Estimating the sample variance from the sample size and range

For meta-analysis studies and systematic reviews, it is important to pool the data from a set of similar clinical trials. To pool the data, one needs to know their SD. Many trial reports, however, contain only the median, the minimum and maximum values, and the sample size. It is therefore important to be able to estimate the SD S from the sample size n and range r. For small n ≤ 100 , we improve existing estimators of r/S , the “divisor,” denoted by $\xi (n)$. This in turn yields improved estimators of the SD in the form $\widehat{S}=\text{◂+▸}r/\widehat{\xi }(n)$ on simulated as well as real datasets. We provide numerical values of the proposed estimator as well as approximation by a simple formula $\text{◂,▸}3\text{◂√▸}\sqrt{\ln (n)}-1.4025$. Furthermore, for large n, we provide estimators $\widehat{\xi }(n)$ of the divisor $\xi (n)$ for the normal, exponential, and other bounded and unbounded distributions.     

Improved two-step testing of genome-wide gene–environment interactions

Two-step tests for gene–environment ($G\times E$) interactions exploit marginal single-nucleotide polymorphism (SNP) effects to improve the power of a genome-wide interaction scan. They combine a screening step based on marginal effects used to “bin” SNPs for weighted hypothesis testing in the second step to deliver greater power over single-step tests while preserving the genome-wide Type I error. However, the presence of many SNPs with detectable marginal effects on the trait of interest can reduce power by “displacing” true interactions with weaker marginal effects and by adding to the number of tests that need to be corrected for multiple testing. We introduce a new significance-based allocation into bins for Step-2 $G\times E$ testing that overcomes the displacement issue and propose a computationally efficient approach to account for multiple testing within bins. Simulation results demonstrate that these simple improvements can provide substantially greater power than current methods under several scenarios. An application to a multistudy collaboration for understanding colorectal cancer reveals a G × Sex interaction located near the SMAD7 gene.     

An improved closed procedure for testing multiple hypotheses

Clinical trials routinely involve multiple hypothesis testing. The closed testing procedure (CTP) is a fundamental principle in testing multiple hypotheses. This article presents an improved CTP in which intersection hypotheses can be tested at a level greater than $\alpha$ such that the control of the familywise error rate at level $\alpha$ remains. Consequently, our method uniformly improves the power of discovering false hypotheses over the original CTP. We illustrate that an improvement by our method exists for many commonly used tests. An empirical study on the effectiveness of a glucose-lowering drug is provided.     

Two-sample test with g-modeling and its applications

Many real data analyses involve two-sample comparisons in location or in distribution. Most existing methods focus on problems where observations are independently and identically distributed in each group. However, in some applications the observed data are not identically distributed but associated with some unobserved parameters which are identically distributed. To address this challenge, we propose a novel two-sample testing procedure as a combination of the $g$-modeling density estimation introduced by Efron and the two-sample Kolmogorov-Smirnov test. We also propose efficient bootstrap algorithms to estimate the statistical significance for such tests. We demonstrate the utility of the proposed approach with two biostatistical applications: the analysis of surgical nodes data with binomial model and differential expression analysis of single-cell RNA sequencing data with zero-inflated Poisson model.     

Group sequential designs for clinical trials with bivariate endpoints

Although all clinical trials are designed and monitored using more than one endpoint, methods are needed to assure that decision criteria are chosen to reflect the clinically relevant tradeoffs that assure the trial's scientific integrity. This article presents a framework for the design and monitoring clinical trials in a bivariate outcome space. The framework uses a rectangular hyperbola to define a bivariate null curve that divides outcome space into regions of benefit and lack of benefit. The curve is shown to be a flexible mapping of bivariate space that allows a continuous tradeoff between the two endpoints in a manner that captures many previous bivariate designs. The curve is extended to a distance function in bivariate space that allows different decisions in each of the four quadrants that comprise bivariate space. The distance function forms a statistic ($\delta$); the distribution of its estimate is derived and used as a basis for trial design and group sequential monitoring plans in bivariate space. A recursive form of the bivariate group sequential density is used to evaluate and control operating characteristics for the proposed design. The bivariate designs are shown to meet or exceed the usual standards for size and power. The proposed design is illustrated in the ongoing NHLBI-sponsored Kids-DOTT multinational randomized controlled trial comparing shortened versus conventional anticoagulation for the treatment of venous thromboembolism in patients less than 21 years of age. The proposed methods are broadly applicable to a wide range of clinical settings and trial designs.     

