Meta-analysis combines pertinent information from existing studies to provide an overall estimate of population parameters/effect sizes, as well as to quantify and explain the differences between studies. However, testing the between-study heterogeneity is one of the most troublesome topics in meta-analysis research. Additionally, no methods have been proposed to test whether the size of the heterogeneity is larger than a specific level. The existing methods, such as the Q test and likelihood ratio (LR) tests, are criticized for their failure to control the Type I error rate and/or failure to attain enough statistical power. Although better reference distribution approximations have been proposed in the literature, the expression is complicated and the application is limited. In this article, we propose bootstrap based heterogeneity tests combining the restricted maximum likelihood (REML) ratio test or Q test with bootstrap procedures, denoted as B-REML-LRT and B-Q respectively. Simulation studies were conducted to examine and compare the performance of the proposed methods with the regular LR tests, the regular Q test, and the improved Q test in both the random-effects meta-analysis and mixed-effects meta-analysis. Based on the results of Type I error rates and statistical power, B-Q is recommended. An R package \mathtt{boot.heterogeneity} is provided to facilitate the implementation of the proposed method.
Case-mix heterogeneity across studies complicates meta-analyses. As a result of this, treatments that are equally effective on patient subgroups may appear to have different effectiveness on patient populations with different case mix. It is therefore important that meta-analyses be explicit for what patient population they describe the treatment effect. To achieve this, we develop a new approach for meta-analysis of randomized clinical trials, which use individual patient data (IPD) from all trials to infer the treatment effect for the patient population in a given trial, based on direct stan…
Meta-regression is often used to form hypotheses about what is associated with heterogeneity in a meta-analysis and to estimate the extent to which effects can vary between cohorts and other distinguishing factors. However, study-level variables, called moderators, that are available and used in the meta-regression analysis will rarely explain all of the heterogeneity. Therefore, measuring and trying to understand residual heterogeneity is still important in a meta-regression, although it is not clear how some heterogeneity measures should be used in the meta-regression context. The coefficien…
In meta-analysis, the fraction of variance that is due to heterogeneity is known as I2. We show that the usual estimator of I2 is biased. The bias is largest when a meta-analysis has few studies and little heterogeneity. For example, with 7 studies and the true value of I2 at 0, the average estimate of I2 is .124. Estimates of I2 should be interpreted cautiously when the meta-analysis is small and the null hypothesis of homogeneity (I2=0) has not been rejected. In small meta-analyses, confidence intervals may be preferable to point estimates for I2.
Empirical prior distributions for treatment-by-subgroup interaction heterogeneity in random-effects meta-analysis
University Medical Center Göttingen（哥廷根大学医学中心）；DZHK (German Center for Cardiovascular Research)（德国心血管研究中心）；DZKJ (German Center for Child and Adolescent Health)（德国儿童与青少年健康中心）
The normal-normal hierarchical model (NNHM) constitutes a simple and widely used framework for meta-analysis. In the common case of only few studies contributing to the meta-analysis, standard approaches to inference tend to perform poorly, and Bayesian meta-analysis has been suggested as a potential solution. The Bayesian approach, however, requires the sensible specification of prior distributions. While non-informative priors are commonly used for the overall mean effect, the use of weakly informative priors has been suggested for the heterogeneity parameter, in particular in the setting of…
Random-effects meta-analyses are used to combine evidence of treatment effects from multiple studies. Since treatment effects may vary across trials due to differences in study characteristics, heterogeneity in treatment effects between studies must be accounted for to achieve valid inference. The standard model for random-effects meta-analysis assumes approximately normal effect estimates and a normal random-effects model. However, standard methods based on this model ignore the uncertainty in estimating the between-trial heterogeneity. In the special setting of only two studies and in the pr…
Cochran's $Q$ statistic is routinely used for testing heterogeneity in meta-analysis. Its expected value is also used for estimation of between-study variance $τ^2$. Cochran's $Q$, or $Q_{IV}$, uses estimated inverse-variance weights which makes approximating its distribution rather complicated. As an alternative, we are investigating a new $Q$ statistic, $Q_F$, whose constant weights use only the studies' effective sample sizes. For standardized mean difference as the measure of effect, we study, by simulation, approximations to distributions of $Q_{IV}$ and $Q_F$, as the basis for tests of h…
A number of popular estimators of the between-study variance, $τ^2$, are based on the Cochran's $Q$ statistic for testing heterogeneity in meta analysis. We introduce new point and interval estimators of $τ^2$ for log-odds-ratio. These include new DerSimonian-Kacker-type moment estimators based on the first moment of $Q_F$, the $Q$ statistic with effective-sample-size weights, and novel median-unbiased estimators. We study, by simulation, bias and coverage of these new estimators of $τ^2$ and, for comparative purposes, bias and coverage of a number of well-known estimators based on the $Q$ sta…
An absolute measure of heterogeneity for lnOR in meta-analysis
Beijing University of Technology（北京工业大学）；Yunnan University of Finance and Economics（云南财经大学）；Renmin University of China（中国人民大学）；Hong Kong Baptist University（香港浸会大学）
Popular measures of meta-analysis heterogeneity, such as $I^2$, cannot be considered measures of population heterogeneity since they are dependant on samples sizes within studies. The coefficient of variation (CV) recently introduced and defined to be the heterogeneity variance divided by the absolute value of the overall mean effect does not suffer such shortcomings. However, very large CV values can occur when the effect is small making interpretation difficult. The purpose of this paper is two-fold. Firstly, we consider variants of the CV that exist in the interval (0, 1] making interpretat…
Genetic association analyses often involve data from multiple potentially-heterogeneous subgroups. The expected amount of heterogeneity can vary from modest (e.g., a typical meta-analysis) to large (e.g., a strong gene--environment interaction). However, existing statistical tools are limited in their ability to address such heterogeneity. Indeed, most genetic association meta-analyses use a "fixed effects" analysis, which assumes no heterogeneity. Here we develop and apply Bayesian association methods to address this problem. These methods are easy to apply (in the simplest case, requiring on…
