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 P_{j}. 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 (*SE*_{j}) is large compared with the largest effect that might realistically exist (*K*_{j}). The standard BC procedure gives an Overall P-value, or Overall Attained Significance Level, OASL_{BC} = *k**min_{j}{*P*_{j}}. As an extension, let *f*_{j}(.) be an increasing, preferably continuous, function through the origin, and *g*_{j}(.) its inverse. When applied to summary statistic *T* = min_{j}{f_{j}(P_{j})}, the Bonferroni inequality implies *P*_{0}{*T* =< *t*} =< SIGMA_{J} g_{J}(*t*), so the associated OASL = SIGMA_{J} *g*_{J}(min_{j}{*f*_{j}(P_{j})}). This OASL reduces to OASL_{BC} 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 *H*_{0j}: *m*_{j} = 0 is *H*_{(alt)j}: 0 < *m*_{j} < *K*_{j}, K_{j} being small relative to the associated *SE*_{j}. In most models the likelihood ratio, *LR*_{j}(data *y*) = *p*{*y*|*H*_0j} / sup_{(alt)j}{p{y|m_{j}}}, is a natural starting point for combined inference and implicitly is a monotone function of (the conventional one-sided) *Pj*. This suggests defining *T* = min_{j}{*LR*_{j}} and using the extended Bonferroni rule. The (one-sided) BC procedure is reproduced as long as *K*_{j}'s are effectively infinite. As *K*_{j}/SE_{j} approaches zero for a certain part of the *j*'s, i.e. as the hope of useful information about these *m*_{j}'s dwindles, the procedure focusses on the remaining, fewer, subtests, thereby recuperating power. The extended Bonferroni scheme can be built into sequential rejection schemes.