Table 1 Definitions of key concepts and associated statistical parameters, which are used in formulas in the main text

Effect size

A measurement of effect (usually state of a single group, comparison between groups, or association, see Table 2). In a meta-analytic model, it becomes the response variable (noted as z_i in the formulas)

Sampling variance

A measure of uncertainty in effect size (noted as v_i). Its inverse is often called ‘weight’ (the square-root of weight is ‘precision’, and the square root of sampling variance is ‘sampling standard error’)

Meta-analysis

A statistical method to aggregate effect sizes from studies on the same or similar topics, by assigning different weights based on sampling variance of effect sizes. Strictly speaking, in a formal (weighted) meta-analysis, sampling variance needs to be incorporated and it is assumed to be known (Table 2)

Overall mean (effect)

An average effect size based on a meta-analytic model (noted as \({\beta }_{0}\) and its standard errors, se(\({\beta }_{0}\)))

Heterogeneity

An indicator of consistency among effect sizes, or an extent of variation around the overall effect (\({\beta }_{0}\)); heterogeneity can be quantified by absolute measures, such as \({\tau }^{2}\), or relative measures, such as I²

Meta-regression

A regression model which extends a meta-analytic model with a moderator(s), aiming to explain heterogeneity (quantified as R²) and quantifying the effect of a moderator (noted as, for example, \({\beta }_{1}\))

Publication bias tests

A set of statistical methodologies to detect and correct for publication bias, where a subset of results (positive findings) is more likely to be published and present in the meta-analytic dataset than otherwise

Sensitivity analysis

A set of statistical analyses that checks the robustness of one’s main analysis; if sensitivity analysis shows different results (qualitatively and/or quantitively), then we must doubt the robustness of the main findings