A Method of Moments Estimator for Random Effect Multivariate Meta‐Analysis

Summary 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.     

Inference for meta‐analysis with a suspected temporal trend

There is sometimes a clear evidence of a strong secular trend in the treatment effect of studies included in a meta‐analysis. In such cases, estimating the present‐day treatment effect by meta‐regression is both reasonable and straightforward. We however consider the more common situation where a secular trend is suspected, but is not strongly statistically significant. Typically, this lack of significance is due to the small number of studies included in the analysis, so that a meta‐regression could give wild point estimates. We introduce an empirical Bayes meta‐analysis methodology, which shrinks the secular trend toward zero. This has the effect that treatment effects are adjusted for trend, but where the evidence from data is weak, wild results are not obtained. We explore several frequentist approaches and a fully Bayesian method is also implemented. A measure of trend analogous to I² is described, and exact significance tests for trend are given. Our preferred method is one based on penalized or h‐likelihood, which is computationally simple, and allows invariance of predictions to the (arbitrary) choice of time origin. We suggest that a trendless standard random effects meta‐analysis should routinely be supplemented with an h‐likelihood analysis as a sensitivity analysis.     

Multivariate meta‐analysis with an increasing number of parameters

Meta‐analysis can average estimates of multiple parameters, such as a treatment's effect on multiple outcomes, across studies. Univariate meta‐analysis (UVMA) considers each parameter individually, while multivariate meta‐analysis (MVMA) considers the parameters jointly and accounts for the correlation between their estimates. The performance of MVMA and UVMA has been extensively compared in scenarios with two parameters. Our objective is to compare the performance of MVMA and UVMA as the number of parameters, p, increases. Specifically, we show that (i) for fixed‐effect (FE) meta‐analysis, the benefit from using MVMA can substantially increase as p increases; (ii) for random effects (RE) meta‐analysis, the benefit from MVMA can increase as p increases, but the potential improvement is modest in the presence of high between‐study variability and the actual improvement is further reduced by the need to estimate an increasingly large between study covariance matrix; and (iii) when there is little to no between‐study variability, the loss of efficiency due to choosing RE MVMA over FE MVMA increases as p increases. We demonstrate these three features through theory, simulation, and a meta‐analysis of risk factors for non‐Hodgkin lymphoma.     

A population‐averaged approach to diagnostic test meta‐analysis

The meta‐analysis of diagnostic accuracy studies is often of interest in screening programs for many diseases. The typical summary statistics for studies chosen for a diagnostic accuracy meta‐analysis are often two dimensional: sensitivities and specificities. The common statistical analysis approach for the meta‐analysis of diagnostic studies is based on the bivariate generalized linear‐mixed model (BGLMM), which has study‐specific interpretations. In this article, we present a population‐averaged (PA) model using generalized estimating equations (GEE) for making inference on mean specificity and sensitivity of a diagnostic test in the population represented by the meta‐analytic studies. We also derive the marginalized counterparts of the regression parameters from the BGLMM. We illustrate the proposed PA approach through two dataset examples and compare performance of estimators of the marginal regression parameters from the PA model with those of the marginalized regression parameters from the BGLMM through Monte Carlo simulation studies. Overall, both marginalized BGLMM and GEE with sandwich standard errors maintained nominal 95% confidence interval coverage levels for mean specificity and mean sensitivity in meta‐analysis of 25 of more studies even under misspecification of the covariance structure of the bivariate positive test counts for diseased and nondiseased subjects.     

Comparison of One‐Step and Two‐Step Meta‐Analysis Models Using Individual Patient Data

The problem of combining information from separate trials is a key consideration when performing a meta‐analysis or planning a multicentre trial. Although there is a considerable journal literature on meta‐analysis based on individual patient data (IPD), i.e. a one‐step IPD meta‐analysis, versus analysis based on summary data, i.e. a two‐step IPD meta‐analysis, recent articles in the medical literature indicate that there is still confusion and uncertainty as to the validity of an analysis based on aggregate data. In this study, we address one of the central statistical issues by considering the estimation of a linear function of the mean, based on linear models for summary data and for IPD. The summary data from a trial is assumed to comprise the best linear unbiased estimator, or maximum likelihood estimator of the parameter, along with its covariance matrix. The setup, which allows for the presence of random effects and covariates in the model, is quite general and includes many of the commonly employed models, for example, linear models with fixed treatment effects and fixed or random trial effects. For this general model, we derive a condition under which the one‐step and two‐step IPD meta‐analysis estimators coincide, extending earlier work considerably. The implications of this result for the specific models mentioned above are illustrated in detail, both theoretically and in terms of two real data sets, and the roles of balance and heterogeneity are highlighted. Our analysis also shows that when covariates are present, which is typically the case, the two estimators coincide only under extra simplifying assumptions, which are somewhat unrealistic in practice.     

