Between-study Standard Deviation Research Articles

How trace plots help interpret meta-analysis results.

The trace plot is seldom used in meta-analysis, yet it is a very informative plot. In this article, we define and illustrate what the trace plot is, and discuss why it is important. The Bayesian version of the plot combines the posterior density of , the between-study standard deviation, and the shrunken estimates of the study effects as a function of . With a small or moderate number of studies, is not estimated with much precision, and parameter estimates and shrunken study effect estimates can vary widely depending on the correct value of . The trace plot allows visualization of the sensitivity to along with a plot that shows which values of are plausible and which are implausible. A comparable frequentist or empirical Bayes version provides similar results. The concepts are illustrated using examples in meta-analysis and meta-regression; implementation in R is facilitated in a Bayesian or frequentist framework using the bayesmeta and metafor packages, respectively.

Dealing with censoring in a network meta-analysis of time-to-event data

Background The Health Technology Assessment agencies typically require an economic evaluation considering a lifetime horizon for interventions affecting survival. However, survival data are often censored and are typically analyzed assuming the censoring mechanism independent of the event process. This assumption may lead to biased results when the censoring mechanism is informative. Methods We propose a flexible approach to jointly model the participants experiencing an event and censored participants by incorporating the pattern-mixture (PM) model in the fractional polynomial (FP) model within the network meta-analysis (NMA) framework. We introduce the informative censoring hazard ratio parameter that quantifies the departure from the censored at random assumption. The FP-PM model is exemplified in an NMA of the overall survival from non-small cell lung carcinoma studies using Bayesian methods. Results The results on hazard ratio and survival from the FP-PM model are similar to those from the FP model. However, the posterior standard deviation of the hazard ratio is slightly greater when censored data are modeled because the uncertainty induced by censoring is naturally accounted for in the FP-PM model. The between-study standard deviation is almost identical in both models due to the low censoring rate across the studies. At the end of the corresponding studies, the informative censoring hazard ratio demonstrated a possible departure from the censored at random assumption for gefitinib and best supportive care. Conclusions The proposed method offers a comprehensive sensitivity analysis framework to examine the robustness of the NMA results to clinically plausible censoring scenarios.

Effects of Ashwagandha (Withania somnifera) on Physical Performance: Systematic Review and Bayesian Meta-Analysis.

Ashwagandha (Withania somnifera) is considered a potent adaptogen and anti-stress agent that could have some potential to improve physical performance. This preferred reporting items for systematic reviews and meta-analyses (PRISMA)-based comprehensive systematic review and Bayesian meta-analysis aimed to evaluate clinical trials up to 2020 from PubMed, ScienceDirect, and Google Scholar databases regarding the effect of Ashwagandha supplementation on physical performance in healthy individuals. Besides implementing estimation statistics analysis, we developed Bayesian hierarchical models for a pre-specified subgroup meta-analysis on strength/power, cardiorespiratory fitness and fatigue/recovery variables. A total of 13 studies met the requirements of this systematic review, although only 12 were included in the quantitative analysis. A low-to-moderate overall risk of bias of the trials included in this study was detected. All Bayesian hierarchical models converged to a target distribution (Ȓ = 1) for both meta-analytic effect size (μ) and between-study standard deviation (τ). The meta-analytic approaches of the included studies revealed that Ashwagandha supplementation was more efficacious than placebo for improving variables related to physical performance in healthy men and female. In fact, the Bayesian models showed that future interventions might be at least in some way beneficial on the analyzed outcomes considering the 95% credible intervals for the meta-analytic effect size. Several practical applications and future directions are discussed, although more comparable studies are needed in exercise training, and athletic populations are needed to derive a more stable estimate of the true underlying effect.

On the treatment effect heterogeneity of antidepressants in major depression: A Bayesian meta-analysis and simulation study.

