Bonferroni Technique

Development

The basis of the Bonferroni technique is Bonferroni's inequality: αf ≤ α1 + α2 + ··· + αm. Because it is αf that the researcher wishes to control, then choosing values for the several αj terms that make α1 + α2 +···+ αm = αj will result in control over αf, the familywise Type I error rate. Usually, the several αj values are set equal to each other, and each test is run at αt = αf/m. The value of αf is therefore, at most, mαt.

Application

In order to apply the Bonferroni technique, we must have a family (i.e., set) of m (at least two) significance tests and will define two significance (α) levels: the per-test Type I error rate and the familywise Type I error rate. The per-test Type I error rate, αt, is the significance level of each statistical test (i.e., the null hypothesis is rejected if the p value of the test falls below αt). The familywise Type I error rate, αf, is the probability that one or more of the m tests in the family is declared significant as a Type I error (i.e., when its null hypothesis is true).

Use of the Bonferroni technique involves adjusting the per-test Type I error rate such that αt = αf/m. In practice, the familywise Type I error rate is usually held to .05. Thus, αt is usually set at .05/m.

Although tables of these unusual percentiles of various distributions (e.g., t, chi-square) exist, in practice, the Bonferroni technique is most easily applied using the p values that are standard output in statistical packages. The p value of each statistical test is compared with the calculated αt and if p ≤ αt, the result is declared statistically significant; otherwise, it is not.

Example

A questionnaire with four items has been distributed to a group of participants who have also been identified by gender. The intent is to test, for each item, whether there is a difference between the distributions of males and females. CHI-SQUARE TESTS based on the four two-way tables (gender by item response) will be used. However, if each test were conducted at the α = .05 level, the probability of at least one of them reaching statistical significance by chance (assuming all null hypotheses true) is greater than .05 because there are four opportunities for a Type I error, not just one. In this example, there are m = 4 significance tests, so instead of comparing the p value for each test with .05, it is compared with .05/4 = .0125.

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