Combination of p-values from multiple independent tests has been widely studied since 1930's. To find the optimal combination methods, various combiners such as Fisher's method, inverse normal transformation, maximal p-value, minimal p-value, etc. have been compared by different criteria. In this work, we focus on the criterion of Bahadur efficiency, and compare various methods under the TFisher. As a recently developed general family of combiners, TFisher cover Fisher's method, the rank truncated product method (RTP), the truncation product method (TPM, or the hard-thresholding method), soft-thresholding method, minimal p-value method, etc. Through the Bahadur asymptotics, we better understand the relative performance of these methods. In particular, through calculating the Bahadur exact slopes for the problem of detecting sparse signals, we reveal the relative advantages of truncation versus non-truncation, hard-thresholding versus soft-thresholding. As a result, the soft thresholding method is shown superior when signal strength is relatively weak and the ratio between the sample size of each p-value and the number of combining p-values is small.

The method of combination p-values from multiple tests is the foundation for some studies like meta-analysis and detection of signal. There are tremendous methods have been developed and applied like minimum p-values, Cauchy Combination, goodness-of-fit combination and Fisher’s combination. In this paper, I tested their ability to detect signals which is related to real case in biology to find out significant single-nucleotide polymorphisms (SNPs). I simulated p-values for SNPs logistics regression model and test 7 combination methods’ power performance in different setting conditions. I compared sparse or dense signals, dependent or independent and combine them in gene-level or pathway-level. One method based on Fisher’s combination called Omni-TFisher is ideal for most of the situations. Recent years, genome-wide association studies (GWASs) focused on BMD-related SNPs at gene significance level. In this paper I used Omni-TFisher to analyses real data on haplotype blocks. As a result, haplotype blocks can find more SNPs in non-coding and intergeneric regions than gene-based and save computational complexity. It finds out not only known genes, but also other genes need further verification.