Fig. 1

Three-tiered meta-analysis pipeline. a Data preparation: Select the GEO series relevant to ASD and co-morbid diseases. b Three tiers: (1) For each disease, select significant genes from differential expression analysis of GEO series with a Fisher’s combined test with p<0.05 after Benjamini–Yekutieli (BY) FDR adjustment. (2) For each disease, select significant pathways from hypergeometric enrichment analysis with p<0.05. (3) Identify significant shared pathways across diseases using Fisher’s combined test with p<0.05 after Bonferroni FDR correction. Exclude the non-significant pathways in ASD. c Post analysis. (1) Using the gene expression data from a healthy cohort, generate a null distribution of pathway p values and calculate prior probabilities of pathways being significant by chance. (2.1) Using the prior probabilities, pathway p values in each individual disease, and the Fisher’s combined p values of significant pathways across diseases, calculate minimum Bayes factors and minimum posterior probabilities of null hypotheses for each significant pathway in each disease and in the combined case. (2.2) Combine the pathway p value distribution of each disease with the average null distribution of p values using Fisher’s combined probability test and compare the combined p value distribution with the background chi-squared distribution using a QQ plot for significance. Identify the significant pathways using the combined p values, minimum posterior probabilities, and QQ plots. ASD autism spectrum disorder, BY Benjamini–Yekutieli correction, FDR false discovery rate, GEO Gene Expression Omnibus, QQ plot, quantile–quantile plot