Jørgen Hilden, Michael Weis Bentzon (University of Copenhagen, Danmark)

Bonferroni with contextual P-value transforms: a means to gain power

Consider a k-faced null hypothesis, the jth subtest P-value being Pj. Bonferroni corrections (BC), when needed, are wasteful of power. Our present aim is to suggest a method of improving power by exploiting the subject-matter context: we deemphasize those subtests that provide little hope of revealing something interesting because the standard error (SEj) is large compared with the largest effect that might realistically exist (Kj). The standard BC procedure gives an Overall P-value, or Overall Attained Significance Level, OASLBC = k*minj{Pj}. As an extension, let fj(.) be an increasing, preferably continuous, function through the origin, and gj(.) its inverse. When applied to summary statistic T = minj{fj(Pj)}, the Bonferroni inequality implies P0{T =< t} =< SIGMAJ gJ(t), so the associated OASL = SIGMAJ gJ(minj{fj(Pj)}). This OASL reduces to OASLBC when the transform functions are all the same (or k = 1). Now, for some index j, the context may indicate that the realistic alternative to H0j: mj = 0 is H(alt)j: 0 < mj < Kj, Kj being small relative to the associated SEj. In most models the likelihood ratio, LRj(data y) = p{y|H_0j} / sup(alt)j{p{y|mj}}, is a natural starting point for combined inference and implicitly is a monotone function of (the conventional one-sided) Pj. This suggests defining T = minj{LRj} and using the extended Bonferroni rule. The (one-sided) BC procedure is reproduced as long as Kj's are effectively infinite. As Kj/SEj approaches zero for a certain part of the j's, i.e. as the hope of useful information about these mj's dwindles, the procedure focusses on the remaining, fewer, subtests, thereby recuperating power. The extended Bonferroni scheme can be built into sequential rejection schemes.