The average treatment effect of antidepressants in major depression was found to be about 2 points on the 17-item Hamilton Depression Rating Scale, which lies below clinical relevance. Here, we searched for evidence of a relevant treatment effect heterogeneity that could justify the usage of antidepressants despite their low average treatment effect. Bayesian meta-analysis of 169 randomized, controlled trials including 58,687 patients. We considered the effect sizes log variability ratio (lnVR) and log coefficient of variation ratio (lnCVR) to analyze the difference in variability of active and placebo response. We used Bayesian random-effects meta-analyses (REMA) for lnVR and lnCVR and fitted a random-effects meta-regression (REMR) model to estimate the treatment effect variability between antidepressants and placebo. The variability ratio was found to be very close to 1 in the best fitting models (REMR: 95% highest density interval (HDI) [0.98, 1.02], REMA: 95% HDI [1.00, 1.02]). The between-study standard deviation τ under the REMA with respect to lnVR was found to be low (95% HDI [0.00, 0.02]). Simulations showed that a large treatment effect heterogeneity is only compatible with the data if a strong correlation between placebo response and individual treatment effect is assumed. The published data from RCTs on antidepressants for the treatment of major depression is compatible with a near-constant treatment effect. Although it is impossible to rule out a substantial treatment effect heterogeneity, its existence seems rather unlikely. Since the average treatment effect of antidepressants falls short of clinical relevance, the current prescribing practice should be re-evaluated.

EVidence Synthesis For Health Technology Assessment With Limited Studies

Pairwise and network meta-analyses using fixed effect and random effects models are commonly applied to synthesise evidence from randomised controlled trials in health technology assessment. 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. When heterogeneity is expected and inferences are required beyond the sample of studies available, then an analysis using a fixed effect model will underestimate uncertainty about the treatment effect. This research aims to provide a Bayesian approach to overcome the problem of too few studies to use a random effects model, by proposing a framework for eliciting an informative prior distribution for the between-study standard deviation in a random effects model to genuinely represent heterogeneity. We developed an elicitation method using external information such as empirical evidence and experts’ beliefs on the ‘range’ of treatment effects in order to infer the prior distribution for the between-study standard deviation. We also developed the method to be implemented in R using the SHELF package. The three-stage elicitation approach allows uncertainty to be represented by a genuine prior distribution to avoid making misleading inferences. It is flexible to what judgments an expert can provide, and is applicable to all types of outcome measure for which we can construct a treatment effect on an additive scale. The choice between using a fixed effect or random effects meta-analysis model depends on the objective of the analysis and knowledge of the included studies, but not on the number of available studies. Our elicitation framework captures external evidence about heterogeneity and overcomes the often implausible assumption that studies are estimating the same treatment effect, thereby improving the quality of inferences in decision making.

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Bayesian estimation in random effects meta-analysis using a non-informative prior.

Pooling information from multiple, independent studies (meta-analysis) adds great value to medical research. Random effects models are widely used for this purpose. However, there are many different ways of estimating model parameters, and the choice of estimation procedure may be influential upon the conclusions of the meta-analysis. In this paper, we describe a recently proposed Bayesian estimation procedure and compare it with a profile likelihood method and with the DerSimonian-Laird and Mandel-Paule estimators including the Knapp-Hartung correction. The Bayesian procedure uses a non-informative prior for the overall mean and the between-study standard deviation that is determined by the Berger and Bernardo reference prior principle. The comparison of these procedures focuses on the frequentist properties of interval estimates for the overall mean. The results of our simulation study reveal that the Bayesian approach is a promising alternative producing more accurate interval estimates than those three conventional procedures for meta-analysis. The Bayesian procedure is also illustrated using three examples of meta-analysis involving real data. Copyright © 2016 John Wiley & Sons, Ltd.

Commentary: Priors, parameters, and probability: a bayesian perspective on sensitivity analysis.

Commentary: Priors, parameters, and probability: a bayesian perspective on sensitivity analysis.

Avoiding zero between‐study variance estimates in random‐effects meta‐analysis

Fixed-effects meta-analysis has been criticized because the assumption of homogeneity is often unrealistic and can result in underestimation of parameter uncertainty. Random-effects meta-analysis and meta-regression are therefore typically used to accommodate explained and unexplained between-study variability. However, it is not unusual to obtain a boundary estimate of zero for the (residual) between-study standard deviation, resulting in fixed-effects estimates of the other parameters and their standard errors. To avoid such boundary estimates, we suggest using Bayes modal (BM) estimation with a gamma prior on the between-study standard deviation. When no prior information is available regarding the magnitude of the between-study standard deviation, a weakly informative default prior can be used (with shape parameter 2 and rate parameter close to 0) that produces positive estimates but does not overrule the data, leading to only a small decrease in the log likelihood from its maximum. We review the most commonly used estimation methods for meta-analysis and meta-regression including classical and Bayesian methods and apply these methods, as well as our BM estimator, to real datasets. We then perform simulations to compare BM estimation with the other methods and find that BM estimation performs well by (i) avoiding boundary estimates; (ii) having smaller root mean squared error for the between-study standard deviation; and (iii) better coverage for the overall effects than the other methods when the true model has at least a small or moderate amount of unexplained heterogeneity.

