Implementing informative priors for heterogeneity in meta‐analysis using meta‐regression and pseudo data

Many meta‐analyses combine results from only a small number of studies, a situation in which the between‐study variance is imprecisely estimated when standard methods are applied. Bayesian meta‐analysis allows incorporation of external evidence on heterogeneity, providing the potential for more robust inference on the effect size of interest. We present a method for performing Bayesian meta‐analysis using data augmentation, in which we represent an informative conjugate prior for between‐study variance by pseudo data and use meta‐regression for estimation. To assist in this, we derive predictive inverse‐gamma distributions for the between‐study variance expected in future meta‐analyses. These may serve as priors for heterogeneity in new meta‐analyses. In a simulation study, we compare approximate Bayesian methods using meta‐regression and pseudo data against fully Bayesian approaches based on importance sampling techniques and Markov chain Monte Carlo (MCMC). We compare the frequentist properties of these Bayesian methods with those of the commonly used frequentist DerSimonian and Laird procedure. The method is implemented in standard statistical software and provides a less complex alternative to standard MCMC approaches. An importance sampling approach produces almost identical results to standard MCMC approaches, and results obtained through meta‐regression and pseudo data are very similar. On average, data augmentation provides closer results to MCMC, if implemented using restricted maximum likelihood estimation rather than DerSimonian and Laird or maximum likelihood estimation. The methods are applied to real datasets, and an extension to network meta‐analysis is described. The proposed method facilitates Bayesian meta‐analysis in a way that is accessible to applied researchers. © 2016 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

Harmonization of Study and Reference Data by PhaseLift: Saving Time When Imputing Study Data

Genome‐wide association studies are usually accompanied by imputation techniques to complement genome‐wide SNP chip genotypes. Current imputation approaches separate the phasing of study data from imputing, which makes the phasing independent from the reference data. The two‐step approach allows for updating the imputation for a new reference panel without repeating the tedious phasing step. This advantage, however, does no longer hold, when the build of the study data differs from the build of the reference data. In this case, the current approach is to harmonize the study data annotation with the reference data (prephasing lift‐over), requiring rephasing and re‐imputing. As a novel approach, we propose to harmonize study haplotypes with reference haplotypes (postphasing lift‐over). This allows for updating imputed study data for new reference panels without requiring rephasing. With continuously updated reference panels, our approach can save considerable computing time of up to 1 month per re‐imputation. We evaluated the rephasing and postphasing lift‐over approaches by using data from 1,644 unrelated individuals imputed by both approaches and comparing it with directly typed genotypes. On average, both approaches perform equally well with mean concordances of 93% between imputed and typed genotypes for both approaches. Also, imputation qualities are similar (mean difference in RSQ < 0.1%). We demonstrate that our novel postphasing lift‐over approach is a practical and time‐saving alternative to the prephasing lift‐over. This might encourage study partners to accommodate updated reference builds and ultimately improve the information content of study data. Our novel approach is implemented in the software PhaseLift.     

A non‐parametric approach to estimate the total deviation index for non‐normal data

Concordance indices are used to assess the degree of agreement between different methods that measure the same characteristic. In this context, the total deviation index (TDI) is an unscaled concordance measure that quantifies to which extent the readings from the same subject obtained by different methods may differ with a certain probability. Common approaches to estimate the TDI assume data are normally distributed and linearity between response and effects (subjects, methods and random error). Here, we introduce a new non‐parametric methodology for estimation and inference of the TDI that can deal with any kind of quantitative data. The present study introduces this non‐parametric approach and compares it with the already established methods in two real case examples that represent situations of non‐normal data (more specifically, skewed data and count data). The performance of the already established methodologies and our approach in these contexts is assessed by means of a simulation study. Copyright © 2015 John Wiley & Sons, Ltd.     

