Quantitative evidence synthesis: a practical guide on meta-analysis, meta-regression, and publication bias tests for environmental sciences

Abstract

Meta-analysis is a quantitative way of synthesizing results from multiple studies to obtain reliable evidence of an intervention or phenomenon. Indeed, an increasing number of meta-analyses are conducted in environmental sciences, and resulting meta-analytic evidence is often used in environmental policies and decision-making. We conducted a survey of recent meta-analyses in environmental sciences and found poor standards of current meta-analytic practice and reporting. For example, only ~ 40% of the 73 reviewed meta-analyses reported heterogeneity (variation among effect sizes beyond sampling error), and publication bias was assessed in fewer than half. Furthermore, although almost all the meta-analyses had multiple effect sizes originating from the same studies, non-independence among effect sizes was considered in only half of the meta-analyses. To improve the implementation of meta-analysis in environmental sciences, we here outline practical guidance for conducting a meta-analysis in environmental sciences. We describe the key concepts of effect size and meta-analysis and detail procedures for fitting multilevel meta-analysis and meta-regression models and performing associated publication bias tests. We demonstrate a clear need for environmental scientists to embrace multilevel meta-analytic models, which explicitly model dependence among effect sizes, rather than the commonly used random-effects models. Further, we discuss how reporting and visual presentations of meta-analytic results can be much improved by following reporting guidelines such as PRISMA-EcoEvo (Preferred Reporting Items for Systematic Reviews and Meta-Analyses for Ecology and Evolutionary Biology). This paper, along with the accompanying online tutorial, serves as a practical guide on conducting a complete set of meta-analytic procedures (i.e., meta-analysis, heterogeneity quantification, meta-regression, publication bias tests and sensitivity analysis) and also as a gateway to more advanced, yet appropriate, methods.

Keywords: Hierarchical models; Meta-analysis of variance; Missing data; Multivariate meta-analysis; Network meta-analysis; Robust variance estimation; Spatial dependency; Variance–covariance matrix.

Fig. 1

Visualisation of the three statistical models of meta-analysis: A a fixed-effect model (1-level), B a random-effects model (2-level), and C a multilevel model (3-level; see the text for what symbols mean)

Fig. 2

Visualisation of the three types of non-independence among effect sizes: A due to shared study identities (effect sizes from the same study), B due to shared measurements (effect sizes come from the same group of individuals/plots but are based on different types of measurements), and C due to shared control (effect sizes are calculated using the same control group and multiple treatment groups; see the text for more details)

Fig. 3

Visualisation of variation (heterogeneity) partitioned into different variance components: A quantifying different types of I² from a multilevel model (3-level; see Fig. 1C) and B variance explained, R², by moderators. Note that different levels of variances would be explained, depending on which level a moderator belongs to (study level and effect-size level)

Fig. 4

Different types of plots useful for a meta-analysis using data from Midolo et al. [133]: A a typical forest plot with the overall mean shown as a diamond at the bottom (20 effect sizes from 20 studies are used), B a caterpillar plot (100 effect sizes from 24 studies are used), C an orchard plot of categorical moderator with seven levels (all effect sizes are used), and D a bubble plot of a continuous moderator. Note that the first two only show confidence intervals, while the latter two also show prediction intervals (see the text for more details)

Fig. 5

Different types of plots for publication bias tests: A a funnel plot using model residuals, showing a funnel (white) that shows the region of statistical non-significance (30 effect sizes from 30 studies are used; note that we used the inverse of standard errors for the y-axis, but for some effect sizes, sample size or ‘effective’ sample size may be more appropriate), B a bubble plot visualising a multilevel meta-regression that tests for the small study effect (note that the slope was non-significant: b = 0.120, 95% CI = [− 0.095, 0.334]; all effect sizes are used), and C a bubble plot visualising a multilevel meta-regression that tests for the decline effect (the slope was non-significant: b = 0.003, 95%CI = [− 0.002, 0.008])