p-Value Adjustments

The MULTTEST Procedure

p-Value Adjustments

The FISHER_C option requests adjusted p-values using closed tests, based on the idea of Fisher's combination test. The Fisher combination test for a joint test of any set of S hypotheses with p-values uses the Chi-Square statistic $\chi^2=-2 \sum \log (p_i)$ , with 2S degrees of freedom. The FISHER_C adjusted p-value for test j is the maximum of all p-values for the combination tests, taken over all joint tests that include j as one of their components. Independence of p-values is absolutely required for this method.

Hommel's (1988) method is a closed testing procedure based on Simes' (1986) test. The Simes p-value for a joint test of any set of S hypotheses with p-values $p_1 \leq p2 \leq ... \leq p_S$ is min((S/1)p₁,(S/2)p₂, ... ,(S/S)p_S). The Hommel adjusted p-value for test j is the maximum of all such Simes p-values, taken over all joint tests that include j as one of their components. Hochberg adjusted p-values are always as large or larger than Hommel adjusted p-values. Sarkar and Chang (1997) showed that Simes' method is valid under independent or positively dependent p-values, so Hommel's and Hochberg's methods also are valid in such cases by the closure principle.

Westfall et al. (1999) and Westfall and Wolfinger (2000) are new references dealing with multiplicity issues and PROC MULTTEST.

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