Using cases to strengthen inference on the association between single nucleotide polymorphisms and a secondary phenotype in genome‐wide association studies

Case‐control genome‐wide association studies provide a vast amount of genetic information that may be used to investigate secondary phenotypes. We study the situation in which the primary disease is rare and the secondary phenotype and genetic markers are dichotomous. An analysis of the association between a genetic marker and the secondary phenotype based on controls only (CO) is valid, whereas standard methods that also use cases result in biased estimates and highly inflated type I error if there is an interaction between the secondary phenotype and the genetic marker on the risk of the primary disease. Here we present an adaptively weighted (AW) method that combines the case and control data to study the association, while reducing to the CO analysis if there is strong evidence of an interaction. The possibility of such an interaction and the misleading results for standard methods, but not for the AW or CO approaches, are illustrated by data from a case‐control study of colorectal adenoma. Simulations and asymptotic theory indicate that the AW method can reduce the mean square error for estimation with a prespecified SNP and increase the power to discover a new association in a genome‐wide study, compared to CO analysis. Further experience with genome‐wide studies is needed to determine when methods that assume no interaction gain precision and power, thereby can be recommended, and when methods such as the AW or CO approaches are needed to guard against the possibility of nonzero interactions. Genet. Epidemiol. 34:427–433, 2010. Published 2010 Wiley‐Liss, Inc.     

Estimation of relative risk and prevalence ratio

Relative risks (RRs) and prevalence ratios (PRs) are measures of association that are more intuitively interpretable than odds ratios (ORs). Many health science studies report OR estimates, however, even when their designs permit and study questions target RRs and/or PRs. This is, partially, attributable to the popularity and technical advantage (i.e. no restriction on the parameter space) of logistic regression for estimating ORs. To improve this practice, several biostatistical approaches for estimating RR/PR, adjusting for potential confounders, have been proposed. In this paper, we consider two RR/PR estimating methods: (1) the modification of log‐binomial regression with the COPY method; and (2) an inverse‐probability‐of‐treatment‐weighted (IPTW) log‐binomial regression we newly propose. For the COPY method, we rigorously establish the existence and uniqueness of the maximum‐likelihood estimator, provided certain degeneracies in the data do not occur. Moreover, the global maximum of the COPY‐modified likelihood is shown to occur at an interior point of the restricted parameter space. This result explains why the COPY method avoids convergence problems of log‐binomial models frequently. For the IPTW estimator, we show that its simple procedure results in standardized estimates of RR/PR, and discuss its potential challenges, extensions, and an improvement through propensity‐score‐based grouping of observations. Furthermore, we compare the performances of four RR/PR estimation methods, including the COPY method and IPTW log‐binomial regression, on simulated data. We demonstrate a lack of robustness of the COPY method against misspecification of the true relationship between binary outcome and explanatory variables, and show robustness of the IPTW approach in this regard. Copyright © 2010 John Wiley & Sons, Ltd.     

A semiparametric probit model for case 2 interval‐censored failure time data

Interval‐censored data occur naturally in many fields and the main feature is that the failure time of interest is not observed exactly, but is known to fall within some interval. In this paper, we propose a semiparametric probit model for analyzing case 2 interval‐censored data as an alternative to the existing semiparametric models in the literature. Specifically, we propose to approximate the unknown nonparametric nondecreasing function in the probit model with a linear combination of monotone splines, leading to only a finite number of parameters to estimate. Both the maximum likelihood and the Bayesian estimation methods are proposed. For each method, regression parameters and the baseline survival function are estimated jointly. The proposed methods make no assumptions about the observation process and can be applicable to any interval‐censored data with easy implementation. The methods are evaluated by simulation studies and are illustrated by two real‐life interval‐censored data applications. Copyright © 2010 John Wiley & Sons, Ltd.     

Estimating long‐term treatment effects in observational data: A comparison of the performance of different methods under real‐world uncertainty

In the presence of time‐dependent confounding, there are several methods available to estimate treatment effects. With correctly specified models and appropriate structural assumptions, any of these methods could provide consistent effect estimates, but with real‐world data, all models will be misspecified and it is difficult to know if assumptions are violated. In this paper, we investigate five methods: inverse probability weighting of marginal structural models, history‐adjusted marginal structural models, sequential conditional mean models, g‐computation formula, and g‐estimation of structural nested models. This work is motivated by an investigation of the effects of treatments in cystic fibrosis using the UK Cystic Fibrosis Registry data focussing on two outcomes: lung function (continuous outcome) and annual number of days receiving intravenous antibiotics (count outcome). We identified five features of this data that may affect the performance of the methods: misspecification of the causal null, long‐term treatment effects, effect modification by time‐varying covariates, misspecification of the direction of causal pathways, and censoring. In simulation studies, under ideal settings, all five methods provide consistent estimates of the treatment effect with little difference between methods. However, all methods performed poorly under some settings, highlighting the importance of using appropriate methods based on the data available. Furthermore, with the count outcome, the issue of non‐collapsibility makes comparison between methods delivering marginal and conditional effects difficult. In many situations, we would recommend using more than one of the available methods for analysis, as if the effect estimates are very different, this would indicate potential issues with the analyses.     