One‐Sided Test to Assess Correlation in Linear Logistic Models using Estimating Equations

A score‐type test is proposed for testing the hypothesis of independent binary random variables against positive correlation in linear logistic models with sparse data and cluster specific covariates. The test is developed for univariate and multivariate one‐sided alternatives. The main advantage of using score test is that it requires estimation of the model only under the null hypothesis, that in this case corresponds to the binomial maximum likelihood fit. The score‐type test is developed from a class of estimating equations with block‐diagonal structure in which the coefficients of the linear logistic model are estimated simultaneously with the correlation. The simplicity of the score test is illustrated in two particular examples.     

Nonindependence and sensitivity analyses in ecological and evolutionary meta‐analyses

Meta‐analysis is an important tool for synthesizing research on a variety of topics in ecology and evolution, including molecular ecology, but can be susceptible to nonindependence. Nonindependence can affect two major interrelated components of a meta‐analysis: (i) the calculation of effect size statistics and (ii) the estimation of overall meta‐analytic estimates and their uncertainty. While some solutions to nonindependence exist at the statistical analysis stages, there is little advice on what to do when complex analyses are not possible, or when studies with nonindependent experimental designs exist in the data. Here we argue that exploring the effects of procedural decisions in a meta‐analysis (e.g. inclusion of different quality data, choice of effect size) and statistical assumptions (e.g. assuming no phylogenetic covariance) using sensitivity analyses are extremely important in assessing the impact of nonindependence. Sensitivity analyses can provide greater confidence in results and highlight important limitations of empirical work (e.g. impact of study design on overall effects). Despite their importance, sensitivity analyses are seldom applied to problems of nonindependence. To encourage better practice for dealing with nonindependence in meta‐analytic studies, we present accessible examples demonstrating the impact that ignoring nonindependence can have on meta‐analytic estimates. We also provide pragmatic solutions for dealing with nonindependent study designs, and for analysing dependent effect sizes. Additionally, we offer reporting guidelines that will facilitate disclosure of the sources of nonindependence in meta‐analyses, leading to greater transparency and more robust conclusions.     

Rapid maximum likelihood ancestral state reconstruction of continuous characters: A rerooting‐free algorithm

Ancestral state reconstruction is a method used to study the evolutionary trajectories of quantitative characters on phylogenies. Although efficient methods for univariate ancestral state reconstruction under a Brownian motion model have been described for at least 25 years, to date no generalization has been described to allow more complex evolutionary models, such as multivariate trait evolution, non‐Brownian models, missing data, and within‐species variation. Furthermore, even for simple univariate Brownian motion models, most phylogenetic comparative R packages compute ancestral states via inefficient tree rerooting and full tree traversals at each tree node, making ancestral state reconstruction extremely time‐consuming for large phylogenies. Here, a computationally efficient method for fast maximum likelihood ancestral state reconstruction of continuous characters is described. The algorithm has linear complexity relative to the number of species and outperforms the fastest existing R implementations by several orders of magnitude. The described algorithm is capable of performing ancestral state reconstruction on a 1,000,000‐species phylogeny in fewer than 2 s using a standard laptop, whereas the next fastest R implementation would take several days to complete. The method is generalizable to more complex evolutionary models, such as phylogenetic regression, within‐species variation, non‐Brownian evolutionary models, and multivariate trait evolution. Because this method enables fast repeated computations on phylogenies of virtually any size, implementation of the described algorithm can drastically alleviate the computational burden of many otherwise prohibitively time‐consuming tasks requiring reconstruction of ancestral states, such as phylogenetic imputation of missing data, bootstrapping procedures, Expectation‐Maximization algorithms, and Bayesian estimation. The described ancestral state reconstruction algorithm is implemented in the Rphylopars functions anc.recon and phylopars.     

