Testing for Homogeneity in Meta‐Analysis I. The One‐Parameter Case: Standardized Mean Difference

Summary Meta‐analysis seeks to combine the results of several experiments in order to improve the accuracy of decisions. It is common to use a test for homogeneity to determine if the results of the several experiments are sufficiently similar to warrant their combination into an overall result. Cochran’s Q statistic is frequently used for this homogeneity test. It is often assumed that Q follows a chi‐square distribution under the null hypothesis of homogeneity, but it has long been known that this asymptotic distribution for Q is not accurate for moderate sample sizes. Here, we present an expansion for the mean of Q under the null hypothesis that is valid when the effect and the weight for each study depend on a single parameter, but for which neither normality nor independence of the effect and weight estimators is needed. This expansion represents an order O(1/n) correction to the usual chi‐square moment in the one‐parameter case. We apply the result to the homogeneity test for meta‐analyses in which the effects are measured by the standardized mean difference (Cohen’s d‐statistic). In this situation, we recommend approximating the null distribution of Q by a chi‐square distribution with fractional degrees of freedom that are estimated from the data using our expansion for the mean of Q. The resulting homogeneity test is substantially more accurate than the currently used test. We provide a program available at the Paper Information link at the Biometrics website http://www.biometrics.tibs.org for making the necessary calculations.     

Assessing Equivalence Tests with Respect to their Expected p‐Value

Monte‐Carlo simulation methods are commonly used for assessing the performance of statistical tests under finite sample scenarios. They help us ascertain the nominal level for tests with approximate level, e.g. asymptotic tests. Additionally, a simulation can assess the quality of a test on the alternative. The latter can be used to compare new tests and established tests under certain assumptions in order to determinate a preferable test given characteristics of the data. The key problem for such investigations is the choice of a goodness criterion. We expand the expected p‐value as considered by Sackrowitz and Samuel‐Cahn (1999) to the context of univariate equivalence tests. This presents an effective tool to evaluate new purposes for equivalence testing because of its independence of the distribution of the test statistic under null‐hypothesis. It helps to avoid the often tedious search for the distribution under null‐hypothesis for test statistics which have no considerable advantage over yet available methods. To demonstrate the usefulness in biometry a comparison of established equivalence tests with a nonparametric approach is conducted in a simulation study for three distributional assumptions.     

Population genomics reveals multiple drivers of population differentiation in a sex‐role‐reversed pipefish

A major goal of molecular ecology is to identify the causes of genetic and phenotypic differentiation among populations. Population genomics is suitably poised to tackle these key questions by diagnosing the evolutionary mechanisms driving divergence in nature. Here, we set out to investigate the evolutionary processes underlying population differentiation in the Gulf pipefish, Syngnathus scovelli. We sampled approximately 50 fish from each of 12 populations distributed from the Gulf coast of Texas to the Atlantic coast of Florida and performed restriction‐site‐associated DNA sequencing to identify SNPs throughout the genome. After imposing quality and stringency filters, we selected a panel of 6348 SNPs present in all 12 populations, 1753 of which were not physically linked. We identified a genome‐wide pattern of isolation by distance, in addition to a more substantial genetic break separating populations in the Gulf of Mexico from those in the Atlantic. We also used several divergence outlier approaches and tests for genotype–environment correlations to identify 400 SNPs putatively involved in local adaptation. Patterns of phenotypic differentiation and variation diverged from the overall genomic pattern, suggesting that selection, phenotypic plasticity or demographic factors may be shaping phenotypes in distinct populations. Overall, our results suggest that population divergence is driven by a variety of factors in S. scovelli, including neutral processes and selection on multiple traits.     

Nonparametric association analysis of bivariate left‐truncated competing risks data

We develop time‐varying association analyses for onset ages of two lung infections to address the statistical challenges in utilizing registry data where onset ages are left‐truncated by ages of entry and competing‐risk censored by deaths. Two types of association estimators are proposed based on conditional cause‐specific hazard function and cumulative incidence function that are adapted from unconditional quantities to handle left truncation. Asymptotic properties of the estimators are established by using the empirical process techniques. Our simulation study shows that the estimators perform well with moderate sample sizes. We apply our methods to the Cystic Fibrosis Foundation Registry data to study the relationship between onset ages of Pseudomonas aeruginosa and Staphylococcus aureus infections.     

