# The calculation of Bonferroni-adjusted p-values

## Problem

How does SPSS calculate the Bonferroni-corrected p-values for pairwise comparisons?

## Resolving The Problem

SPSS offers Bonferroni-adjusted significance tests for pairwise comparisons. This adjustment is available as an option for post hoc tests and for the estimated marginal means feature.

Statistical textbooks often present Bonferroni adjustment (or correction) in the following terms. First, divide the desired alpha-level by the number of comparisons. Second, use the number so calculated as the p-value for determining significance. So, for example, with alpha set at .05, and three comparisons, the LSD p-value required for significance would be .05/3 = .0167.

SPSS and some other major packages employ a mathematically equivalent adjustment. Here's how it works. Take the observed (uncorrected) p-value and multiply it by the number of comparisons made. What does this mean in the context of the previous example, in which alpha was set at .05 and there were three pairwise comparisons? It's very simple. Suppose the LSD p-value for a pairwise comparison is .016. This is an unadjusted p-value. To obtain the corrected p-value, we simply multiply the uncorrected p-value of .016 by 3, which equals .048. Since this value is less than .05, we would conclude that the difference was significant.

Finally, it's important to understand what happens when the product of the LSD p-value and the number of comparisons exceeds 1. In such cases, the Bonferroni-corrected p-value reported by SPSS will be 1.000. The reason for this is that probabilities cannot exceed 1. With respect to the previous example, this means that if an LSD p-value for one of the contrasts were .500, the Bonferroni-adjusted p-value reported would be 1.000 and not 1.500, which is the product of .5 multiplied by 3

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Modified date:
16 April 2020

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