A novel meta‐analytical approach to improve systematic review of rates and patterns of microevolution

A classic topic in ecology and evolution, phenotypic microevolution of quantitative traits has received renewed attention in the face of rapid global environmental change. However, for plants, synthesis has been hampered by the limited use of standard metrics, which makes it difficult to systematize empirical information. Here we demonstrate the advantages of incorporating meta‐analysis tools to the review of microevolutionary rates. We perform a systematic survey of the plant literature on microevolution of quantitative traits over known periods of time, based on the scopus database. We quantify the amount of change by standard mean difference and develop a set of effect sizes to analyze such data. We show that applying meta‐analysis tools to a systematic literature review allows the extraction of a much larger volume of information than directly calculating microevolutionary rates. We also propose derived meta‐analysis effect sizes (h, LG and LR) which are appropriate for the study of evolutionary patterns, the first being similar to haldanes, the second and third allowing the application of a preexisting analytical framework for the inference of evolutionary mechanisms. This novel methodological development is applicable to the study of microevolution in any taxa. To pilot test it, we built an open‐access database of 1,711 microevolutionary rates of 152 angiosperm species from 128 studies documenting population changes in quantitative traits following an environmental novelty with a known elapsed time (<260 years). The performance of the metrics proposed (h, LG and LR) is similar to that of preexisting ones, and at the same time they bring the advantages of lower estimation bias and higher number of usable observations typical of meta‐analysis.     

Widespread evidence for incipient ecological speciation: a meta‐analysis of isolation‐by‐ecology

Ecologically mediated selection has increasingly become recognised as an important driver of speciation. The correlation between neutral genetic differentiation and environmental or phenotypic divergence among populations, to which we collectively refer to as isolation‐by‐ecology (IBE), is an indicator of ecological speciation. In a meta‐analysis framework, we determined the strength and commonality of IBE in nature. On the basis of 106 studies, we calculated a mean effect size of IBE with and without controlling for spatial autocorrelation among populations. Effect sizes were 0.34 (95% CI 0.24–0.42) and 0.26 (95% CI 0.13–0.37), respectively, indicating that an average of 5% of the neutral genetic differentiation among populations was explained purely by ecological contrast. Importantly, spatial autocorrelation reduced IBE correlations for environmental variables, but not for phenotypes. Through simulation, we showed how the influence of isolation‐by‐distance and spatial autocorrelation of ecological variables can result in false positives or underestimated correlations if not accounted for in the IBE model. Collectively, this meta‐analysis showed that ecologically induced genetic divergence is pervasive across time‐scales and taxa, and largely independent of the choice of molecular marker. We discuss the importance of these results in the context of adaptation and ecological speciation and suggest future research avenues.     

When and where plant‐soil feedback may promote plant coexistence: a meta‐analysis

Plant‐soil feedback (PSF) theory provides a powerful framework for understanding plant dynamics by integrating growth assays into predictions of whether soil communities stabilise plant–plant interactions. However, we lack a comprehensive view of the likelihood of feedback‐driven coexistence, partly because of a failure to analyse pairwise PSF, the metric directly linked to plant species coexistence. Here, we determine the relative importance of plant evolutionary history, traits, and environmental factors for coexistence through PSF using a meta‐analysis of 1038 pairwise PSF measures. Consistent with eco‐evolutionary predictions, feedback is more likely to mediate coexistence for pairs of plant species (1) associating with similar guilds of mycorrhizal fungi, (2) of increasing phylogenetic distance, and (3) interacting with native microbes. We also found evidence for a primary role of pathogens in feedback‐mediated coexistence. By combining results over several independent studies, our results confirm that PSF may play a key role in plant species coexistence, species invasion, and the phylogenetic diversification of plant communities.     

