Holm–Bonferroni method

Holm–Bonferroni method is a statistical method used to counteract the problem of multiple comparisons. It is an adjustment procedure developed by Sture Holm in 1979 to control the Familywise error rate (FWER) when performing multiple pairwise tests. The Holm–Bonferroni method is an improvement over the simple Bonferroni correction, as it tends to be less conservative while still controlling the FWER, thus allowing for more powerful statistical tests.

Overview

When conducting multiple statistical tests simultaneously, the chance of incorrectly rejecting at least one true null hypothesis (i.e., making at least one Type I error) increases. The Holm–Bonferroni method provides a stepwise procedure to adjust p-values to account for this increase in error rate, thereby maintaining the overall confidence level. The method is applicable to a wide range of research fields, including medicine, psychology, and biology, where multiple comparisons are common.

Procedure

The Holm–Bonferroni method involves the following steps:

1. Order the individual p-values from the multiple tests in ascending order.
2. For each p-value, multiply it by the number of tests (N) minus its rank (i) plus one. The adjusted p-value is the maximum of this value and the adjusted p-value of the previous test.
3. Compare each adjusted p-value with the desired significance level (α). Reject the null hypothesis for any test whose adjusted p-value is less than or equal to α.

This procedure ensures that the FWER is controlled at a level of α, providing a more accurate assessment of statistical significance when multiple comparisons are made.

Advantages and Limitations

The main advantage of the Holm–Bonferroni method is its increased power compared to the original Bonferroni correction, especially when dealing with a large number of comparisons. It is a simple, non-parametric method that does not assume independence among tests.

However, the method can still be conservative, particularly with a very large number of tests, potentially leading to Type II errors (false negatives). Additionally, it does not account for the potential correlation between test statistics, which can be addressed by more complex procedures like the Benjamini-Hochberg procedure.

Applications

The Holm–Bonferroni method is widely used in fields requiring the control of FWER in multiple hypothesis testing, such as in clinical trials to adjust for multiple secondary endpoints, in genomics for gene expression analysis, and in psychology for experiments involving multiple groups or conditions.

See Also

• Multiple comparisons
• Familywise error rate
• Bonferroni correction
• Benjamini-Hochberg procedure
• False discovery rate