Meta-analysis was performed centrally using the software package rareGWAMA (see URLs). All statistical tests in the meta-analysis or secondary analyses of the meta-analytic results (e.g., polygenic risk scoring, functional enrichment, MTAG, Genomic SEM, etc.) were two-sided. Given that rarer variants and/or behavioral phenotypes may show between-study heterogeneity in allele frequencies, imputation qualities, or genetic architecture, we extended existing methods and developed a novel fixed effects approach that accounts for between-study heterogeneity. Specifically, the methods aggregated weighted Z-score statistics, i.e. ZMETA=∑kwkZk(∑kwk2)1/2, where Zk is the Z-score statistic in study k. The weight wk is defined by wk=Nkpk(1−pk)Rk2, where pk is the variant allele frequency, Rk2 is the imputation quality, and Nk is the sample size for study k. Under the null and with the present sample sizes, ZMETA is normally distributed. The weights are proportional to the sample genotype variance. When the trait is uniformly measured and the allele frequency is similar, the method is approximately equivalent to meta-analysis of sample-size-weighted Z-scores. Yet, the method accounts for between-study heterogeneity in imputation accuracy and allele frequencies. The use of a fixed effects
