Meta-analysis quantitatively aggregates effect size data collected from existing research. The effect size one uses for a study question needs to be chosen carefully. Each make certain assumptions about the underlying data. It’s important to be weary of these assumptions and identify when something might have gone wrong.

The effect size chosen must satisfy two important features. First, the effect size must be comparable across studies. This means that they must be placed on the same scale so that they can be aggregated. Some effect size estimates can be inter-converted to each other (e.g., Zr, Hedges’ g), but this isn’t the case for all of them. Second, we must be able to derive or approximate the sampling variance for a given effect size. As already indicated, the sampling variance is very important if one wishes to use meta-analytic models that account for sampling variance and is needed to report certain measures of heterogeneity.

Meta-analyses in comparative physiology, and indeed, ecology and evolution more generally, often use highly heterogeneous data (Noble et al., 2022; Senior et al., 2016). This poses other challenges in aggregating effect size data that make their comparability questionable. We’ll discuss some of these issues in later tutorials and identify some ways in which some of this heterogeneity can be dealt with when deriving the effect size itself.

By the end of this tutorial, we hope that you will feel comfortable:

  1. Calculating different effect sizes from your data.
  2. Interpreting the meaning of different effect size statistics
  3. Transforming effect size estimates back to their original scales when needed

Common Effect Sizes

What is an effect size? The answer to this question depends on what kind of study one wishes to synthesise. For example, an effect size could be a contrast between experimental treatments. We might extract mean differences from experiments manipulating, say, bisphenol-A (BPA) and looking at the effect of BPA on aquatic ectotherm phenotype relative to some control (Wu and Seebacher, 2020). Alternatively, the effect of interest might be the magnitude and direction of a correlation between two observed variables, such as metabolism and behaviour (Holtmann et al., 2016). At times, we might even just be interested in analysing the mean or variance itself.

“An effect size is a statistical parameter that can be used to compare, on the same scale, the results of different studies in which a common effect of interest has been measured” (Rosenberg et al., 2013)

There are many types of effect sizes that are commonplace in ecological and evolutionary research (Table 1 describes many of the more common ones from Noble et al., 2022).

We’ll use metafor to demonstrate how to calculate commonly used effect size estimates in the field of comparative physiology, including Z-transformed correlation coefficients (Zr), log response ratios and Hedge’s g.

Setting up

To start, well need to load some packages and data. The data is all stored on GitHub. It’s downloaded directly from the web. You won’t need to have these on hand. We’ll first load the packages that we need.

# Load packages install.packages('pacman') # If you haven't already installed
# the pacman package do so with this code
pacman::p_load(metafor, flextable, tidyverse, orchaRd, pander, mathjaxr, equatags)
# To use mathjaxr you need to run equatags::mathjax_install()

Correlation between physiology and movement patterns: Z-transformed Correlation Coefficient (Zr)