Ratio estimators of intervention effects on event rates in cluster randomized trials

We consider five asymptotically unbiased estimators of intervention effects on event rates in non-matched and matched-pair cluster randomized trials, including ratio of mean counts $\left(r_1\right)$, ratio of mean cluster-level event rates $\left(r_2\right)$, ratio of event rates $\left(r_3\right)$, double ratio of counts $\left(r_4\right)$, and double ratio of event rates $\left(r_5\right)$. In the absence of an indirect effect, they all estimate the direct effect of the intervention. Otherwise, $r_1$, $r_2,$ and $r_3$ estimate the total effect, which comprises the direct and indirect effects, whereas $r_4$ and $r_5$ estimate the direct effect only. We derive the conditions under which each estimator is more precise or powerful than its alternatives. To control bias in studies with a small number of clusters, we propose a set of approximately unbiased estimators. We evaluate their properties by simulation and apply the methods to a trial of seasonal malaria chemoprevention. The approximately unbiased estimators are practically unbiased and their confidence intervals usually have coverage probability close to the nominal level; the asymptotically unbiased estimators perform well when the number of clusters is approximately 32 or more per trial arm. Despite its simplicity, $r_1$ performs comparably with $r_2$ and $r_3$ in trials with a large but realistic number of clusters. When the variability of baseline event rate is large and there is no indirect effect, $r_4$ and $r_5$ tend to offer higher power than $r_1$, $r_2,$ and $r_3$. We discuss the implications of these findings to the planning and analysis of cluster randomized trials.     

Joint inference about the AUC and Youden index for paired biomarkers

It is common to compare biomarkers' diagnostic or prognostic performance using some summary ROC measures such as the area under the ROC curve (AUC) or the Youden index. We propose to compare two paired biomarkers using both the AUC and the Youden index since the two indices describe different aspects of the ROC curve. This comparison can be made by estimating the joint confidence region (an elliptical area) of the differences of the paired AUCs and the Youden indices. Furthermore, for deciding if one marker is better than the other in terms of both the $AUC$ and the Youden index (J), we can test $H_0:AU{C}_a\le AU{C}_b$ or $J_a\le J_b$ against $H_a:AU{C}_a>AU{C}_b$ and $J_a>J_b$ using the paired differences. The construction of such a joint hypothesis is an example of the multivariate order-restricted hypotheses. For such a hypothesis, we propose and compare three testing procedures: (1) the intersection-union test ($IUT$); (2) the conditional test; and (3) the joint test. The performance of the proposed inference methods was evaluated and compared through simulations. The simulation results demonstrate that the proposed joint confidence region maintains the desired confidence level, and all three tests maintain the type I error under the null. Furthermore, among the three proposed testing methods, the conditional test is the preferred approach with markedly larger power consistently than the other two competing methods.     

A flexible and robust method for assessing conditional association and conditional concordance

When analyzing bivariate outcome data, it is often of scientific interest to measure and estimate the association between the bivariate outcomes. In the presence of influential covariates for one or both of the outcomes, conditional association measures can quantify the strength of association without the disturbance of the marginal covariate effects, to provide cleaner and less-confounded insights into the bivariate association. In this work, we propose estimation and inferential procedures for assessing the conditional Kendall's tau coefficient given the covariates, by adopting the quantile regression and quantile copula framework to handle marginal covariate effects. The proposed method can flexibly accommodate right censoring and be readily applied to bivariate survival data. It also facilitates an estimator of the conditional concordance measure, namely, a conditional index, where the unconditional index is commonly used to assess the predictive capacity for survival outcomes. The proposed method is flexible and robust and can be easily implemented using standard software. The method performed satisfactorily in extensive simulation studies with and without censoring. Application of our methods to two real-life data examples demonstrates their desirable practical utility.     

One-step validation method for surrogate endpoints using data from multiple randomized cancer clinical trials with failure-time endpoints

A surrogate endpoint can be used instead of the most relevant clinical endpoint to assess the efficiency of a new treatment. Before being used, a surrogate endpoint must be validated based on appropriate methods. Numerous validation approaches have been proposed with the most popular used in a context of meta-analysis, based on a two-step analysis strategy. For two failure-time endpoints, two association measurements are usually used, Kendall's τ at the individual level and the adjusted coefficient of determination ( ) at the trial level. However, is not always available due to model estimation constraints. We propose a one-step validation approach based on a joint frailty model, including both individual-level and trial-level random effects. Parameters have been estimated using a semiparametric penalized marginal log-likelihood method, and various numerical integration approaches were considered. Both individual- and trial-level surrogacy were evaluated using a new definition of Kendall's τ and the coefficient of determination. Estimators' performances were evaluated using simulation studies and satisfactory results were found. The model was applied to individual patient data meta-analyses in gastric cancer to assess disease-free survival as a surrogate for overall survival, as part of the evaluation of adjuvant therapy.     

