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Network meta-analysis is used to simultaneously compare multiple treatments in a single analysis. However, network meta-analyses may exhibit inconsistency, where direct and different forms of indirect evidence are not in agreement with each other, even after allowing for between-study heterogeneity. Models for network meta-analysis with random inconsistency effects have the dual aim of allowing for inconsistencies and estimating average treatment effects across the whole network. To date, two classical estimation methods for fitting this type of model have been developed: a method of moments that extends DerSimonian and Laird's univariate method and maximum likelihood estimation. However, the Paule and Mandel estimator is another recommended classical estimation method for univariate meta-analysis. In this paper, we extend the Paule and Mandel method so that it can be used to fit models for network meta-analysis with random inconsistency effects. We apply all three estimation methods to a variety of examples that have been used previously and we also examine a challenging new dataset that is highly heterogenous. We perform a simulation study based on this new example. We find that the proposed Paule and Mandel method performs satisfactorily and generally better than the previously proposed method of moments because it provides more accurate inferences. Furthermore, the Paule and Mandel method possesses some advantages over likelihood-based methods because it is both semiparametric and requires no convergence diagnostics. Although restricted maximum likelihood estimation remains the gold standard, the proposed methodology is a fully viable alternative to this and other estimation methods., (© 2017 The Authors. Research Synthesis Methods Published by John Wiley & Sons Ltd.)

The modified method for random-effects meta-analysis, usually attributed to Hartung and Knapp and also proposed by Sidik and Jonkman, is easy to implement and is becoming advocated for general use. Here, we examine a range of potential concerns about the widespread adoption of this method. Motivated by these issues, a variety of different conventions can be adopted when using the modified method in practice. We describe and investigate the use of a variety of these conventions using a new taxonomy of meta-analysis datasets. We conclude that the Hartung and Knapp modification may be a suitable replacement for the standard method. Despite this, analysts who advocate the modified method should be ready to defend its use against the possible objections to it that we present. We further recommend that the results from more conventional approaches should be used as sensitivity analyses when using the modified method. It has previously been suggested that a common-effect analysis should be used for this purpose but we suggest amending this recommendation and argue that a standard random-effects analysis should be used instead., (© 2017 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.)

One of the reasons for the popularity of meta-analysis is the notion that these analyses will possess more power to detect effects than individual studies. This is inevitably the case under a fixed-effect model. However, the inclusion of the between-study variance in the random-effects model, and the need to estimate this parameter, can have unfortunate implications for this power. We develop methods for assessing the power of random-effects meta-analyses, and the average power of the individual studies that contribute to meta-analyses, so that these powers can be compared. In addition to deriving new analytical results and methods, we apply our methods to 1991 meta-analyses taken from the Cochrane Database of Systematic Reviews to retrospectively calculate their powers. We find that, in practice, 5 or more studies are needed to reasonably consistently achieve powers from random-effects meta-analyses that are greater than the studies that contribute to them. Not only is statistical inference under the random-effects model challenging when there are very few studies but also less worthwhile in such cases. The assumption that meta-analysis will result in an increase in power is challenged by our findings., (Copyright © 2017 The Authors. Research Synthesis Methods Published by John Wiley & Sons Ltd.)

Background: Confidence intervals for the between study variance are useful in random-effects meta-analyses because they quantify the uncertainty in the corresponding point estimates. Methods for calculating these confidence intervals have been developed that are based on inverting hypothesis tests using generalised heterogeneity statistics. Whilst, under the random effects model, these new methods furnish confidence intervals with the correct coverage, the resulting intervals are usually very wide, making them uninformative., Methods: We discuss a simple strategy for obtaining 95 % confidence intervals for the between-study variance with a markedly reduced width, whilst retaining the nominal coverage probability. Specifically, we consider the possibility of using methods based on generalised heterogeneity statistics with unequal tail probabilities, where the tail probability used to compute the upper bound is greater than 2.5 %. This idea is assessed using four real examples and a variety of simulation studies. Supporting analytical results are also obtained., Results: Our results provide evidence that using unequal tail probabilities can result in shorter 95 % confidence intervals for the between-study variance. We also show some further results for a real example that illustrates how shorter confidence intervals for the between-study variance can be useful when performing sensitivity analyses for the average effect, which is usually the parameter of primary interest., Conclusions: We conclude that using unequal tail probabilities when computing 95 % confidence intervals for the between-study variance, when using methods based on generalised heterogeneity statistics, can result in shorter confidence intervals. We suggest that those who find the case for using unequal tail probabilities convincing should use the '1-4 % split', where greater tail probability is allocated to the upper confidence bound. The 'width-optimal' interval that we present deserves further investigation.

