Multiple hypotheses testing with weights

Yoav Benjamini, Yosef Hochberg

Research output: Contribution to journalArticlepeer-review


In this paper we offer a multiplicity of approaches and procedures for multiple testing problems with weights. Some rationale for incorporating weights in multiple hypotheses testing are discussed. Various type-I error-rates and different possible formulations are considered, for both the intersection hypothesis testing and the multiple hypotheses testing problems. An optimal per family weighted error-rate controlling procedure a lá Spjøtvoll (1972) is obtained. This model serves as a vehicle for demonstrating the different implications of the approaches to weighting. Alternative approaches to that of Holm (1979) for family-wise error-rate control with weights are discussed, one involving an alternative procedure for family-wise error-rate control, and the other involving the control of a weighted family-wise error-rate. Extensions and modifications of the procedures based on Simes (1986) are given. These include a test of the overall intersection hypothesis with general weights, and weighted sequentially rejective procedures for testing the individual hypotheses. The false discovery rate controlling approach and procedure of Benjamini & Hochberg (1995) are extended to allow for different weights.

Original languageEnglish
Pages (from-to)407-418
Number of pages12
JournalScandinavian Journal of Statistics
Issue number3
StatePublished - Sep 1997


  • Control weights
  • False discovery rate
  • Family-wise error-rate
  • P-values
  • Per-family error-rate
  • Procedural weights


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