very flexible and allows results to be combined even when effect size estimates are not available or the β-coefficients and standard errors from individual studies are in different units. The second approach implemented in METAL weights the effect size estimates, or β-coefficients, by their estimated standard errors. This second approach requires effect size estimates and their standard errors to be in consistent units across studies. Asymptotically, the two approaches are equivalent when the trait distribution is identical across samples (such that standard errors are a predictable function of sample size). Key formulae for both approaches are in Table 1. Table 1.Formulae for meta-analysisAnalytical strategySample size basedInverse variance basedInputsNi - sample size for study iβi- effect size estimate for study iPi−P-value for study iΔi - direction of effect for study isei - standard error for study iIntermediate StatisticsZi = Φ−1(Pi/2) * sign(Δi)wi = 1/SEi2Overall Z-ScoreZ=β/SEOverall P-valueP=2Φ(|−Z|)
