Introduction to Multi-level Meta-Regression

Meta-analysis often reveals substantial heterogeneity among effect estimates included in the analysis (Senior et al., 2016). While many meta-analysts are interested in ‘overall mean effects’ we need to couch these effects in context by reporting on their variability and work hard to understand what factors drive effect variability and why (Lajeunesse, 2010; Noble et al., 2022). For meta-analyses in comparative physiology, explaining variation in effects should probably be the main goal of every analysis. Sampling variance often explains only a little amount of the total heterogeneity, so it’s important that we think hard, a priori, about what factors are likely to explain effect size variation.

As already indicated, some variation in effects could be explained by nuisance heterogeneity, which may or may not be of interest. Other sources of variability could be methodological (Noble et al., 2022). Do different chemicals that result in oxidative stress induce stronger or weaker responses? Understanding how methodological choices impact upon effect sizes can change how research is done in a field.

While methodological moderators are important biological variables are usually our main interest. We might be interested in understanding how much variation among species exist or how different life-histories, body sizes, ecological niches, feeding habits or thermal strategies (e.g., ectotherms vs endotherms) differ in their response to some treatment or the association between two variables.

Exploring population-level / average changes in effect size magnitude (and direction) as a function of ‘moderators’ (what are typically called predictors or fixed effects in other literature) is a critical aspects of meta-analysis in ecology and evolution. The models used to test how average effects change as some function of a moderator is called meta-regression.

Extending our Multi-level Meta-analytic Model to a Multi-level Meta-regression Model

We can extend our multi-level meta-analytic model, which only has an intercept, to include moderators quite easily:

\[ y_{i} = \mu + \sum_{i = 1}^{N_{m}}\beta_{m}x_{m} + s_{j[i]} + spp_{k[i]} + e_{i} + m_{i} \\ m_{i} \sim N(0, v_{i}) \\ s_{j} \sim N(0, \tau^2) \\ s_{k} \sim N(0, \sigma_{k}^2) \\ e_{i} \sim N(0, \sigma_{e}^2) \]

Here, \(\beta_{m}\) is the slope (or contrast) for the m th moderator (where m = 1, …, \(N_{moderator}\) or \(N_{m}\)). We can include different types of continuous and categorical moderators in our model. Some of these moderators might control for nuisance hetereogenity, others might be to understand how different experimental designs affect the magnitude and direction of mean effect size and some may simply be biological in nature – they might compare ectotherms to endotherms, for example.

Phylogenetic Multi-level Meta-regression