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• Meta-analyses enable synthesis of results from globally distributed experiments to draw general conclusions about the impacts of global change factors on ecosystem function. Traditional meta-analyses, however, are challenged by the complexity and diversity of experimental results. We illustrate how several key issues can be addressed by a multivariate, hierarchical Bayesian meta-analysis (MHBM) approach applied to information extracted from published studies. • We applied an MHBM to log-response ratios for aboveground biomass (AB, n = 300), belowground biomass (BB, n = 205) and soil CO₂ exchange (SCE, n = 544), representing 100 studies. The MHBM accounted for study duration, climate effects and covariation among the AB, BB and SCE responses to elevated CO₂ (eCO₂) and/or warming. • The MHBM revealed significant among-study covariation in the AB and BB responses to experimental treatments. The MHBM imputed missing duration (4.2%) and climate (6%) data, and revealed that climate context governs how eCO₂ and warming impact ecosystem function. Predictions identified biomes that may be particularly sensitive to eCO₂ or warming, but that are under-represented in global change experiments. • The MHBM approach offers a flexible and powerful tool for synthesising disparate experimental results reported across multiple studies, sites and response variables.

Meta‐analysis is a powerful approach to combine evidence from multiple studies to make inference about one or more parameters of interest, such as regression coefficients. The validity of the fixed effect model meta‐analysis depends on the underlying assumption that all studies in the meta‐analysis share the same effect size. In the presence of heterogeneity, the fixed effect model incorrectly ignores the between‐study variance and may yield false positive results. The random effect model takes into account both within‐study and between‐study variances. It is more conservative than the fixed effect model and should be favored in the presence of heterogeneity. In this paper, we develop a noniterative method of moments estimator for the between‐study covariance matrix in the random effect model multivariate meta‐analysis. To our knowledge, it is the first such method of moments estimator in the matrix form. We show that our estimator is a multivariate extension of DerSimonian and Laird’s univariate method of moments estimator, and it is invariant to linear transformations. In the simulation study, our method performs well when compared to existing random effect model multivariate meta‐analysis approaches. We also apply our method in the analysis of a real data example.

Multivariate meta-analysis is gaining prominence in evidence synthesis research because it enables simultaneous synthesis of multiple correlated outcome data, and random-effects models have generally been used for addressing between-studies heterogeneities. However, coverage probabilities of confidence regions or intervals for standard inference methods for randomeffects models (eg, restricted maximum likelihood estimation) cannot retain their nominal confidence levels in general, especially when the number of synthesized studies is small because their validities depend on large sample approximations. In this article, we provide permutation-based inference methods that enable exact joint inferences for average outcome measures without large sample approximations. We also provide accurate marginal inference methods under general settings of multivariate meta-analyses. We propose effective approaches for permutation inferences using optimal weighting based on the efficient score statistic. The effectiveness of the proposed methods is illustrated via applications to bivariate meta-analyses of diagnostic accuracy studies for airway eosinophilia in asthma and a network meta-analysis for antihypertensive drugs on incident diabetes, as well as through simulation experiments. In numerical evaluations performed via simulations, our methods generally provided accurate confidence regions or intervals under a broad range of settings, whereas the current standard inference methods exhibited serious undercoverage properties.

Using a meta-analysis of 142 experimental observations, this work examines the influence of different research design and outcome measures on extremeness aversion (i.e., the propensity to avoid extreme options in choice situations). The results indicate that extremeness aversion is a robust phenomenon: middle options are significantly more often selected than other options. However, the strength of this behavioral effect exhibits substantial variation (up to three times the average magnitude) across methodological decisions: employing price-quality tradeoffs, nondurable categories, or binary-trinary choice-set comparisons tend to reduce the probability of extremeness aversion among consumers, whereas using a larger number of tradeoff dimensions, non-numeric attributes, high-quality extensions, or utilitarian products increase its likelihood. Because extremeness aversion has been assessed using three different measurement paradigms (absolute-share changes, relative-share shifts, and middle-option proportions), we discuss their characteristics and investigate their degree of agreement. We find that the three measures can lead to rather different effect magnitudes and even contradictory conclusions about the effect of moderators.

Univariate meta-analysis concerns a single outcome of interest measured across a number of independent studies. However, many research studies will have also measured secondary outcomes. Multivariate meta-analysis allows us to take these secondary outcomes into account and can also include studies where the primary outcome is missing. We define the efficiency E as the variance of the overall estimate from a multivariate meta-analysis relative to the variance of the overall estimate from a univariate meta-analysis. The extra information gained from a multivariate meta-analysis of n studies is then similar to the extra information gained if a univariate meta-analysis of the primary effect had a further n(1-E)/E studies. The variance contribution of a study's secondary outcomes (its borrowing of strength) can be thought of as a contrast between the variance matrix of the outcomes in that study and the set of variance matrices of all the studies in the meta-analysis. In the bivariate case this is given a simple graphical interpretation as the borrowing-of-strength plot. We discuss how these findings can also be used in the context of random-effects meta-analysis. Our discussion is motivated by a published meta-analysis of 10 antihypertension clinical trials.

