Evidently, our plain old meta-analysis model already has multilevel properties “built in.” It exhibits this property because we assume that participants are nested within studies in our data. The fixed-effects model can also be written in this way–we only have to set \(\zeta_k\) to zero. What we then obtain is a formula exactly identical to the one of the random-effects model from before. You might have already detected that we can substitute \(\theta_k\) in the first equation with its definition in the second equation.
If we do this, we get the following result: To make this more obvious, we have to split the equation into two formulas, where each corresponds to one of the two levels. Implicitly, this formula already describes the multilevel structure of our meta-analysis data. Let us go back to the random-effects model formula in equation (10.1). Traditionally, such type of data is called nested: one can say that participants are “nested” within studies. Pooling on level 2, the study level, however, has to be performed as part of the meta-analysis. When we conduct a meta-analysis, data on level 1 usually already reaches us in a “pooled” form (e.g. the authors of the paper provide us with the mean and standard deviation of their studied sample instead of the raw data). This overlying layer of studies constitutes our second level. These participants are part of larger units: the studies included in our meta-analysis. Figure 10.1 below symbolizes this structure.įigure 10.1: Multilevel structure of the conventional random-effects model.Īt the lowest level (level 1) we have the participants (or patients, specimens, etc., depending on the research field). The two error terms \(\epsilon_k\) and \(\zeta_k\) correspond with the two levels in our meta-analysis data: the “participant” level (level 1) and the “study” level (level 2). Therefore, our aim in the random-effects model is to estimate the mean of the distribution of true effect sizes, denoted with \(\mu\). This distribution is from where the individual true effect size \(\theta_k\) was drawn. This heterogeneity is caused by the fact that the true effect size of some study \(k\) is again only part of an overarching distribution of true effect sizes. The second one, \(\zeta_k\), represents the between-study heterogeneity. The first one is caused by the sampling error ( \(\epsilon_k\)) of individual studies, which leads effect size estimates to deviate from the true effect size \(\theta_k\). We discussed that the terms \(\epsilon_k\) and \(\zeta_k\) are introduced in a random-effects model because we assume that there are two sources of variability.
\hat\theta_k = \mu + \epsilon_k + \zeta_k To see why meta-analysis has multiple levels by default, let us go back to the formula of the random-effects model that we discussed in Chapter 4.1.2: As always, we will also have a look at how such models can be fitted in R using a hands-on example.ġ0.1 The Multilevel Nature of Meta-Analysis In this chapter, we will therefore first describe why meta-analysis naturally implies a multilevel structure of our data, and how we can extend a conventional meta-analysis to a three-level model. Such models are indeed somewhat different to the fixed-effect and random-effects model we already know. When people talk about multilevel meta-analysis, what they think of are three-level meta-analysis models.
In the chapters before, we have already fitted a multilevel (meta-analysis) model several times–without even knowing. Every meta-analytic model presupposes a multilevel structure of our data to pool results ( Pastor and Lazowski 2018). Describing a study as a “multilevel” meta-analysis insinuates that this is something special or extraordinary compared to “standard” meta-analyses. You probably wonder why we put the word “multilevel” into quotation marks. This first chapter deals with the topic of “multilevel” meta-analyses. Where useful, we will therefore also provide literature for further reading. Many of the following topics merit books of their own, and what we cover here should only be considered as a brief introduction. However, if you have worked yourself through the previous chapters of the guide, you should be more than well equipped to understand and implement the contents that are about to follow. We consider the following methods “advanced” because their mathematical underpinnings are more involved, or because of their implementation in R. With this background, we can now proceed to somewhat more advanced techniques. In the previous part of the guide, we took a deep dive into topics that we consider highly relevant for almost every meta-analysis.
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