Using simulations to evaluate Mantel‐based methods for assessing landscape resistance to gene flow

Mantel‐based tests have been the primary analytical methods for understanding how landscape features influence observed spatial genetic structure. Simulation studies examining Mantel‐based approaches have highlighted major challenges associated with the use of such tests and fueled debate on when the Mantel test is appropriate for landscape genetics studies. We aim to provide some clarity in this debate using spatially explicit, individual‐based, genetic simulations to examine the effects of the following on the performance of Mantel‐based methods: (1) landscape configuration, (2) spatial genetic nonequilibrium, (3) nonlinear relationships between genetic and cost distances, and (4) correlation among cost distances derived from competing resistance models. Under most conditions, Mantel‐based methods performed poorly. Causal modeling identified the true model only 22% of the time. Using relative support and simple Mantel r values boosted performance to approximately 50%. Across all methods, performance increased when landscapes were more fragmented, spatial genetic equilibrium was reached, and the relationship between cost distance and genetic distance was linearized. Performance depended on cost distance correlations among resistance models rather than cell‐wise resistance correlations. Given these results, we suggest that the use of Mantel tests with linearized relationships is appropriate for discriminating among resistance models that have cost distance correlations <0.85 with each other for causal modeling, or <0.95 for relative support or simple Mantel r. Because most alternative parameterizations of resistance for the same landscape variable will result in highly correlated cost distances, the use of Mantel test‐based methods to fine‐tune resistance values will often not be effective.     

A Bayesian Chi‐Squared Goodness‐of‐Fit Test for Censored Data Models

Summary We propose a Bayesian chi‐squared model diagnostic for analysis of data subject to censoring. The test statistic has the form of Pearson's chi‐squared test statistic and is easy to calculate from standard output of Markov chain Monte Carlo algorithms. The key innovation of this diagnostic is that it is based only on observed failure times. Because it does not rely on the imputation of failure times for observations that have been censored, we show that under heavy censoring it can have higher power for detecting model departures than a comparable test based on the complete data. In a simulation study, we show that tests based on this diagnostic exhibit comparable power and better nominal Type I error rates than a commonly used alternative test proposed by Akritas (1988, Journal of the American Statistical Association 83, 222–230). An important advantage of the proposed diagnostic is that it can be applied to a broad class of censored data models, including generalized linear models and other models with nonidentically distributed and nonadditive error structures. We illustrate the proposed model diagnostic for testing the adequacy of two parametric survival models for Space Shuttle main engine failures.     

Interim analysis incorporating short‐ and long‐term binary endpoints

Designs incorporating more than one endpoint have become popular in drug development. One of such designs allows for incorporation of short‐term information in an interim analysis if the long‐term primary endpoint has not been yet observed for some of the patients. At first we consider a two‐stage design with binary endpoints allowing for futility stopping only based on conditional power under both fixed and observed effects. Design characteristics of three estimators: using primary long‐term endpoint only, short‐term endpoint only, and combining data from both are compared. For each approach, equivalent cut‐off point values for fixed and observed effect conditional power calculations can be derived resulting in the same overall power. While in trials stopping for futility the type I error rate cannot get inflated (it usually decreases), there is loss of power. In this study, we consider different scenarios, including different thresholds for conditional power, different amount of information available at the interim, different correlations and probabilities of success. We further extend the methods to adaptive designs with unblinded sample size reassessments based on conditional power with inverse normal method as the combination function. Two different futility stopping rules are considered: one based on the conditional power, and one from P‐values based on Z‐statistics of the estimators. Average sample size, probability to stop for futility and overall power of the trial are compared and the influence of the choice of weights is investigated.     

Nonparametric All‐Pairs Multiple Comparisons

Nonparametric all‐pairs multiple comparisons based on pairwise rankings can be performed in the one‐way design with the Steel‐Dwass procedure. To apply this test, Wilcoxon's rank sum statistic is calculated for all pairs of groups; the maximum of the rank sums is the test statistic. We provide exact calculations of the asymptotic critical values (and P‐values, respectively) even for unbalanced designs. We recommend this asymptotic method whenever large sample sizes are present. For small sample sizes we recommend the use of the new statistic according to Baumgartner , Weiss , and Schindler (1998, Biometrics 54 , 1129–1135) instead of Wilcoxon's rank sum for the multiple comparisons. We show that the resultant procedure can be less conservative and, according to simulation results, more powerful than the original Steel‐Dwass procedure. We illustrate the methods with a practical data set.     