Multivariate meta-analysis allows the joint synthesis of summary estimates from multiple end points and accounts for their within-study and between-study correlation. Yet practitioners usually meta-analyse each end point independently. I examine the role of within-study correlation in multivariate meta-analysis, to elicit the consequences of ignoring it. Using analytic reasoning and a simulation study, the within-study correlation is shown to influence the 'borrowing of strength' across end points, and wrongly ignoring it gives meta-analysis results with generally inferior statistical properties; for example, on average it increases the mean-square error and standard error of pooled estimates, and for non-ignorable missing data it increases their bias. The influence of within-study correlation is only negligible when the within-study variation is small relative to the between-study variation, or when very small differences exist across studies in the within-study covariance matrices. The findings are demonstrated by applied examples within medicine, dentistry and education. Meta-analysts are thus encouraged to account for the correlation between end points. To facilitate this, I conclude by reviewing options for multivariate meta-analysis when within-study correlations are unknown; these include obtaining individual patient data, using external information, performing sensitivity analyses and using alternatively parameterized models.

Many randomized controlled trials report more than one primary outcome. As a result, multivariate meta-analytic methods for the assimilation of treatment effects in systematic reviews of randomized controlled trials have received increasing attention in the literature. These methods show promise with respect to bias reduction and efficiency gain compared with univariate meta-analysis. However, most methods for multivariate meta-analysis have focused on pairwise treatment comparisons (i.e. when the number of treatments is 2). Current methods for mixed treatment comparisons meta-analysis (i.e. when the number of treatments is more than 2) have focused on univariate or, very recently, bivariate outcomes. To broaden their application, we propose a framework for mixed treatment comparisons meta-analysis of multivariate (two or more) outcomes where the correlations between multivariate outcomes within and between studies are accounted for through copulas, and the joint modelling of multivariate random effects respectively. We consider a Bayesian hierarchical model using Markov chain Monte Carlo methods for estimation. An important feature of the framework proposed is that it allows for borrowing of information across correlated outcomes. We show via simulation that our approach reduces the effect of outcome reporting bias in a variety of missing outcome scenarios. We apply the method to a systematic review of randomized controlled trials of pharmacological treatments for alcohol dependence, which tends to report multiple outcomes potentially subject to outcome reporting bias.

Conventional meta-analytic procedures assume that effect sizes are independent. When effect sizes are not independent, conclusions based on these conventional procedures can be misleading or even wrong. Traditional approaches, such as averaging the effect sizes and selecting one effect size per study, are usually used to avoid the dependence of the effect sizes. These ad-hoc approaches, however, may lead to missed opportunities to utilize all available data to address the relevant research questions. Both multivariate meta-analysis and three-level meta-analysis have been proposed to handle non-independent effect sizes. This paper gives a brief introduction to these new techniques for applied researchers. The first objective is to highlight the benefits of using these methods to address non-independent effect sizes. The second objective is to illustrate how to apply these techniques with real data in R and Mplus. Researchers may modify the sample R and Mplus code to fit their data.

Meta-analysis has changed the way researchers conduct literature reviews not only in medical and social sciences but also in biological sciences. Meta-analysis in biological sciences, especially in ecology and evolution (which we refer to as ‘biological’ meta-analysis) faces somewhat different methodological problems from its counterparts in medical and social sciences, where meta-analytic techniques were originally developed. The main reason for such differences is that biological meta-analysis often integrates complex data composed of multiple strata with, for example, different measurements and a variety of species. Here, we review methodological issues and advancements in biological meta-analysis, focusing on three topics: (1) non-independence arising from multiple effect sizes obtained in single studies and from phylogenetic relatedness, (2) detecting and accounting for heterogeneity, and (3) identifying publication bias and measuring its impact. We show how the marriage between mixed-effects (hierarchical/multilevel) models and phylogenetic comparative methods has resolved most of the issues under discussion. Furthermore, we introduce the concept of across-study and within-study meta-analysis, and propose how the use of within-study meta-analysis can improve many empirical studies typical of ecology and evolution.

Meta-analysis is a quantitative way of synthesizing results from multiple studies to obtain reliable evidence of an intervention or phenomenon. Indeed, an increasing number of meta-analyses are conducted in environmental sciences, and resulting meta-analytic evidence is often used in environmental policies and decision-making. We conducted a survey of recent meta-analyses in environmental sciences and found poor standards of current meta-analytic practice and reporting. For example, only ~ 40% of the 73 reviewed meta-analyses reported heterogeneity (variation among effect sizes beyond sampling error), and publication bias was assessed in fewer than half. Furthermore, although almost all the meta-analyses had multiple effect sizes originating from the same studies, non-independence among effect sizes was considered in only half of the meta-analyses. To improve the implementation of meta-analysis in environmental sciences, we here outline practical guidance for conducting a meta-analysis in environmental sciences. We describe the key concepts of effect size and meta-analysis and detail procedures for fitting multilevel meta-analysis and meta-regression models and performing associated publication bias tests. We demonstrate a clear need for environmental scientists to embrace multilevel meta-analytic models, which explicitly model dependence among effect sizes, rather than the commonly used random-effects models. Further, we discuss how reporting and visual presentations of meta-analytic results can be much improved by following reporting guidelines such as PRISMA-EcoEvo (Preferred Reporting Items for Systematic Reviews and Meta-Analyses for Ecology and Evolutionary Biology). This paper, along with the accompanying online tutorial, serves as a practical guide on conducting a complete set of meta-analytic procedures (i.e., meta-analysis, heterogeneity quantification, meta-regression, publication bias tests and sensitivity analysis) and also as a gateway to more advanced, yet appropriate, methods.