# Hervé Abdi:

# The Bonferonni and Šidák Corrections

7, 868

0

0

1, 907

1

1, 907

192

2

384

20

3

60

13

4

52

0

5

0

Table 1: Results of a Monte Carlo simulation. Numbers of Type 1 errors when performing C = 5 tests for 10,000 families when H_{0 is }true. How to read the table? For example, 192 families over 10,000 have 2 Type 1 errors, this gives 2 × 192 = 384 Type 1 errors.

10, 000

2, 403

Number of families with X Type I errors

X : Number of Type I errors per family

Number of Type I errors

# 2.2 A Monte Carlo illustration

A “Monte Carlo" simulation can illustrate the difference between α[PT ] and α[PF ]. The Monte Carlo technique consists of running a simulated experiment many times using random data. This gives the pattern of results that happens on the basis of chance.

Here 6 groups with 100 observations per group were created with data randomly sampled from the same normal population. By construction, H_{0 is true (i.e., all population means are equal). }Call that procedure an experiment. We performed 5 independent tests from these 6 groups. For each test, we computed an F-test. If its probability was smaller than α = .05, the test was declared sig- nificant (i.e., α[PT ] is used). We performed this experiment 10,000 times. Therefore, there were 10,000 experiments, 10,000 families, and 5 × 10,000 = 50,000 tests. The results of this simulation are given in Table 1.

Table 1 shows that H_{0 is rejected for 2,403 tests over 50,000 }tests performed. From these data, an estimation of α[PT ] is com- puted as:

4