The Choice of a Noninformative Prior on Between-Study Variance Strongly Affects Predictions of Future Treatment Effect

Bayesian random-effects meta-analyses require the analyst to specify the prior distribution for between-study variance of the treatment effect. We assessed the sensitivity of prediction and other outputs of the meta-analysis to the choice of this prior. We reanalyzed 7 published meta-analyses (5-14 trials) with rare (event rates <5%), moderate (15%-50%), and frequent binary outcomes (>50%). We examined 10 noninformative priors: inverse gamma on between-study variance (τ (2)), 2 uniforms on each of the between-study standard deviation (τ) and τ (2), uniform shrinkage on τ (2), DuMouchel shrinkage on τ, half-normal on τ (2), and half-normal priors on τ with large and small variances. For each analysis, we calculated the posterior distributions for τ, the population treatment effect in current studies, and the predicted treatment effect in a future study. We assessed goodness of fit using total residual deviance, the deviance information criterion, and predictive deviance (by cross-validations). According to total residual deviance, the best-fitting priors were uniform on τ (2). According to predictive deviance, half-normal on τ (2) and the shrinkage priors were optimal. Across analyses with the 10 priors, there were no important differences in the posteriors for the population treatment effect, but there were substantial differences in the posteriors for τ and predictions. The priors that fitted best according to predictive deviance resulted in less uncertainty around predictions of future treatment effect. In this sample of Bayesian meta-analyses with binary outcomes, the choice of noninformative prior for between-study variance affected model fit and the predictions of future treatment effect. When the predictive distribution is of interest, we highly recommend examination of multiple prior distributions for between-study variance, especially the half-normal on τ (2) and the shrinkage priors.

Growth Hormone Receptor Polymorphism and Growth Hormone Therapy Response in Children: A Bayesian Meta-Analysis

Recombinant human growth hormone (rhGH) therapy is used in the long-term treatment of children with growth disorders, but there is considerable treatment response variability. The exon 3-deleted growth hormone receptor polymorphism (GHR(d3)) may account for some of this variability. The authors performed a systematic review (to April 2011), including investigator-only data, to quantify the effects of the GHR(fl-d3) and GHR(d3-d3) genotypes on rhGH therapy response and used a recently established Bayesian inheritance model-free approach to meta-analyze the data. The primary outcome was the 1-year change-in-height standard-deviation score for the 2 genotypes. Eighteen data sets from 12 studies (1,527 children) were included. After several prior assumptions were tested, the most appropriate inheritance model was codominant (posterior probability = 0.93). Compared with noncarriers, carriers had median differences in 1-year change-in-height standard-deviation score of 0.09 (95% credible interval (CrI): 0.01, 0.17) for GHR(fl-d3) and of 0.14 (95% CrI: 0.02, 0.26) for GHR(d3-d3). However, the between-study standard deviation of 0.18 (95% CrI: 0.10, 0.33) was considerable. The authors tested by meta-regression for potential modifiers and found no substantial influence. They conclude that 1) the GHR(d3) polymorphism inheritance is codominant, contrasting with previous reports; 2) GHR(d3) genotypes account for modest increases in rhGH effects in children; and 3) considerable unexplained variability in responsiveness remains.

Bayesian implementation of a genetic model‐free approach to the meta‐analysis of genetic association studies

A genetic model-free method for the meta-analysis of genetic association studies is described that estimates the mode of inheritance from the data rather than assuming that it is known. For a bi-allelic polymorphism, with G as risk allele and g as wild-type, the genetic model depends on the ratio of the two log odds ratios, lambda = log OR(Gg)/log OR(GG), where OR(GG) compares GG with gg and OR(Gg) compares Gg with gg. Modelling log OR(GG) as a random effect creates a hierarchical model that can be implemented within a Bayesian framework. In Bayesian modelling, vague prior distributions have to be specified for all unknown parameters when no external information is available. When the data are sparse even supposedly vague prior distributions may have an influence on the posterior estimates. We investigate the impact of different vague prior distributions for the between-study standard deviation of log OR(GG) and for lambda, by considering three published meta-analyses and associated simulations. Our results show that depending on the characteristics of the meta-analysis the results may indeed be sensitive to the choice of vague prior distribution for either parameter. Genetic association studies usually use a case-control design that should be analysed by the corresponding retrospective likelihood. However, under some circumstances the prospective likelihood has been shown to produce identical results and it is usually preferred for its simplicity. In our meta-analyses the two likelihoods give very similar results.