Set-regression with applications to subgroup analysis

Regression is a commonly used statistical model. It is the conditional mean of the response given covariates $\mu (\mathit{x})=E(Y|\mathit{X}=\mathit{x})$. However, in some practical problems, the interest is the conditional mean of the response given the covariates belonging to some set A. Notably, in precision medicine and subgroup analysis in clinical trials, the aim is to identify subjects who benefit the most from the treatment, or identify an optimal set in the covariate space which manifests treatment favoritism if a subject's covariates fall in this set and the subject is classified to the favorable treatment subgroup. Existing methods for subgroup analysis achieve this indirectly by using classical regression. This motivates us to develop a new type of regression: set-regression, defined as $\mu (A)=E(Y|\mathit{X}\in A)$ which directly addresses the subgroup analysis problem. This extends not only the classical regression model but also improves recursive partitioning and support vector machine approaches, and is particularly suitable for objectives involving optimization of the regression over sets, such as subgroup analysis. We show that the new versatile set-regression identifies the subgroup with increased accuracy. It is easy to use. Simulation studies also show superior performance of the proposed method in finite samples.     

Beyond the traditional simulation design for evaluating type 1 error control: From the “theoretical” null to “empirical” null

When evaluating a newly developed statistical test, an important step is to check its type 1 error (T1E) control using simulations. This is often achieved by the standard simulation design S0 under the so-called “theoretical” null of no association. In practice, the whole-genome association analyses scan through a large number of genetic markers (s) for the ones associated with an outcome of interest (), where comes from an alternative while the majority of s are not associated with ; the relationships are under the “empirical” null. This reality can be better represented by two other simulation designs, where design S1.1 simulates from analternative model based on , then evaluates its association with independently generated ; while design S1.2 evaluates the association between permutated and . More than a decade ago, Efron (2004) has noted the important distinction between the “theoretical” and “empirical” null in false discovery rate control. Using scale tests for variance heterogeneity, direct univariate, and multivariate interaction tests as examples, here we show that not all null simulation designs are equal. In examining the accuracy of a likelihood ratio test, while simulation design S0 suggested the method being accurate, designs S1.1 and S1.2 revealed its increased empirical T1E rate if applied in real data setting. The inflation becomes more severe at the tail and does not diminish as sample size increases. This is an important observation that calls for new practices for methods evaluation and T1E control interpretation.     

Linear mixed models for association analysis of quantitative traits with next-generation sequencing data

We develop linear mixed models (LMMs) and functional linear mixed models (FLMMs) for gene-based tests of association between a quantitative trait and genetic variants on pedigrees. The effects of a major gene are modeled as a fixed effect, the contributions of polygenes are modeled as a random effect, and the correlations of pedigree members are modeled via inbreeding/kinship coefficients. -statistics and χ ² likelihood ratio test (LRT) statistics based on the LMMs and FLMMs are constructed to test for association. We show empirically that the -distributed statistics provide a good control of the type I error rate. The -test statistics of the LMMs have similar or higher power than the FLMMs, kernel-based famSKAT (family-based sequence kernel association test), and burden test famBT (family-based burden test). The -statistics of the FLMMs perform well when analyzing a combination of rare and common variants. For small samples, the LRT statistics of the FLMMs control the type I error rate well at the nominal levels and . For moderate/large samples, the LRT statistics of the FLMMs control the type I error rates well. The LRT statistics of the LMMs can lead to inflated type I error rates. The proposed models are useful in whole genome and whole exome association studies of complex traits.     

Methods for large-scale single mediator hypothesis testing: Possible choices and comparisons

