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# 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 H0 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, H0 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 H0 is rejected for 2,403 tests over 50,000 tests performed. From these data, an estimation of α[PT ] is com- puted as:

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