Statistical methodology for estimating the mean difference in a meta‐analysis without study‐specific variance information

Statistical inference for analyzing the results from several independent studies on the same quantity of interest has been investigated frequently in recent decades. Typically, any meta‐analytic inference requires that the quantity of interest is available from each study together with an estimate of its variability. The current work is motivated by a meta‐analysis on comparing two treatments (thoracoscopic and open) of congenital lung malformations in young children. Quantities of interest include continuous end‐points such as length of operation or number of chest tube days. As studies only report mean values (and no standard errors or confidence intervals), the question arises how meta‐analytic inference can be developed. We suggest two methods to estimate study‐specific variances in such a meta‐analysis, where only sample means and sample sizes are available in the treatment arms. A general likelihood ratio test is derived for testing equality of variances in two groups. By means of simulation studies, the bias and estimated standard error of the overall mean difference from both methodologies are evaluated and compared with two existing approaches: complete study analysis only and partial variance information. The performance of the test is evaluated in terms of type I error. Additionally, we illustrate these methods in the meta‐analysis on comparing thoracoscopic and open surgery for congenital lung malformations and in a meta‐analysis on the change in renal function after kidney donation. Copyright © 2017 John Wiley & Sons, Ltd.     

Inference for correlated effect sizes using multiple univariate meta‐analyses

Multivariate meta‐analysis, which involves jointly analyzing multiple and correlated outcomes from separate studies, has received a great deal of attention. One reason to prefer the multivariate approach is its ability to account for the dependence between multiple estimates from the same study. However, nearly all the existing methods for analyzing multivariate meta‐analytic data require the knowledge of the within‐study correlations, which are usually unavailable in practice. We propose a simple non‐iterative method that can be used for the analysis of multivariate meta‐analysis datasets, that has no convergence problems, and does not require the use of within‐study correlations. Our approach uses standard univariate methods for the marginal effects but also provides valid joint inference for multiple parameters. The proposed method can directly handle missing outcomes under missing completely at random assumption. Simulation studies show that the proposed method provides unbiased estimates, well‐estimated standard errors, and confidence intervals with good coverage probability. Furthermore, the proposed method is found to maintain high relative efficiency compared with conventional multivariate meta‐analyses where the within‐study correlations are known. We illustrate the proposed method through two real meta‐analyses where functions of the estimated effects are of interest. © 2015 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

Hartung–Knapp method is not always conservative compared with fixed‐effect meta‐analysis

A widely used method in classic random‐effects meta‐analysis is the DerSimonian–Laird method. An alternative meta‐analytical approach is the Hartung–Knapp method. This article reports results of an empirical comparison and a simulation study of these two methods and presents corresponding analytical results. For the empirical evaluation, we took 157 meta‐analyses with binary outcomes, analysed each one using both methods and performed a comparison of the results based on treatment estimates, standard errors and associated P‐values. In several simulation scenarios, we systematically evaluated coverage probabilities and confidence interval lengths. Generally, results are more conservative with the Hartung–Knapp method, giving wider confidence intervals and larger P‐values for the overall treatment effect. However, in some meta‐analyses with very homogeneous individual treatment results, the Hartung–Knapp method yields narrower confidence intervals and smaller P‐values than the classic random‐effects method, which in this situation, actually reduces to a fixed‐effect meta‐analysis. Therefore, it is recommended to conduct a sensitivity analysis based on the fixed‐effect model instead of solely relying on the result of the Hartung–Knapp random‐effects meta‐analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

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

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

Bayesian modeling of the covariance structure for irregular longitudinal data using the partial autocorrelation function