UMP(U)‐Tests for a Binomial Parameter: A Paradox

We consider uniformly most powerful (UMP) as well as uniformly most powerful unbiased (UMPU) tests and their non‐randomized versions for certain hypotheses concerning a binomial parameter. It will be shown that the power function of a UMP(U)‐test based on sample size n can coincide on the entire parameter space with the power function of the corresponding test based on sample size n + 1. A complete characterization of this paradox will be derived. Apart some exceptional cases for two‐sided tests and equivalence tests the paradox appears if and only if a test based on sample size n is non‐randomized.     

Genetic sex assignment in wild populations using genotyping‐by‐sequencing data: A statistical threshold approach

Establishing the sex of individuals in wild systems can be challenging and often requires genetic testing. Genotyping‐by‐sequencing (GBS) and other reduced‐representation DNA sequencing (RRS) protocols (e.g., RADseq, ddRAD) have enabled the analysis of genetic data on an unprecedented scale. Here, we present a novel approach for the discovery and statistical validation of sex‐specific loci in GBS data sets. We used GBS to genotype 166 New Zealand fur seals (NZFS, Arctocephalus forsteri) of known sex. We retained monomorphic loci as potential sex‐specific markers in the locus discovery phase. We then used (i) a sex‐specific locus threshold (SSLT) to identify significantly male‐specific loci within our data set; and (ii) a significant sex‐assignment threshold (SSAT) to confidently assign sex in silico the presence or absence of significantly male‐specific loci to individuals in our data set treated as unknowns (98.9% accuracy for females; 95.8% for males, estimated via cross‐validation). Furthermore, we assigned sex to 86 individuals of true unknown sex using our SSAT and assessed the effect of SSLT adjustments on these assignments. From 90 verified sex‐specific loci, we developed a panel of three sex‐specific PCR primers that we used to ascertain sex independently of our GBS data, which we show amplify reliably in at least two other pinniped species. Using monomorphic loci normally discarded from large SNP data sets is an effective way to identify robust sex‐linked markers for nonmodel species. Our novel pipeline can be used to identify and statistically validate monomorphic and polymorphic sex‐specific markers across a range of species and RRS data sets.     

A Robust Genome‐Wide Scan Statistic of the Wellcome Trust Case–Control Consortium

Summary In genome‐wide association (GWA) studies, test statistics that are efficient and robust across various genetic models are preferable, particularly for studying multiple diseases in the Wellcome Trust Case–Control Consortium ( WTCCC, 2007 , Nature 447 , 661–678). A new test statistic, the minimum of the p‐values of the trend test and Pearson's test, was considered by the WTCCC. It is referred to here as MIN2. Because the minimum of two p‐values is no longer a valid p‐value itself, the WTCCC only used it to rank single nucleotide polymorphisms (SNPs) but did not report the p‐values of the associated SNPs when MIN2 was used for ranking. Given its importance in practice, we derive the asymptotic null distribution of MIN2, study some of its analytical properties related to GWA studies, and compare it with existing methods (the trend test, Pearson's test, MAX3, and the constrained likelihood ratio test [CLRT]) by simulations across a wide range of possible genetic models: the recessive (REC), additive (ADD), multiplicative (MUL), dominant (DOM), and overdominant models. The results show that MAX3 and CLRT have greater efficiency robustness than other tests when the REC, ADD/MUL, and DOM models are possible, whereas Pearson's test and MIN2 have greater efficiency robustness if the possible genetic models also include the overdominant model. We conclude that robust tests (MAX3, MIN2, CLRT, and Pearson's test) are preferable to a single trend test for initial GWA studies. The four robust tests are applied to more than 100 SNPs associated with 11 common diseases identified by the two WTCCC GWA studies.     

Order‐restricted Scores Test for the Evaluation of Population‐based Case–control Studies when the Genetic Model is Unknown

The Cochran–Armitage (CA) linear trend test for proportions is often used for genotype‐based analysis of candidate gene association. Depending on the underlying genetic mode of inheritance, the use of model‐specific scores maximises the power. Commonly, the underlying genetic model, i.e. additive, dominant or recessive mode of inheritance, is a priori unknown. Association studies are commonly analysed using permutation tests, where both inference and identification of the underlying mode of inheritance are important. Especially interesting are tests for case–control studies, defined by a maximum over a series of standardised CA tests, because such a procedure has power under all three genetic models. We reformulate the test problem and propose a conditional maximum test of scores‐specific linear‐by‐linear association tests. For maximum‐type, sum and quadratic test statistics the asymptotic expectation and covariance can be derived in a closed form and the limiting distribution is known. Both the limiting distribution and approximations of the exact conditional distribution can easily be computed using standard software packages. In addition to these technical advances, we extend the area of application to stratified designs, studies involving more than two groups and the simultaneous analysis of multiple loci by means of multiplicity‐adjusted p‐values for the underlying multiple CA trend tests. The new test is applied to reanalyse a study investigating genetic components of different subtypes of psoriasis. A new and flexible inference tool for association studies is available both theoretically as well as practically since already available software packages can be easily used to implement the suggested test procedures.     

