Type 1 Error Rate | Statistics

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Type 1 Error Rate | Statistics

Testing multiple variables inflates the type 1 error rate or the false positive rate. This is called the multiple comparison problem. Correcting for this alpha-inflation is not hard. There are two main ways, namely the Bonferroni correction and the Holm correction.

Bonferroni correction

The Bonferroni correction is simple but quite conservative. You divide your alpha level by the number of tests you are about to perform. This will be the new significance level. So in this case:

ɑ / n

ɑ: alpha or significance level

n: number of tests

0.05 / 10 = 0.005

You can thus do this quite easily yourself when reading a paper. If five variables are tested, you know the alpha level should be about 0.01 instead of 0.05 (0.05 / 5). This is under the assumption that the researchers did not perform a boatload of tests “behind the scenes” while not reporting them. This is called data-dredging or p-hacking.

Another way is by simply multiplying the p-value in the paper by the number of tests.

Eg.

P-value = 0.03

0.03 * 10 = 0.3

This means that the previously significant p-value now became insignificant if 10 variables were tested.

Bonferroni correction limitations

The Bonferroni correction is a widely used method to adjust the significance level for multiple comparisons in order to control the overall Type I error rate. However, it has several limitations.

One of the main issues is that it can be overly stringent, which may lead to a loss of statistical power. Additionally, it assumes that all comparisons are independent, which may not be the case in real-world data, potentially leading to higher Type II error rates.

Another limitation of the Bonferroni correction is that it increases the chance of false negatives or Type II errors, meaning that there is a higher chance of missing a true effect.

Finally, the Bonferroni correction is most appropriate for situations where the number of comparisons is relatively small, as it may not be as effective when the number of comparisons is very large. Therefore, researchers should carefully consider the appropriateness of the Bonferroni correction for their research question and data set, and be aware of its limitations.

Holm correction

A second way to correct the alpha inflation is the Holm correction. Let’s say the researchers did five tests and thus became five p-values. For the Holm correction to work, they should be ranked from lowest to highest.

Eg.

0,0004
0,0130
0,0172
0,0460
0,0600

The Holm formula is as follows:

p-value * (m + 1 – k)

m = number of p-values

k = the rank of the p-value

So for the third p-value we get…

0,0172 * (5 + 1 – 3) = 0,0516

… making the results insignificant.

Holm Correction Limitations

One limitation is that Holm’s correction assumes that all tests are independent, meaning that the results of one test do not affect the results of another. However, in some cases, the tests may be dependent, such as when testing multiple outcomes from the same sample or when testing different time points from the same intervention. In such cases, Holm’s correction may be too conservative or too liberal, leading to incorrect conclusions. Another limitation of Holm’s correction is that it does not take into account the correlation between the tests, which can affect the false positive rate. For example, if multiple tests are related to the same underlying construct, the probability of detecting a significant effect increases, and Holm’s correction may not adequately account for this. While Holm’s correction is a useful method for adjusting p-values in multiple comparison testing, it is important to consider its limitations, particularly when tests are dependent or correlated. Other methods such as False Discovery Rate control or Bayesian methods may be more appropriate in some cases.