Improving the Power of Bonferroni Adjustments under Joint Normality and Exchangeability

Abstract

Bonferroni's correction is a popular tool to address multiplicity but is notorious for its low power when tests are dependent. This paper proposes a practical modification of Bonferroni's correction when test statistics are jointly normal and exchangeable. This method is intuitive to practitioners and achieves higher power in sparse alternatives, as our simulations suggest. We also prove that this method successfully controls the family-wise error rate at any significance level.

Figures (3)

• Figure 1: Comparison of the power of \ref{['gnpa']} against various methods when a single test statistic attains a non-zero mean. Size is indicated by the horizontal black line. Note the higher power achieved by \ref{['gnpa']} compared to the other methods.
• Figure 2: Comparison of the power of \ref{['gnpa']} against various methods as the proportion of test statistics attaining non-zero mean increases. Size is indicated by the horizontal black line. Note the higher power achieved by \ref{['gnpa']} compared to the other methods when the proportion of non-zero mean test statistics is less than 10%.
• Figure 3: Absolute test statistics plotted based on index, where indices less than 500 attain mean 3 while others attain mean 0. The red line indicates the critical value while the blue line indicates the separation between mean 3 and mean 0. Note the conservative nature of the critical value when detecting other test statistics which are not the maximum.

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 1
• proof : Proof of Lemma \ref{['lemma1']}
• Lemma 2
• proof : Proof of Lemma \ref{['lemma2']}
• Lemma 3
• proof : Proof of Lemma \ref{['lemma3']}
• proof : Proof of Theorem \ref{['thm1']}