Incorporating Correlation for Multivariate Failure Time Data When Cluster Size Is Large

Summary We propose a new estimation method for multivariate failure time data using the quadratic inference function (QIF) approach. The proposed method efficiently incorporates within‐cluster correlations. Therefore, it is more efficient than those that ignore within‐cluster correlation. Furthermore, the proposed method is easy to implement. Unlike the weighted estimating equations in Cai and Prentice (1995, Biometrika 82 , 151–164), it is not necessary to explicitly estimate the correlation parameters. This simplification is particularly useful in analyzing data with large cluster size where it is difficult to estimate intracluster correlation. Under certain regularity conditions, we show the consistency and asymptotic normality of the proposed QIF estimators. A chi‐squared test is also developed for hypothesis testing. We conduct extensive Monte Carlo simulation studies to assess the finite sample performance of the proposed methods. We also illustrate the proposed methods by analyzing primary biliary cirrhosis (PBC) data.     

Minimum‐Norm Estimation for Binormal Receiver Operating Characteristic (ROC) Curves

The receiver operating characteristic (ROC) curve is often used to assess the usefulness of a diagnostic test. We present a new method to estimate the parameters of a popular semi‐parametric ROC model, called the binormal model. Our method is based on minimization of the functional distance between two estimators of an unknown transformation postulated by the model, and has a simple, closed‐form solution. We study the asymptotics of our estimators, show via simulation that they compare favorably with existing estimators, and illustrate how covariates may be incorporated into the norm minimization framework.     

Confidence Intervals of the Attributable Risk under Cross‐Sectional Sampling with Confounders

Since it can account for both the strength of the association between exposure to a risk factor and the underlying disease of interest and the prevalence of the risk factor, the attributable risk (AR) is probably the most commonly used epidemiologic measure for public health administrators to locate important risk factors. This paper discusses interval estimation of the AR in the presence of confounders under cross‐sectional sampling. This paper considers four asymptotic interval estimators which are direct generalizations of those originally proposed for the case of no confounders, and employs Monte Carlo simulation to evaluate the finite‐sample performance of these estimators in a variety of situations. This paper finds that interval estimators using Wald's test statistic and a quadratic equation suggested here can consistently perform reasonably well with respect to the coverage probability in all the situations considered here. This paper notes that the interval estimator using the logarithmic transformation, that is previously found to consistently perform well for the case of no confounders, may have the coverage probability less than the desired confidence level when the underlying common prevalence rate ratio (RR) across strata between the exposure and the non‐exposure is large (≥4). This paper further notes that the interval estimator using the logit transformation is inappropriate for use when the underlying common RR ≐ 1. On the other hand, when the underlying common RR is large (≥4), this interval estimator is probably preferable to all the other three estimators. When the sample size is large (≥400) and the RR ≥ 2 in the situations considered here, this paper finds that all the four interval estimators developed here are essentially equivalent with respect to both the coverage probability and the average length.     

Robust Prediction of t‐Year Survival with Data from Multiple Studies

Summary Recently meta‐analysis has been widely utilized to combine information across multiple studies to evaluate a common effect. Integrating data from similar studies is particularly useful in genomic studies where the individual study sample sizes are not large relative to the number of parameters of interest. In this article, we are interested in developing robust prognostic rules for the prediction of t ‐year survival based on multiple studies. We propose to construct a composite score for prediction by fitting a stratified semiparametric transformation model that allows the studies to have related but not identical outcomes. To evaluate the accuracy of the resulting score, we provide point and interval estimators for the commonly used accuracy measures including the time‐specific receiver operating characteristic curves, and positive and negative predictive values. We apply the proposed procedures to develop prognostic rules for the 5‐year survival of breast cancer patients based on five breast cancer genomic studies.     