Formalizing the definition of meta‐analysis in Molecular Ecology

Meta‐analysis, the statistical synthesis of pertinent literature to develop evidence‐based conclusions, is relatively new to the field of molecular ecology, with the first meta‐analysis published in the journal Molecular Ecology in 2003 (Slate & Phua 2003). The goal of this article is to formalize the definition of meta‐analysis for the authors, editors, reviewers and readers of Molecular Ecology by completing a review of the meta‐analyses previously published in this journal. We also provide a brief overview of the many components required for meta‐analysis with a more specific discussion of the issues related to the field of molecular ecology, including the use and statistical considerations of Wright's F_ST and its related analogues as effect sizes in meta‐analysis. We performed a literature review to identify articles published as ‘meta‐analyses’ in Molecular Ecology, which were then evaluated by at least two reviewers. We specifically targeted Molecular Ecology publications because as a flagship journal in this field, meta‐analyses published in Molecular Ecology have the potential to set the standard for meta‐analyses in other journals. We found that while many of these reviewed articles were strong meta‐analyses, others failed to follow standard meta‐analytical techniques. One of these unsatisfactory meta‐analyses was in fact a secondary analysis. Other studies attempted meta‐analyses but lacked the fundamental statistics that are considered necessary for an effective and powerful meta‐analysis. By drawing attention to the inconsistency of studies labelled as meta‐analyses, we emphasize the importance of understanding the components of traditional meta‐analyses to fully embrace the strengths of quantitative data synthesis in the field of molecular ecology.     

Comparing six antihypertensive medication classes for preventing new‐onset diabetes mellitus among hypertensive patients: a network meta‐analysis

Hypertensive patients usually have a higher risk of new‐onset diabetes mellitus (NOD) which may trigger cardiovascular diseases. In this study, the effectiveness of six antihypertensive agents with respect to NOD prevention in hypertensive patients was assessed. A network meta‐analysis was conducted to compare the efficacy of specific drug classes. PubMed and Embase databases were searched for relevant articles. Results of the pairwised meta‐analysis were illustrated by odd ratios (OR) and a corresponding 95% confidence interval (CI). The probabilities and outcome of each treatment were ranked and summarized using the surface under the cumulative ranking curve (SUCRA).Twenty‐three trials were identified, including 224,832 patients with an average follow‐up period of 3.9 ± 1.0 years. The network meta‐analysis showed that patients treated by angiotensin II receptor blockers (ARBs) were associated with a lower risk of NOD compared to placebo (PCB), calcium channel blockers (CCBs) and β‐blockers, while diuretic appeared to be ineffective for NOD prevention. Network meta‐analysis results of specific drugs showed that enalapril exhibited distinct advantages and hydrochlorothiazide also exhibited a reliable performance. Our results suggested that both ARBs and angiotensin converse enzyme inhibitors (ACEIs), especially candesartan and enalapril, were preferable for NOD prevention in hypertensive patients. Hydrochlorothiazide also exhibited a reliable performance in comparison with other agents.     

The efficacy and safety of nutrient supplements in the treatment of mental disorders: a meta‐review of meta‐analyses of randomized controlled trials

The role of nutrition in mental health is becoming increasingly acknowledged. Along with dietary intake, nutrition can also be obtained from “nutrient supplements”, such as polyunsaturated fatty acids (PUFAs), vitamins, minerals, antioxidants, amino acids and pre/probiotic supplements. Recently, a large number of meta‐analyses have emerged examining nutrient supplements in the treatment of mental disorders. To produce a meta‐review of this top‐tier evidence, we identified, synthesized and appraised all meta‐analyses of randomized controlled trials (RCTs) reporting on the efficacy and safety of nutrient supplements in common and severe mental disorders. Our systematic search identified 33 meta‐analyses of placebo‐controlled RCTs, with primary analyses including outcome data from 10,951 individuals. The strongest evidence was found for PUFAs (particularly as eicosapentaenoic acid) as an adjunctive treatment for depression. More nascent evidence suggested that PUFAs may also be beneficial for attention‐deficit/hyperactivity disorder, whereas there was no evidence for schizophrenia. Folate‐based supplements were widely researched as adjunctive treatments for depression and schizophrenia, with positive effects from RCTs of high‐dose methylfolate in major depressive disorder. There was emergent evidence for N‐acetylcysteine as a useful adjunctive treatment in mood disorders and schizophrenia. All nutrient supplements had good safety profiles, with no evidence of serious adverse effects or contraindications with psychiatric medications. In conclusion, clinicians should be informed of the nutrient supplements with established efficacy for certain conditions (such as eicosapentaenoic acid in depression), but also made aware of those currently lacking evidentiary support. Future research should aim to determine which individuals may benefit most from evidence‐based supplements, to further elucidate the underlying mechanisms.     

