# Bayesian Analysis in Systematic Reviews and Meta-analyses

A **Bayesian** approach can be applied in a systematic review and meta-analysis so that you can incorporate prior beliefs. A systematic review is a comprehensive search of all available literature on a particular research question, whose findings become synthesized with meta-analysis to obtain an overall effect size. Bayesian analysis is a statistical approach that uses Bayes’ theorem to update the probability of a hypothesis as new evidence becomes available.

This post explores a recent study by Yamashina et al. (2022), which examined what factors affect heart rate-reducing treatment in a systematic review and meta-analysis using Bayesian analysis. It breaks down the value of this type of use of Bayesian analysis and gives other examples.

## What is Bayesian analysis?

Bayesian analysis is a type of statistical inference that views probabilities as the likelihood of an event happening. It differs from classical (frequentist) statistics in that probabilities are treated as degrees of belief, rather than frequencies. These beliefs can be updated as new data is acquired. The result is a posterior distribution. This represents the hypothesis’ updated probabilities based on the data and prior information.

In Bayesian analysis, you first set a prior belief, which is the probability of an event occurring before the experiment is conducted. This belief then gets updated through Bayes’ theorem to incorporate new evidence. The new evidence creates the posterior belief, which results in the posterior distribution.

Bayes’ theorem is summarized as:

...where *P(A)* is the probability of *A* occurring, *P(B)* is the probability of *B* occurring, *P(A|B)* is the probability of *A* given event *B*, and *P(B|A)* is the probability of *B* given event *A*. In Bayesian statistics, *A* is the parameters and *B* is the data.

Importantly, Bayesian analysis helps you calculate the ratio of evidence for one model (i.e., hypothesis) over another. You calculate this evidence using the posterior probability, quantified with the Bayes factor. The Bayes factor gives you a ratio of the two hypotheses’ likelihoods.

This, in turn, allows you get the *strength* of evidence of one hypothesis over the other, as opposed to point estimates. And that helps you take uncertainty into account.

Empirical Bayes is another type of Bayesian analysis. This statistical technique uses observed data to estimate a prior distribution’s parameters. It then combines this prior information (Bayesian) with a maximum likelihood estimation (frequentist), making it a hybrid of both Bayesian and frequentist statistical inference.

Empirical Bayes is often used when the amount of data is limited or when there’s a need to flexibly incorporate prior information into the analysis. It lets you incorporate data-driven information into the prior distribution. That can improve the precision and stability of Bayesian estimates.

## Why use Bayesian analysis in a systematic review?

Bayesian analyses are used in systematic reviews and meta-analyses because they let you incorporate prior beliefs, such as sample size or variability. They also let you take uncertainty into account better than some other approaches. Notably, with a heterogeneous data set, some outliers may lead to an incorrect estimate if performing a conventional meta-analysis.

Bayesian analysis can also be easily incorporated into a random-effects model. This is a statistical model in which the model parameters are random variables – the data within them are assumed to be drawn from different populations rather than from the same population, as in fixed-effect models. A random-effects model is typically used to pool the overall effect size of the articles you include in the meta-analysis.

Two other important measures in any meta-analysis are the pooled effect size, or the overall effect of interest, and between-study heterogeneity, which is how different studies in the meta-analysis may vary.

When a Bayesian approach is applied to a meta-analysis, prior information can be incorporated into these factors with a prior distribution. This lets you specify *a priori* how you think the true overall effect size and between-study heterogeneity might look.

The ability to incorporate a prior belief about the meta-analysis is useful if you have limited data or limited studies. For meta-analyses, this prior can be estimated from expert opinion, previous meta-analyses or articles, or directly from the data in your selected articles.

Bayesian analyses also help you quantify uncertainty better in your meta-analysis by giving a full posterior distribution of your overall effect size and heterogeneity measures. From there, you calculate the probability that the overall effect or measure of heterogeneity is smaller or larger than the value you initially thought in your prior.