This article proposes different tests for treatment effect heterogeneity when the outcome of interest, typically a duration variable, may be right-censored. The proposed tests study whether a policy 1) has zero distributional (average) effect for all subpopulations defined by covariate values, and 2) has homogeneous average effect across different subpopulations. The proposed tests are based on two-step Kaplan-Meier integrals and do not rely on parametric distributional assumptions, shape restrictions, or on restricting the potential treatment effect heterogeneity across different subpopulatio…
Background: Pairwise and network meta-analyses using fixed effect and random effects models are commonly applied to synthesise evidence from randomised controlled trials. The models differ in their assumptions and the interpretation of the results. The model choice depends on the objective of the analysis and knowledge of the included studies. Fixed effect models are often used because there are too few studies with which to estimate the between-study standard deviation from the data alone. Objectives: The aim is to propose a framework for eliciting an informative prior distribution for the be…
A new method for clustered survival data]{A new method for clustered survival data: Estimation of treatment effect heterogeneity and variable selection
Common causal estimands include the average treatment effect (ATE), the average treatment effect of the treated (ATT), and the average treatment effect on the controls (ATC). Using augmented inverse probability weighting methods, parametric models are judiciously leveraged to yield doubly robust estimators, i.e., estimators that are consistent when at least one the parametric models is correctly specified. Three sources of uncertainty are associated when we evaluate these estimators and their variances, i.e., when we estimate the treatment and outcome regression models as well as the desired t…
Recent work has made important contributions in the development of causally-interpretable meta-analysis. These methods transport treatment effects estimated in a collection of randomized trials to a target population of interest. Ideally, estimates targeted toward a specific population are more interpretable and relevant to policy-makers and clinicians. However, between-study heterogeneity not arising from differences in the distribution of treatment effect modifiers can raise difficulties in synthesizing estimates across trials. The existence of such heterogeneity, including variations in tre…
Meta-analyses in orphan diseases and small populations generally face particular problems including small numbers of studies, small study sizes, and heterogeneity of results. However, the heterogeneity is difficult to estimate if only very few studies are included. Motivated by a systematic review in immunosuppression following liver transplantation in children we investigate the properties of a range of commonly used frequentist and Bayesian procedures in extensive simulation studies. Furthermore, the consequences for interval estimation of the common treatment effect in random effects meta-a…
In this paper, we study the problem of testing the mean vectors of high dimensional data in both one-sample and two-sample cases. The proposed testing procedures employ maximum-type statistics and the parametric bootstrap techniques to compute the critical values. Different from the existing tests that heavily rely on the structural conditions on the unknown covariance matrices, the proposed tests allow general covariance structures of the data and therefore enjoy wide scope of applicability in practice. To enhance powers of the tests against sparse alternatives, we further propose two-step pr…
A variety of problems in random-effects meta-analysis arise from the conventional $Q$ statistic, which uses estimated inverse-variance (IV) weights. In previous work on standardized mean difference and log-odds-ratio, we found superior performance with an estimator of the overall effect whose weights use only group-level sample sizes. The $Q$ statistic with those weights has the form proposed by DerSimonian and Kacker. The distribution of this $Q$ and the $Q$ with IV weights must generally be approximated. We investigate approximations for those distributions, as a basis for testing and estima…
We outline a Bayesian model-averaged meta-analysis for standardized mean differences in order to quantify evidence for both treatment effectiveness $δ$ and across-study heterogeneity $τ$. We construct four competing models by orthogonally combining two present-absent assumptions, one for the treatment effect and one for across-study heterogeneity. To inform the choice of prior distributions for the model parameters, we used 50% of the Cochrane Database of Systematic Reviews to specify rival prior distributions for $δ$ and $τ$. The relative predictive performance of the competing models and riv…
Simulations for estimation of heterogeneity variance and overall effect with constant and inverse-variance weights in meta-analysis of difference in standardized means (DSM)
Bayesian hierarchical bootstrap framework for causal subgroup estimation with a time-to-event outcome
Dalla Lana School of Public Health, University of Toronto（多伦多大学达拉·兰纳公共卫生学院）；Division of Gastroenterology, Hepatology Nutrition, The Hospital for Sick Children（多伦多加急儿童医院胃肠病、肝病与营养科）；University of Utah and Intermountain Healthcare Primary Children’s Hospital（犹他大学和盐湖城初级儿童医院）；Institute of Health Policy, Management and Evaluation, University of Toronto（多伦多大学卫生政策、管理与评估研究所）
Standard random-effects meta-analysis methods perform poorly when applied to few studies only. Such settings however are commonly encountered in practice. It is unclear, whether or to what extent small-sample-size behaviour can be improved by more sophisticated modeling. We consider several likelihood-based inference methods. Confidence intervals are based on normal or Student-t approximations. We extract an empirical data set of 40 meta-analyses from recent reviews published by the German Institute for Quality and Efficiency in Health Care (IQWiG). Methods are then compared empirically as wel…
Random-effects meta-analyses are very commonly used in medical statistics. Recent methodological developments include multivariate (multiple outcomes) and network (multiple treatments) meta-analysis. Here we provide a new model and corresponding estimation procedure for multivariate network meta-analysis, so that multiple outcomes and treatments can be included in a single analysis. Our new multivariate model is a direct extension of a univariate model for network meta-analysis that has recently been proposed. We allow two types of unknown variance parameters in our model, which represent betw…
