When the adjusted p-value would exceed 1

Question

You ran 30 hypothesis tests, and one returned a raw p-value of 0.05. You apply the Bonferroni adjustment by multiplying: $30 \times 0.05 = 1.5$ . But a probability cannot exceed 1.

What is the correct Bonferroni-adjusted p-value to report?

Accepted Answer

The multiplication overshoots. The rule "adjusted p-value equals m 	imes p" is only the first step. Here it gives 30 	imes 0.05 = 1.5, which is not a valid p-value, no probability can exceed 1. Cap at 1. The full definition of the Bonferroni-adjusted p-value takes the smaller of m 	imes p and 1: p{	ext{adj}} = \min(m 	imes p,\ 1) = \min(1.5,\ 1) = 1. So you report an adjusted p-value of 1. What it means. An adjusted p-value of 1 is the correction's way of saying "this result carries no evidence at all once we account for 30 tests." A raw p of 0.05 is exactly the average outcome of a sing…

When the adjusted p-value would exceed 1

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