Mixed‐Effects State‐Space Models for Analysis of Longitudinal Dynamic Systems

Summary The rapid development of new biotechnologies allows us to deeply understand biomedical dynamic systems in more detail and at a cellular level. Many of the subject‐specific biomedical systems can be described by a set of differential or difference equations that are similar to engineering dynamic systems. In this article, motivated by HIV dynamic studies, we propose a class of mixed‐effects state‐space models based on the longitudinal feature of dynamic systems. State‐space models with mixed‐effects components are very flexible in modeling the serial correlation of within‐subject observations and between‐subject variations. The Bayesian approach and the maximum likelihood method for standard mixed‐effects models and state‐space models are modified and investigated for estimating unknown parameters in the proposed models. In the Bayesian approach, full conditional distributions are derived and the Gibbs sampler is constructed to explore the posterior distributions. For the maximum likelihood method, we develop a Monte Carlo EM algorithm with a Gibbs sampler step to approximate the conditional expectations in the E‐step. Simulation studies are conducted to compare the two proposed methods. We apply the mixed‐effects state‐space model to a data set from an AIDS clinical trial to illustrate the proposed methodologies. The proposed models and methods may also have potential applications in other biomedical system analyses such as tumor dynamics in cancer research and genetic regulatory network modeling.     

Enabling regional management in a changing climate through Bayesian meta‐analysis of a large‐scale disturbance

Aim Quantifying and predicting change in large ecosystems is an important research objective for applied ecologists as human disturbance effects become increasingly evident at regional and global scales. However, studies used to make inferences about large‐scale change are frequently of uneven quality and few in number, having been undertaken to study local, rather than global, change. Our aim is to improve the quality of inferences that can be made in meta‐analyses of large‐scale disturbance by integrating studies of varying quality in a unified modelling framework that is informative for both local and regional management. Innovation Here we improve conventionally structured meta‐analysis methods by including imputation of unknown study variances and the use of Bayesian factor potentials. The approach is a coherent framework for integrating data of varying quality across multiple studies while facilitating belief statements about the uncertainty in parameter estimates and the probable outcome of future events. The approach is applied to a regional meta‐analysis of the effects of loss of coral cover on species richness and the abundance of coral‐dependent fishes in the western Indian Ocean (WIO) before and after a mass bleaching event in 1998. Main conclusions Our Bayesian approach to meta‐analysis provided greater precision of parameter estimates than conventional weighted linear regression meta‐analytical techniques, allowing us to integrate all available data from 66 available study locations in the WIO across multiple scales. The approach thereby: (1) estimated uncertainty in site‐level estimates of change, (2) provided a regional estimate for future change at any given site in the WIO, and (3) provided a probabilistic belief framework for future management of reef resources at both local and regional scales.     

Ignoring non‐English‐language studies may bias ecological meta‐analyses

Meta‐analysis plays a crucial role in syntheses of quantitative evidence in ecology and biodiversity conservation. The reliability of estimates in meta‐analyses strongly depends on unbiased sampling of primary studies. Although earlier studies have explored potential biases in ecological meta‐analyses, biases in reported statistical results and associated study characteristics published in different languages have never been tested in environmental sciences. We address this knowledge gap by systematically searching published meta‐analyses and comparing effect‐size estimates between English‐ and Japanese‐language studies included in existing meta‐analyses. Of the 40 published ecological meta‐analysis articles authored by those affiliated to Japanese institutions, we find that three meta‐analysis articles searched for studies in the two languages and involved sufficient numbers of English‐ and Japanese‐language studies, resulting in four eligible meta‐analyses (i.e., four meta‐analyses conducted in the three meta‐analysis articles). In two of the four, effect sizes differ significantly between the English‐ and Japanese‐language studies included in the meta‐analyses, causing considerable changes in overall mean effect sizes and even their direction when Japanese‐language studies are excluded. The observed differences in effect sizes are likely attributable to systematic differences in reported statistical results and associated study characteristics, particularly taxa and ecosystems, between English‐ and Japanese‐language studies. Despite being based on a small sample size, our findings suggest that ignoring non‐English‐language studies may bias outcomes of ecological meta‐analyses, due to systematic differences in study characteristics and effect‐size estimates between English‐ and non‐English languages. We provide a list of actions that meta‐analysts could take in the future to reduce the risk of language bias.