The decision tree used to obtain the estimate of the ‘unbiased’ effect (i.e. conditional β 0 ). First, use a two-step procedure to estimate β 0 , β 1 and β 2 from the full model (Equations 2 or 3 ). Then, depending on whether the signs of slopes ( β 1 and β 2 ) 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 β 0 . The first type includes all β 0 regardless of their signs ( β 1 and β 2 ); the second type of estimated β 0 has four scenarios. Scenario 1 = only select β 0 with expected signs of β 1 and β 2 from the full model; Scenario 2 = employ reduced model 1 (Equation 4 ) to re-estimate β 0 where β 1 has an unexpected sign, while β 2 has an expected sign; Scenario 3 = employ reduced model 3 (Equation 5 ) to re-estimate β 0 if β 1 has an expected sign, while β 2 has an unexpected sign; Scenario 4 = use β 0 from the null model (Equation 1 ) when both β 1 and β 2 have unexpected signs (i.e. without the small-study effects or decline effects). The symbols ( β 0 , β 1 , and β 2 ) are as in Fig. 2