Describing Ordinal Odds Ratios for Stratified r × c Tables

For an r × ctable with ordinal responses, odds ratios are commonly used to describe the relationship between the row and column variables. This article shows two types of ordinal odds ratios where local‐global odds ratios are used to compare several groups on a c‐category ordinal response and a global odds ratio is used to measure the global association between a pair of ordinal responses. When there is a stratification factor, we consider Mantel‐Haenszel (MH) type estimators of these odds ratios to summarize the association from several strata. Like the ordinary MH estimator of the common odds ratio for several 2 × 2 contingency tables, the estimators are used when the association is not expected to vary drastically among the strata. Also, the estimators are consistent under the ordinary asymptotic framework in which the number of strata is fixed and also under sparse asymptotics in which the number of strata grows with the sample size. Compared to the maximum likelihood estimators, simulations find that the MH type estimators perform better especially when each stratum has few observations. This article provides variances and covariances formulae for the local‐global odds ratios estimators and applies the bootstrap method to obtain a standard error for the global odds ratio estimator. At the end, we discuss possible ways of testing the homogeneity assumption.     

Pleiotropy informed adaptive association test of multiple traits using genome‐wide association study summary data

Genetic variants associated with disease outcomes can be used to develop personalized treatment. To reach this precision medicine goal, hundreds of large‐scale genome‐wide association studies (GWAS) have been conducted in the past decade to search for promising genetic variants associated with various traits. They have successfully identified tens of thousands of disease‐related variants. However, in total these identified variants explain only part of the variation for most complex traits. There remain many genetic variants with small effect sizes to be discovered, which calls for the development of (a) GWAS with more samples and more comprehensively genotyped variants, for example, the NHLBI Trans‐Omics for Precision Medicine (TOPMed) Program is planning to conduct whole genome sequencing on over 100 000 individuals; and (b) novel and more powerful statistical analysis methods. The current dominating GWAS analysis approach is the “single trait” association test, despite the fact that many GWAS are conducted in deeply phenotyped cohorts including many correlated and well‐characterized outcomes, which can help improve the power to detect novel variants if properly analyzed, as suggested by increasing evidence that pleiotropy, where a genetic variant affects multiple traits, is the norm in genome‐phenome associations. We aim to develop pleiotropy informed powerful association test methods across multiple traits for GWAS. Since it is generally very hard to access individual‐level GWAS phenotype and genotype data for those existing GWAS, due to privacy concerns and various logistical considerations, we develop rigorous statistical methods for pleiotropy informed adaptive multitrait association test methods that need only summary association statistics publicly available from most GWAS. We first develop a pleiotropy test, which has powerful performance for truly pleiotropic variants but is sensitive to the pleiotropy assumption. We then develop a pleiotropy informed adaptive test that has robust and powerful performance under various genetic models. We develop accurate and efficient numerical algorithms to compute the analytical P‐value for the proposed adaptive test without the need of resampling or permutation. We illustrate the performance of proposed methods through application to joint association test of GWAS meta‐analysis summary data for several glycemic traits. Our proposed adaptive test identified several novel loci missed by individual trait based GWAS meta‐analysis. All the proposed methods are implemented in a publicly available R package.     

Heritability and genetic correlations of personality traits in a wild population of yellow‐bellied marmots (Marmota flaviventris)

Describing and quantifying animal personality is now an integral part of behavioural studies because individually distinctive behaviours have ecological and evolutionary consequences. Yet, to fully understand how personality traits may respond to selection, one must understand the underlying heritability and genetic correlations between traits. Previous studies have reported a moderate degree of heritability of personality traits, but few of these studies have either been conducted in the wild or estimated the genetic correlations between personality traits. Estimating the additive genetic variance and covariance in the wild is crucial to understand the evolutionary potential of behavioural traits. Enhanced environmental variation could reduce heritability and genetic correlations, thus leading to different evolutionary predictions. We estimated the additive genetic variance and covariance of docility in the trap, sociability (mirror image stimulation), and exploration and activity in two different contexts (open‐field and mirror image simulation experiments) in a wild population of yellow‐bellied marmots (Marmota flaviventris). We estimated both heritability of behaviours and of personality traits and found nonzero additive genetic variance in these traits. We also found nonzero maternal, permanent environment and year effects. Finally, we found four phenotypic correlations between traits, and one positive genetic correlation between activity in the open‐field test and sociability. We also found permanent environment correlations between activity in both tests and docility and exploration in the MIS test. This is one of a handful of studies to adopt a quantitative genetic approach to explain variation in personality traits in the wild and, thus, provides important insights into the potential variance available for selection.