13.1 The Bayesian Hierarchical Model
To perform a Bayesian meta-analysis, we employ a so-called Bayesian hierarchical model (Röver 2017; Julian Higgins, Thompson, and Spiegelhalter 2009). We already briefly covered this type of model in the network meta-analysis chapter (Chapter 12.3.2).
In Chapter 10, we learned that every meta-analytic model comes with an inherent “multilevel”, and thus hierarchical, structure. On the first level, we have the individual participants. Data on this level usually reaches us in the form of calculated effect sizes (hattheta_k) of each study (k). We assume that participants are nested within studies on the second level and that the true effect sizes (theta_k) of different studies in our meta-analysis follow their own distribution. This distribution of true effects has a mean (mu) (the pooled “true” effect we want to estimate) and variance (tau^2), representing the between-study heterogeneity.
Let us try to formalize this. On the first level, we assume that the observed effect size (hattheta_k) reported in study (k) is an estimate of the “true” effect (theta_k) in this study. The observed effect (hattheta_k) deviates from (theta_k) due to the sampling error (epsilon_k). This is because we assume that (hattheta_k) was drawn (sampled) from the population underlying (k). This population can be seen as a distribution with mean (theta_k), the “true” effect of the study, and a variance (sigma^2).
In the second step, we assume that the true effect sizes (theta_k) themselves are only samples of an overarching distribution of true effect sizes. The mean of this distribution (mu) is the pooled effect size we want to estimate. The study-specific true effects (theta_k) deviate from (mu) because the overarching distribution also has a variance (tau^2), signifying the between-study heterogeneity. Taken together, this gives these two equations:
[begin{align} hattheta_k &sim mathcal{N}(theta_k,sigma_k^2) notag theta_k &sim mathcal{N}(mu,tau^2) tag{13.1} end{align}]Here, we use (mathcal{N}) to indicate that parameters to the left were sampled from a normal distribution. Some may argue that this is an unnecessarily strict assumption for the second equation (Julian Higgins, Thompson, and Spiegelhalter 2009), but the formulation as shown here is the one that is used most of the time. As covered before, the fixed-effect model is simply a special case of this model in which we assume that (tau^2 = 0), meaning that there is no between-study heterogeneity, and that all studies share one single true effect size (i.e. that for all studies (k): (theta_k = mu)).
We can also simplify this formula by using the marginal form:
[begin{equation} hattheta_k sim mathcal{N}(mu,sigma_k^2 + tau^2) tag{13.2} end{equation}]You may have already detected that these formulas look a lot like the ones we defined when discussing the random-effects (Chapter 4.1.2) and three-level meta-analysis (Chapter 10.1) model. Indeed, there is nothing particularly “Bayesian” about this formulation. This changes, however, when we add the following equations (Williams, Rast, and Bürkner 2018):
[begin{align} (mu, tau^2) &sim p(.) notag tau^2 &> 0 tag{13.3} end{align}]The first line is particularly important. It defines prior distributions for the parameters (mu) and (tau^2). This allows us to specify a priori how we think the true pooled effect size (mu) and between-study heterogeneity (tau^2) may look like, and how certain we are about this. The second equation adds the constraint that the between-study heterogeneity variance must be larger than zero. However, this formula does not specify the exact type of prior distribution used for (mu) and (tau^2). It only tells us that some prior distribution is assumed. We will cover reasonable, specific priors for Bayesian meta-analysis models in more detail later.
In the chapter on network meta-analysis, we already covered the method through which Bayesian approaches estimate model parameters. To recap, this involves using Markov Chain Monte Carlo-based sampling procedures, for example Gibbs sampling. In the {brms} package, which we will be using in this chapter, so-called No-U-Turn sampling (NUTS, Hoffman and Gelman 2014) is used64.
In the previous chapters, we primarily used the {meta} and {metafor} packages. These packages allow to conduct meta-analyses based on a non-Bayesian, or frequentist framework. Therefore, you might be wondering why one should start using Bayesian methods, given that we can already resort to such powerful tools using “conventional” approaches. The reason is that Bayesian meta-analysis comes with a few distinct advantages (Williams, Rast, and Bürkner 2018; McNeish 2016; Chung et al. 2013):
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Bayesian methods allow to directly model the uncertainty in our estimate of (tau^2). They can also be superior in estimating pooled effects, particularly when the number of included studies is small (which is very often the case in practice).
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Bayesian methods produce full posterior distributions for both (mu) and (tau^2). This allows to calculate the exact probability that (mu) or (tau^2) is smaller or larger than some specified value. This is in contrast to frequentist methods, where we only calculate confidence intervals. However, (95%) confidence intervals only state that, if data sampling were repeated many, many times, the true value of a population parameter (such as (mu) or (tau^2)) would fall into the range of the confidence interval in 95% of the samples. They do not tell us the probability that the true parameter lies between two specified values.
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Bayesian methods allow us to integrate prior knowledge and assumptions when calculating meta-analyses.