Network meta-analysis is becoming more popular as a way to compare multiple treatments simultaneously. Here, we develop a new estimation method for fitting models for network meta-analysis with random inconsistency effects. This method is an extension of the procedure originally proposed by DerSimonian and Laird. Our methodology allows for inconsistency within the network. The proposed procedure is semi-parametric, non-iterative, fast and highly accessible to applied researchers. The methodology is found to perform satisfactorily in a simulation study provided that the sample size is large enough and the extent of the inconsistency is not very severe. We apply our approach to two real examples., (© 2015 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.)

When combining results across related studies, a multivariate meta-analysis allows the joint synthesis of correlated effect estimates from multiple outcomes. Joint synthesis can improve efficiency over separate univariate syntheses, may reduce selective outcome reporting biases, and enables joint inferences across the outcomes. A common issue is that within-study correlations needed to fit the multivariate model are unknown from published reports. However, provision of individual participant data (IPD) allows them to be calculated directly. Here, we illustrate how to use IPD to estimate within-study correlations, using a joint linear regression for multiple continuous outcomes and bootstrapping methods for binary, survival and mixed outcomes. In a meta-analysis of 10 hypertension trials, we then show how these methods enable multivariate meta-analysis to address novel clinical questions about continuous, survival and binary outcomes; treatment-covariate interactions; adjusted risk/prognostic factor effects; longitudinal data; prognostic and multiparameter models; and multiple treatment comparisons. Both frequentist and Bayesian approaches are applied, with example software code provided to derive within-study correlations and to fit the models., (© 2014 The Authors. Research Synthesis Methods published by John Wiley & Sons, Ltd.)

The process of undertaking a meta-analysis involves a sequence of decisions, one of which is deciding which measure of treatment effect to use. In particular, for comparative binary data from randomised controlled trials, a wide variety of measures are available such as the odds ratio and the risk difference. It is often of interest to know whether important conclusions would have been substantively different if an alternative measure had been used. Here we develop a new type of sensitivity analysis that incorporates standard measures of treatment effect. Thus, rather than examining the implications of a variety of measures in an ad hoc manner, we can simultaneously examine an entire family of possibilities, including the odds ratio, the arcsine difference and the risk difference., (Copyright © 2012 John Wiley & Sons, Ltd.)

Methods for individual participant data meta-analysis of survival outcomes commonly focus on the hazard ratio as a measure of treatment effect. Recently, Siannis et al. (2010, Statistics in Medicine 29:3030-3045) proposed the use of percentile ratios as an alternative to hazard ratios. We describe a novel two-stage method for the meta-analysis of percentile ratios that avoids distributional assumptions at the study level., (Copyright © 2012 John Wiley & Sons, Ltd.)

Measures that quantify the impact of heterogeneity in univariate meta-analysis, including the very popular I(2) statistic, are now well established. Multivariate meta-analysis, where studies provide multiple outcomes that are pooled in a single analysis, is also becoming more commonly used. The question of how to quantify heterogeneity in the multivariate setting is therefore raised. It is the univariate R(2) statistic, the ratio of the variance of the estimated treatment effect under the random and fixed effects models, that generalises most naturally, so this statistic provides our basis. This statistic is then used to derive a multivariate analogue of I(2), which we call I(R)(2). We also provide a multivariate H(2) statistic, the ratio of a generalisation of Cochran's heterogeneity statistic and its associated degrees of freedom, with an accompanying generalisation of the usual I(2) statistic, I(H)(2). Our proposed heterogeneity statistics can be used alongside all the usual estimates and inferential procedures used in multivariate meta-analysis. We apply our methods to some real datasets and show how our statistics are equally appropriate in the context of multivariate meta-regression, where study level covariate effects are included in the model. Our heterogeneity statistics may be used when applying any procedure for fitting the multivariate random effects model., (Copyright © 2012 John Wiley & Sons, Ltd.)

Meta-analyses of clinical trials with continuous outcome data typically report the effect of an intervention as either a mean difference or a standardized mean difference. These results can be difficult to interpret, and re-expressing the effect size in terms of risk may facilitate understanding and applicability. We describe three methods for obtaining risks in such situations. Two of these methods involve direct transformation of a standardized mean difference to an odds ratio. The third entails estimation of risks in the two groups for a specific cut point. We extend this third approach to a completed meta-analysis by expressing the finding in the format of a single 'meta-study'. We compare the methods in two examples of meta-analyses and in a series of simulation studies that examine their properties in individual studies and in meta-analyses. These simulations show that the methods for expressing meta-analysis results from continuous outcomes are sensitive to underlying distributions, sample sizes and cut points but are remarkably robust to the presence of heterogeneity across studies. We offer suggestions of situations in which the various methods may safely be applied. In particular, if the underlying distribution is approximately normal, then estimation of risks for a specific cut point may be used for large sample sizes; direct transformations may be preferable otherwise. However, if the standard deviations in the two groups are notably different, then none of the methods have good properties. Furthermore, absolute risks are safely estimated after direct transformation only if they are in the region of 20% to 80%., (Copyright © 2011 John Wiley & Sons, Ltd.)