Mediation hypothesis testing for a large number of mediators is challenging due to the composite structure of the null hypothesis, $H_0:\alpha \beta =0$ ($\alpha$: effect of the exposure on the mediator after adjusting for confounders; $\beta$: effect of the mediator on the outcome after adjusting for exposure and confounders). In this paper, we reviewed three classes of methods for large-scale one at a time mediation hypothesis testing. These methods are commonly used for continuous outcomes and continuous mediators assuming there is no exposure-mediator interaction so that the product $\alpha \beta$ has a causal interpretation as the indirect effect. The first class of methods ignores the impact of different structures under the composite null hypothesis, namely, (1) $\alpha =0,\beta \ne 0$; (2) $\alpha \ne 0,\beta =0$; and (3) $\alpha =\beta =0$. The second class of methods weights the reference distribution under each case of the null to form a mixture reference distribution. The third class constructs a composite test statistic using the three p values obtained under each case of the null so that the reference distribution of the composite statistic is approximately $U\left(0,1\right)$. In addition to these existing methods, we developed the Sobel-comp method belonging to the second class, which uses a corrected mixture reference distribution for Sobel's test statistic. We performed extensive simulation studies to compare all six methods belonging to these three classes in terms of the false positive rates (FPRs) under the null hypothesis and the true positive rates under the alternative hypothesis. We found that the second class of methods which uses a mixture reference distribution could best maintain the FPRs at the nominal level under the null hypothesis and had the greatest true positive rates under the alternative hypothesis. We applied all methods to study the mediation mechanism of DNA methylation sites in the pathway from adult socioeconomic status to glycated hemoglobin level using data from the Multi-Ethnic Study of Atherosclerosis (MESA). We provide guidelines for choosing the optimal mediation hypothesis testing method in practice and develop an R package medScan available on the CRAN for implementing all the six methods.     

A new measure of between‐studies heterogeneity in meta‐analysis

Assessing the magnitude of heterogeneity in a meta‐analysis is important for determining the appropriateness of combining results. The most popular measure of heterogeneity, I², was derived under an assumption of homogeneity of the within‐study variances, which is almost never true, and the alternative estimator, , uses the harmonic mean to estimate the average of the within‐study variances, which may also lead to bias. This paper thus presents a new measure for quantifying the extent to which the variance of the pooled random‐effects estimator is due to between‐studies variation, , that overcomes the limitations of the previous approach. We show that this measure estimates the expected value of the proportion of total variance due to between‐studies variation and we present its point and interval estimators. The performance of all three heterogeneity measures is evaluated in an extensive simulation study. A negative bias for was observed when the number of studies was very small and became negligible as the number of studies increased, while and I² showed a tendency to overestimate the impact of heterogeneity. The coverage of confidence intervals based upon was good across different simulation scenarios but was substantially lower for and I², especially for high values of heterogeneity and when a large number of studies were included in the meta‐analysis. The proposed measure is implemented in a user‐friendly function available for routine use in r and sas . will be useful in quantifying the magnitude of heterogeneity in meta‐analysis and should supplement the p‐value for the test of heterogeneity obtained from the Q test. Copyright © 2016 John Wiley & Sons, Ltd.     

Estimating cross-population genetic correlations of causal effect sizes

Recent studies have examined the genetic correlations of single-nucleotide polymorphism (SNP) effect sizes across pairs of populations to better understand the genetic architectures of complex traits. These studies have estimated , the cross-population correlation of joint-fit effect sizes at genotyped SNPs. However, the value of depends both on the cross-population correlation of true causal effect sizes () and on the similarity in linkage disequilibrium (LD) patterns in the two populations, which drive tagging effects. Here, we derive the value of the ratio as a function of LD in each population. By applying existing methods to obtain estimates of , we can use this ratio to estimate . Our estimates of were equal to 0.55 ( SE = 0.14) between Europeans and East Asians averaged across nine traits in the Genetic Epidemiology Research on Adult Health and Aging data set, 0.54 ( SE = 0.18) between Europeans and South Asians averaged across 13 traits in the UK Biobank data set, and 0.48 ( SE = 0.06) and 0.65 ( SE = 0.09) between Europeans and East Asians in summary statistic data sets for type 2 diabetes and rheumatoid arthritis, respectively. These results implicate substantially different causal genetic architectures across continental populations.     

Convex combination sequence kernel association test for rare-variant studies

We propose a novel variant set test for rare-variant association studies, which leverages multiple single-nucleotide variant (SNV) annotations. Our approach optimizes a convex combination of different sequence kernel association test (SKAT) statistics, where each statistic is constructed from a different annotation and combination weights are optimized through a multiple kernel learning algorithm. The combination test statistic is evaluated empirically through data splitting. In simulations, we find our method preserves type I error at and has greater power than SKAT(-O) when SNV weights are not misspecified and sample sizes are large (). We utilize our method in the Framingham Heart Study (FHS) to identify SNV sets associated with fasting glucose. While we are unable to detect any genome-wide significant associations between fasting glucose and 4-kb windows of rare variants () in 6,419 FHS participants, our method identifies suggestive associations between fasting glucose and rare variants near ROCK2 () and within CPLX1 (). These two genes were previously reported to be involved in obesity-mediated insulin resistance and glucose-induced insulin secretion by pancreatic beta-cells, respectively. These findings will need to be replicated in other cohorts and validated by functional genomic studies.