Bayesian methods allow for calculating a credible interval (CrI). A Crl is similar to a confidence interval, but is used to quantify the uncertainty around an effect estimate. It’s interpreted as the range of values with a certain credibility (i.e., probability) of containing the true effect.

## See it in action...

Bayesian analysis is valuable in a meta-analysis for many reasons. Yamashina et al. (2022) applied a Bayesian approach in this recent systematic review and meta-analysis.

### Overview of the study

The study investigated what factors affect the effect of heart rate-reducing treatment on mortality in symptomatic heart failure patients with reduced ejection fraction (HFrEF). Specifically, the researchers wanted to evaluate how predictive factors (age, sex, and comorbidities) change the efficacy of heart rate-reducing treatment in HFrEF patients. They also evaluated how these predictive factors affect heart rate-reducing treatment efficacy in subgroups of patients with a heart rate reduction threshold of 10+ beats per minute.

To this aim, the authors conducted a systematic review and meta-analysis to find relevant articles. Eligibility criteria included randomized and placebo-controlled clinical trials of HFrEF patients aged 18+, published in English, and investigating the effect of heart rate therapies on heart rate and clinical outcomes.

They used a Bayesian random-effects model (empirical Bayes) to estimate the overall effect on a clinical outcome, as well as to estimate heterogeneity and posterior distributions. They also applied an empirical Bayes random-effects meta-regression to evaluate the predictive factors of heart rate-reducing therapies on clinical outcomes.

They presented effect sizes in terms of relative risk ratios. This ratio reflects the risk of a health event (in this case, HFrEF) in one group with the risk to another group.

The authors found 20 articles that met their inclusion criteria for meta-analytic synthesis, which included data from 23,564 patients. The Bayesian random-effects model revealed a risk ratio of 0.833 (95% Crl 0.776, 0.890) for the effect of heart rate reduction therapy on all-cause mortality; a risk ratio of 0.836 (95% CrI 0.769, 0.903) for the effect on cardiovascular mortality; and a risk ratio of 0.789 (95% CrI 0.729, 0.849) for the effect of the therapy on rehospitalization due to worsening heart rate (WHF).

These findings suggest that heart rate reduction therapy reduced the risk of all-cause mortality, cardiovascular mortality, and risk of rehospitalization due to WHF by 16%–20% compared with the placebo group.

Additionally, the empirical Bayes random-effects meta-regression showed the presence of type 2 diabetes (T2DM) was a predictor for an increased risk of all-cause mortality and cardiovascular-related mortality in patients treated with heart rate-reducing therapy. The presence of hypertension also showed an increased risk of all-cause mortality, but this was not statistically significant.

### What is meta-regression, and why was it used in this study?

Meta-regression is a statistical technique based on simple linear regression. In a linear regression model, you try to use the value of a variable *x* to predict the value of another variable* y*. In a meta-regression, this same logic applies to all the studies— *x* is the study characteristic (i.e., the year it was conducted), and, based on that information, you try to predict *y* (i.e., the study’s effect size).

Meta-regression lets researchers examine the relationship between the characteristics of the studies and the effect size they report. It aims to identify the factors that influence a particular outcome’s effect size, such as the study design, sample size, or type of intervention.

Yamashina et al. (2022) used meta-regression to examine whether certain medical factors (T2DM, hypertension, and ischemia) predicted clinical outcomes of heart rate-reducing therapies. Traditionally, this means the effect sizes were modeled as a function of these factors. Then, a regression analysis was performed to estimate the relationship between the study characteristics and the effect size.

Specifically, the authors here applied an empirical Bayes random-effects meta-regression. This technique combines a random-effects model with information from the observed data to estimate the distribution of treatment effects across studies. This method allows for the pooling of information across studies while still accounting for the heterogeneity of treatment effects across studies.

