they can make the type I error rate deviate from the nominal level. In order to study such deviations, we introduce the unconditional and conditional across‐trial type I error rates when the non‐inferiority margin is estimated from the historical trial, and investigate how the dependencies affect the type I error rates. We show that the unconditional across‐trial type I error rate increases dramatically as does the correlation between two non‐inferiority tests when a new drug is approved based on the positive results of two non‐inferiority trials. We conclude that the conditional across‐trial type I error rate involves the unknown treatment effect in the historical trial. The formulae of the conditional across‐trial type I error rates provide us with a way of investigating the conditional across‐trial type I error rates for various assumed values of the treatment effect in the historical trial. Copyright © 2010 John Wiley & Sons, Ltd.     

Comparing paired vs non-paired statistical methods of analyses when making inferences about absolute risk reductions in propensity-score matched samples

Propensity-score matching allows one to reduce the effects of treatment-selection bias or confounding when estimating the effects of treatments when using observational data. Some authors have suggested that methods of inference appropriate for independent samples can be used for assessing the statistical significance of treatment effects when using propensity-score matching. Indeed, many authors in the applied medical literature use methods for independent samples when making inferences about treatment effects using propensity-score matched samples. Dichotomous outcomes are common in healthcare research. In this study, we used Monte Carlo simulations to examine the effect on inferences about risk differences (or absolute risk reductions) when statistical methods for independent samples are used compared with when statistical methods for paired samples are used in propensity-score matched samples. We found that compared with using methods for independent samples, the use of methods for paired samples resulted in: (i) empirical type I error rates that were closer to the advertised rate; (ii) empirical coverage rates of 95 per cent confidence intervals that were closer to the advertised rate; (iii) narrower 95 per cent confidence intervals; and (iv) estimated standard errors that more closely reflected the sampling variability of the estimated risk difference. Differences between the empirical and advertised performance of methods for independent samples were greater when the treatment-selection process was stronger compared with when treatment-selection process was weaker. We recommend using statistical methods for paired samples when using propensity-score matched samples for making inferences on the effect of treatment on the reduction in the probability of an event occurring.     

Valid Monte Carlo Permutation Tests for Genetic Case‐Control Studies With Missing Genotypes

Monte Carlo permutation tests can be formally constructed by choosing a set of permutations of individual indices and a real‐valued test statistic measuring the association between genotypes and affection status. In this paper, we develop a rigorous theoretical framework for verifying the validity of these tests when there are missing genotypes. We begin by specifying a nonparametric probability model for the observed genotype data in a genetic case‐control study with unrelated subjects. Under this model and some minimal assumptions about the test statistic, we establish that the resulting Monte Carlo permutation test is exact level α if (1) the chosen set of permutations of individual indices is a group under composition and (2) the distribution of the observed genotype score matrix under the null hypothesis does not change if the assignment of individuals to rows is shuffled according to an arbitrary permutation in this set. We apply these conditions to show that frequently used Monte Carlo permutation tests based on the set of all permutations of individual indices are guaranteed to be exact level α only for missing data processes satisfying a rather restrictive additional assumption. However, if the missing data process depends on covariates that are all identified and recorded, we also show that Monte Carlo permutation tests based on the set of permutations within strata of individuals with identical covariate values are exact level α. Our theoretical results are verified and supplemented by simulations for a variety of missing data processes and test statistics.     

Inferential testing for linkage with GENEHUNTER-MODSCORE: the impact of the pedigree structure on the null distribution of multipoint MOD scores

The asymptotic distribution of [MOD] scores under the null hypothesis of no linkage is only known for affected sib pairs and other types of affected relative pairs. We have extended the GENEHUNTER-MODSCORE program to allow for simulations under the null hypothesis of no linkage to determine the empirical significance of MOD-score results in general situations. We performed simulations with families of different size (one million replicates of 500 families per simulation setting) to thoroughly investigate the impact of the pedigree size on the null distribution of multipoint MOD scores. It is shown that the distribution is dependent on the size and structure of the pedigrees under study. By performing simulations in the context of MOD-score analysis, our new tool efficiently explores the linkage data in a comprehensive way and also provides a valid method to inferentially test for linkage.     

Assessing the impact of selection bias on test decisions in trials with a time‐to‐event outcome

If past treatment assignments are unmasked, selection bias may arise even in randomized controlled trials. The impact of such bias can be measured by considering the type I error probability. In case of a normally distributed outcome, there already exists a model accounting for selection bias that permits calculating the corresponding type I error probabilities. To model selection bias for trials with a time‐to‐event outcome, we introduce a new biasing policy for exponentially distributed data. Using this biasing policy, we derive an exact formula to compute type I error probabilities whenever an F‐test is performed and no observations are censored. Two exemplary settings, with and without random censoring, are considered in order to illustrate how our results can be applied to compare distinct randomization procedures with respect to their performance in the presence of selection bias. © 2017 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.