Objective: Interpretation of meta-analyses of published observational studies is problematic because of numerous sources of bias. We develop bias assessment, elicitation and adjustment methods, and apply them to a systematic review of longitudinal observational studies of the relationship between objectively measured physical activity and subsequent change in adiposity in children., Methods: We separated internal biases that reflect study quality from external biases that reflect generalizability to a target setting. Since published results were presented in different formats, these were all converted to correlation coefficients. Biases were considered as additive or proportional on the correlation scale. Opinions about the extent of each bias in each study, together with its uncertainty, were elicited in a formal process from quantitatively trained assessors for the internal biases and subject-matter specialists for the external biases. Bias-adjusted results for each study were combined across assessors using median pooling, and results combined across studies by random-effects meta-analysis., Results: Before adjusting for bias, the pooled correlation is difficult to interpret because the studies varied substantially in quality and design, and there was considerable heterogeneity. After adjusting for both the internal and external biases, the pooled correlation provides a meaningful quantitative summary of all available evidence, and the confidence interval incorporates the elicited uncertainties about the extent of the biases. In the adjusted meta-analysis, there was no apparent heterogeneity., Conclusion: This approach provides a viable method of bias adjustment for meta-analyses of observational studies, allowing the quantitative synthesis of evidence from otherwise incompatible studies. From the meta-analysis of longitudinal observational studies, we conclude that there is no evidence that physical activity is associated with gain in body fat.

Objective: To illustrate the potential and challenges of the simultaneous analysis of a network of trials, using as a case study the investigation of the relative effectiveness of four topical fluoride treatments and two control interventions (placebo and no treatment) in preventing dental caries in children., Study Design and Setting: We performed multiple-treatments meta-analysis within a Bayesian framework by synthesizing six Cochrane reviews. We explored the compatibility between direct and indirect evidence and adjusted the results using a meta-regression model to take into account differences in the year of randomization across studies., Results: The validity of our conclusions for the superiority of fluoride toothpaste as indicated from the initial network analysis using Bayesian methods was challenged when we adjusted for possible confounders. The network was dominated by studies comparing placebo with toothpaste, which were older and had been carried out in populations with higher baseline risk than studies involving other fluoride modalities., Conclusion: After adjusting for possible differences across studies, we did not find clear evidence that any topical fluoride modality is more effective than any other. Multiple-treatments meta-analysis methods allow for more detailed investigations than naïve methods in the analysis of indirect evidence on treatment effects.

Violation of Hardy-Weinberg equilibrium (HWE) can raise doubts about the validity of the conclusions from genetic association studies. However, for most currently performed gene-disease association studies, the available tests have low power to detect deviations from HWE. We consider this issue from a meta-analysis perspective, and suggest an approach to estimate the deviation and investigate its relationship with the observed genetic effects. Different degrees of deviation from HWE have previously been proposed as a potential source of heterogeneity across studies. We present a hierarchical meta-regression model that can be applied to test this assumption, using the concept of the fixation coefficient. We re-analyse seven meta-analyses to illustrate these methods. The uncertainty in the genetic effect estimate tended to increase once the fixation coefficient was taken into account. Dependence of the genetic effect size on the deviation from HWE was found in one meta-analysis, while in the other six examples, deviations from HWE did not clearly explain between-study heterogeneity in the genetic effects. The proposed hierarchical models allow the synthesis of data across gene-disease association studies with appropriate consideration of HWE issues., (2006 John Wiley & Sons, Ltd.)

Genetic association studies have the potential to advance our understanding of genotype-phenotype relationships, especially for common, complex diseases where other approaches, such as linkage, are less powerful. Unfortunately, many reported studies are not replicated or corroborated. This lack of reproducibility has many potential causes, relating to study design, sample size, and power issues, and from sources of true variability among populations. Genetic association studies can be considered as more similar to randomized trials than other types of observational epidemiological studies because of "Mendelian randomization" (Mendel's second law). The rationale and methodology for synthesizing randomized trials is highly relevant to the meta-analysis of genetic association studies. Nevertheless, there are a number of obstacles to overcome when performing such meta-analyses. In this review, the impacts of Type I error, lack of power, and publication and reporting biases are explored, and the role of multiple testing is discussed. A number of special features of association studies are especially pertinent, because they may lead to true variability among study results. These include population dynamics and structure, linkage disequilibrium, conformity to Hardy-Weinberg Equilibrium, bias, population stratification, statistical heterogeneity, epistatic and environmental interactions, and the choice of statistical models used in the analysis. Approaches to dealing with these issues are outlined. The supreme importance of complete and consistent study reporting and of making data readily available is also highlighted as a prerequisite for sound meta-analysis. We believe that systematic review and meta-analysis has an important role to play in understanding genetic association studies and should help us to separate the wheat from the chaff.