The meta-regression results give insight into the factors that influence the effect size and help researchers to understand the sources of heterogeneity. In this case, they showed that T2DM was a significant predictor of heart rate-reducing treatment on mortality, meaning the effect of this therapy is reduced on HFrEF patients with T2DM.

Applying a meta-regression in this way is useful for making predictions about the effect size in future studies. It can also help researchers identify areas where future work is needed and make recommendations on improving the quality of future studies.

### Why was Bayesian analysis used in this study?

This study likely used Bayesian analysis because it let the researchers incorporate prior knowledge from previous studies into the current analysis. Here, the authors obtained prior distributions directly from the data. This is useful when there’s limited or unreliable previous information, such as when there are only a handful of control trials on the topic.

By doing this, the authors increased the effect size’s robustness. That’s especially important in medical research because it can decrease the rate of false positives or false negatives. It also allowed for a more informed, comprehensive, and flexible analysis of the data. In turn, that led to a better understanding of the factors that influence the effect of heart rate-reducing treatment on mortality in HFrEF patients.

Bayesian analysis also lets you get information on the effects’ strength. Using the credibility intervals, the authors were able to get a range of how likely the results are vs. having one point estimate. This was important in this study because HFrEF can vary on a case-by-case basis.

The study also used Bayesian meta-regression models to incorporate factors such as T2DM and other clinical covariates. This gave a more nuanced understanding of the factors influencing the effect of heart rate-reducing therapies on individuals with HFrEF.

An alternative approach to the Bayesian meta-analysis might have been a traditional frequentist meta-analysis. In a frequentist meta-analysis, the effect size is estimated using traditional statistical methods, such as the inverse variance weighted mean of the effect size estimates from each individual study.

Frequentist meta-analysis’ main advantage is that it’s straightforward and easily understood by many other researchers. Frequentist meta-analyses also often have well-established methods for assessing heterogeneity and publication bias. These are important to consider in any meta-analysis because they could influence the overall effect estimate.

But with a frequentist meta-analysis, you won’t have much flexibility in modeling and prior information. A frequentist meta-analysis also only gives you a point estimate of the effect size rather than in terms of probabilities and uncertainties, which are more transparent and understandable.

## What are some other examples of Bayesian analysis in meta-analyses?

Bayesian analysis in meta-analyses is becoming more popular in medical research because of its statistical advantages – flexibility in modeling and a better understanding of uncertainty. It can also be easily incorporated into several other types of meta-analytic modeling, including hierarchical meta-analyses and network meta-analyses.

Leucht et al. (2017) examined how antipsychotic drug efficacy has changed over the years in patients with schizophrenia. The authors included placebo-controlled trials in patients who had worsening symptoms of schizophrenia, and they examined moderators of drug efficacy. They applied a Bayesian random-effects hierarchical model in their analysis when pooling the overall effect of interest. The hierarchical aspect allowed them to consider multiple sources of uncertainty, such as that of the effect sizes.

Singh et al. (2020) looked at how the choice of vasopressor drugs can affect hypotension during neuraxial anesthesia for Caesarean delivery. They conducted a systematic review of random-controlled trials and conducted a Bayesian network meta-analysis.

A network meta-analysis lets you incorporate both direct and indirect treatment comparisons into one model, or *network*. This is useful when there isn’t as much direct evidence between treatments or when multiple treatment comparisons can be made at the same time.

In all these studies, the Bayesian approach let the researchers integrate prior information, uncertainty, and heterogeneity in the data. This type of flexible modeling is valuable for medical research, where many different factors can influence treatment outcomes and when there isn’t always enough data on specific clinical events.

## Author Bio

Mariela Mihaylova was the main author of this post. She is a PhD researcher in cognitive sciences at the University of Geneva (Switzerland), specializing in memory and learning. Mariela is passionate about scientific communication and has written for a variety of educational blogs and websites. Her favorite topics include statistics, peer review, open science, and best research practices.

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