In long‐term follow‐up studies, irregular longitudinal data are observed when individuals are assessed repeatedly over time but at uncommon and irregularly spaced time points. Modeling the covariance structure for this type of data is challenging, as it requires specification of a covariance function that is positive definite. Moreover, in certain settings, careful modeling of the covariance structure for irregular longitudinal data can be crucial in order to ensure no bias arises in the mean structure. Two common settings where this occurs are studies with ‘outcome‐dependent follow‐up’ and studies with ‘ignorable missing data’. ‘Outcome‐dependent follow‐up’ occurs when individuals with a history of poor health outcomes had more follow‐up measurements, and the intervals between the repeated measurements were shorter. When the follow‐up time process only depends on previous outcomes, likelihood‐based methods can still provide consistent estimates of the regression parameters, given that both the mean and covariance structures of the irregular longitudinal data are correctly specified and no model for the follow‐up time process is required. For ‘ignorable missing data’, the missing data mechanism does not need to be specified, but valid likelihood‐based inference requires correct specification of the covariance structure. In both cases, flexible modeling approaches for the covariance structure are essential. In this paper, we develop a flexible approach to modeling the covariance structure for irregular continuous longitudinal data using the partial autocorrelation function and the variance function. In particular, we propose semiparametric non‐stationary partial autocorrelation function models, which do not suffer from complex positive definiteness restrictions like the autocorrelation function. We describe a Bayesian approach, discuss computational issues, and apply the proposed methods to CD4 count data from a pediatric AIDS clinical trial. © 2015 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

Some drop‐the‐loser designs for monitoring multiple doses

A scenario not uncommon at the end of a Phase II clinical development is that although choices are narrowed down to two to three doses, the project team cannot make a recommendation of one single dose for the Phase III confirmatory study based upon the available data. Several ‘drop‐the‐loser’ designs to monitor multiple doses of an experimental treatment compared with a control in a pivotal Phase III study are considered. Ineffective and/or toxic doses compared with the control may be dropped at the interim analyses as the study continues, and when the accumulated data have demonstrated convincing efficacy and an acceptable safety profile for one dose, the corresponding dose or the study may be stopped to make the experimental treatment available to patients. A decision to drop a toxic dose is usually based upon a comprehensive review of all the available safety data and also a risk/benefit assessment. For dropping ineffective doses, a non‐binding futility boundary may be used as guidance. The desired futility boundary can be derived by using an appropriate combination of risk level (i.e. error rate for accepting null hypothesis when the dose is truly efficacious) and spending strategy (dropping a dose aggressively in early analyses versus late). For establishing convincing evidence of the treatment efficacy, three methods for calculating the efficacy boundary are discussed: the Joint Monitoring (JM) approach, the Marginal Monitoring method with Bonferroni correction (MMB), and the Marginal Monitoring method with Adjustment for correlation (MMA). The JM approach requires intensive computation especially when there are several doses and multiple interim analyses. The marginal monitoring methods are computationally more attractive and also more flexible since each dose is monitored separately by its own alpha‐spending function. The JM and MMB methods control the false positive rate. The MMA method tends to protect the false positive rate and is more powerful than the Bonferroni‐based MMB method. The MMA method offers a practical and flexible solution when there are several doses and multiple interim looks. Copyright © 2010 John Wiley & Sons, Ltd.     

Sparse boosting for high‐dimensional survival data with varying coefficients

Motivated by high‐throughput profiling studies in biomedical research, variable selection methods have been a focus for biostatisticians. In this paper, we consider semiparametric varying‐coefficient accelerated failure time models for right censored survival data with high‐dimensional covariates. Instead of adopting the traditional regularization approaches, we offer a novel sparse boosting (SparseL₂Boosting) algorithm to conduct model‐based prediction and variable selection. One main advantage of this new method is that we do not need to perform the time‐consuming selection of tuning parameters. Extensive simulations are conducted to examine the performance of our sparse boosting feature selection techniques. We further illustrate our methods using a lung cancer data analysis.     

Sample Size Considerations of Prediction‐Validation Methods in High‐Dimensional Data for Survival Outcomes

A variety of prediction methods are used to relate high‐dimensional genome data with a clinical outcome using a prediction model. Once a prediction model is developed from a data set, it should be validated using a resampling method or an independent data set. Although the existing prediction methods have been intensively evaluated by many investigators, there has not been a comprehensive study investigating the performance of the validation methods, especially with a survival clinical outcome. Understanding the properties of the various validation methods can allow researchers to perform more powerful validations while controlling for type I error. In addition, sample size calculation strategy based on these validation methods is lacking. We conduct extensive simulations to examine the statistical properties of these validation strategies. In both simulations and a real data example, we have found that 10‐fold cross‐validation with permutation gave the best power while controlling type I error close to the nominal level. Based on this, we have also developed a sample size calculation method that will be used to design a validation study with a user‐chosen combination of prediction. Microarray and genome‐wide association studies data are used as illustrations. The power calculation method in this presentation can be used for the design of any biomedical studies involving high‐dimensional data and survival outcomes.     

