A modified Bonferroni procedure for multiple tests.

摘要

The Bonferroni adjustment ensures that the overall probability of Type I error is below the nominal level when multiple tests are performed. However, the method results in a Type I error rate that is well below the nominal level. Researchers developed modified Bonferroni methods to improve statistical power. Many of these modifications sequentially test observed p-values. The most common modified Bonferroni method is the Holm method, which tests observed p-values using alpha/(k - 1 + i) beginning with the smallest and testing the largest observed p-value using alpha. The purpose of this research was to establish an improved single step method for multiple tests based on the Bonferroni adjustment. The multiple tests used in this investigation were tests of regression coefficients. This new method was compared to the Bonferroni and Holm methods to evaluate statistical power while maintaining a probability of Type I error that was less than or equal to the nominal level. The data used in this research were simulated using SAS 9.1. Sample data (n = 25, 50, 200) were simulated to have normal distributions for hypothesis testing of k = 3, 4, 5, ..., 20 regression coefficients. Correlation strength between x and y (.3, .5) and among the x's (.0, .1, .3) was varied. The proportion of non-zero relationships between x and y (P_NON) was varied from 0 to a maximum of .75 and was dependent on the number of tests performed. Based on the results, a new denominator was determined that was less than k. The formula to calculate the new value for the denominator was k - (k -1)P_NON. The new method resulted in Type I error that ranged from .0242 to .0579 while the Bonferroni method ranged from .0103 to .0579 and Holm method ranged from .0113 to .0579. The smallest statistical power that occurred was .00451 (New), .00286 (Bonferroni), and .00350 (Holm). Average statistical power was .360 (New), .339 (Bonferroni), and .344 (Holm). In conclusion, a new single step method with improved statistical power was developed for multiple tests when the researcher knows approximately how many independent variables are related to the dependent variable.