Fig. 2.
The workflow showing the data compilation, statistical modelling processes, and our aims. Using the datasets containing 87 independent meta-analyses (36 SMD, 20 lnRR and 31 Zr cases, respectively), we used a two-step modelling procedure to assess (i) the estimated prevalence and severity of publication bias in ecology and evolutionary biology and (ii) how such publication bias affects the estimates of effect size, statistical power, Type M and S errors. In the first step (i.e. within-meta-analysis level), multilevel meta-analytic approaches will be used to estimate the overall mean (used for power and errors calculations), and test and adjust for publication bias for each meta-analytic case. In the second step (i.e. between-meta-analysis level), the estimates from the first step were statistically aggregated using either mixed-effects models or random-effects meta-analytic models (i.e. secondary meta-analysis). β0 is the meta-analytic overall mean (i.e. β0[overall] in Equation 1), which signifies the uncorrected effect size estimate if publication bias exists but is not corrected. β1 and β2 are the indicators of small-study effects and decline effects (equivalent to β1[small − study] and β1[time − lag] in Equation 2). η0[u] is the standardised β0. (i.e. η0[overall]). η0[c] is the standardised bias-corrected meta-analytic overall mean (i.e. η0[bias − corrected] in Equation 6). η1[small − effect], η2[time − lag] are standardised model coefficients corresponding to β0, β1 and β2 (i.e. η1[small − effect] and η2[time − lag] in Equation 6)