Correlated bivariate continuous and binary outcomes: Issues and applications

Increasingly multiple outcomes are collected in order to characterize treatment effectiveness or to evaluate the impact of large policy initiatives. Often the multiple outcomes are non‐commensurate, e.g. measured on different scales. The common approach to inference is to model each outcome separately ignoring the potential correlation among the responses. We describe and contrast several full likelihood and quasi‐likelihood multivariate methods for non‐commensurate outcomes. We present a new multivariate model to analyze binary and continuous correlated outcomes using a latent variable. We study the efficiency gains of the multivariate methods relative to the univariate approach. For complete data, all approaches yield consistent parameter estimates. When the mean structure of all outcomes depends on the same set of covariates, efficiency gains by adopting a multivariate approach are negligible. In contrast, when the mean outcomes depend on different covariate sets, large efficiency gains are realized. Three real examples illustrate the different approaches. Copyright © 2009 John Wiley & Sons, Ltd.     

How to regress and predict in a Bland–Altman plot? Review and contribution based on tolerance intervals and correlated‐errors‐in‐variables models

Two main methodologies for assessing equivalence in method‐comparison studies are presented separately in the literature. The first one is the well‐known and widely applied Bland–Altman approach with its agreement intervals, where two methods are considered interchangeable if their differences are not clinically significant. The second approach is based on errors‐in‐variables regression in a classical (X,Y) plot and focuses on confidence intervals, whereby two methods are considered equivalent when providing similar measures notwithstanding the random measurement errors. This paper reconciles these two methodologies and shows their similarities and differences using both real data and simulations. A new consistent correlated‐errors‐in‐variables regression is introduced as the errors are shown to be correlated in the Bland–Altman plot. Indeed, the coverage probabilities collapse and the biases soar when this correlation is ignored. Novel tolerance intervals are compared with agreement intervals with or without replicated data, and novel predictive intervals are introduced to predict a single measure in an (X,Y) plot or in a Bland–Atman plot with excellent coverage probabilities. We conclude that the (correlated)‐errors‐in‐variables regressions should not be avoided in method comparison studies, although the Bland–Altman approach is usually applied to avert their complexity. We argue that tolerance or predictive intervals are better alternatives than agreement intervals, and we provide guidelines for practitioners regarding method comparison studies. Copyright © 2016 John Wiley & Sons, Ltd.     

Regression analysis of interval‐censored failure time data with possibly crossing hazards

Interval‐censored failure time data occur in many areas, especially in medical follow‐up studies such as clinical trials, and in consequence, many methods have been developed for the problem. However, most of the existing approaches cannot deal with the situations where the hazard functions may cross each other. To address this, we develop a sieve maximum likelihood estimation procedure with the application of the short‐term and long‐term hazard ratio model. In the method, the I‐splines are used to approximate the underlying unknown function. An extensive simulation study was conducted for the assessment of the finite sample properties of the presented procedure and suggests that the method seems to work well for practical situations. The analysis of an motivated example is also provided.     

Bayesian adjustment for covariate measurement errors: A flexible parametric approach

In most epidemiological investigations, the study units are people, the outcome variable (or the response) is a health‐related event, and the explanatory variables are usually environmental and/or socio‐demographic factors. The fundamental task in such investigations is to quantify the association between the explanatory variables (covariates/exposures) and the outcome variable through a suitable regression model. The accuracy of such quantification depends on how precisely the relevant covariates are measured. In many instances, we cannot measure some of the covariates accurately. Rather, we can measure noisy (mismeasured) versions of them. In statistical terminology, mismeasurement in continuous covariates is known as measurement errors or errors‐in‐variables. Regression analyses based on mismeasured covariates lead to biased inference about the true underlying response–covariate associations. In this paper, we suggest a flexible parametric approach for avoiding this bias when estimating the response–covariate relationship through a logistic regression model. More specifically, we consider the flexible generalized skew‐normal and the flexible generalized skew‐t distributions for modeling the unobserved true exposure. For inference and computational purposes, we use Bayesian Markov chain Monte Carlo techniques. We investigate the performance of the proposed flexible parametric approach in comparison with a common flexible parametric approach through extensive simulation studies. We also compare the proposed method with the competing flexible parametric method on a real‐life data set. Though emphasis is put on the logistic regression model, the proposed method is unified and is applicable to the other generalized linear models, and to other types of non‐linear regression models as well. Copyright © 2009 John Wiley & Sons, Ltd.