cochetezgh

2022-09-04

Use of the Bonferroni correction
I am conducting an analysis of survey data.
I have carried out sign tests comparing the answers provided by two different groups to each question. There are several questions for which I am making this analysis.
Am I right in thinking that I do not need to use a Bonferroni (or any other type) of correction?

wegpluktee3

Expert

The Bonferroni correction would be to make all of these confidence intervals (CIs) at the level 100%5%/k if you have created k CIs at the 95% level and wish to use them together to get a comprehensive picture of your data. According to Bonferroni's Inequality for probabilities, the overall 'error rate' of the k CIs cannot exceed 5%. The Bonferroni procedure for CIs is a little too aggressive (in the sense that it can 'over-correct'), but this effect is usually not important for a small (e.g. 5% overall rate, and a small k).
Similar advice holds for testing k related hypotheses. The overall or 'family' significance level for k hypotheses considered together does not exceed 5% if the individual hypotheses are all significant at level 5%/k.
Perhaps this kind of comparison occurs most often in the context of an analysis of variance where a factor with g levels is significant at level α. Then to assess the overall pattern of significant differences for the pairwise comparisions, some sort of 'correction' is needed in order to keep the 'family' type I error rate under reasonable control.
Particularly when all g levels have the same number of replications and the same variance, there are very many possible competitors to the Bonferroni method of controlling the 'family' significance level. Some of the possibilities are 'Fisher LSD', 'Tukey HSD', 'SNK', and 'Duncan'. Which one to use depends on the experimental design, how many comparisons are being made, and exactly what you mean by 'family error rate'.
You do not give enough information about your overall experimental design for me to recommend a specific one of these possible methods of controlling the family error rate for k paired comparisons. Although the Bonferroni method can be too 'conservative' (overall power too low), it is also among the most generally applicable method, not to mention the simplicity of computation. [For example, in R statistical software, the (Welsh) procedure for a one-way ANOVA on levels that may have diverse variances, uses the Bonferroni method as the default multiple-comparison method.]

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