Fig. 3.
The decision tree used to obtain the estimate of the ‘unbiased’ effect (i.e. conditional β0). First, use a two-step procedure to estimate β0, β1 and β2 from the full model (Equations 2 or 3). Then, depending on whether the signs of slopes (β1 and β2) are opposite from what will be expected from publication bias (caused by a high amount of unaccounted heterogeneity), there are two types of estimates of β0. The first type includes all β0 regardless of their signs (β1 and β2); the second type of estimated β0 has four scenarios. Scenario 1 = only select β0 with expected signs of β1 and β2 from the full model; Scenario 2 = employ reduced model 1 (Equation 4) to re-estimate β0 where β1 has an unexpected sign, while β2 has an expected sign; Scenario 3 = employ reduced model 3 (Equation 5) to re-estimate β0 if β1 has an expected sign, while β2 has an unexpected sign; Scenario 4 = use β0 from the null model (Equation 1) when both β1 and β2 have unexpected signs (i.e. without the small-study effects or decline effects). The symbols (β0, β1, and β2) are as in Fig. 2