Bonferroni Test

The Bonferroni test is a statistical comparison test that involves checking multiple tests limiting the chance of failure. It is otherwise known as the Bonferroni correction or Bonferroni adjustment. The test allows for the comparison of several variables to avoid false data appearing statistically significant.

It is a simple method of counteracting multiple comparison problems. When running multiple tests, the chances of an error occurring in the results exist, and the researcher may then declare the result statistically significant. This test avoids data that could appear statistically significant by reducing the alpha value. 

• The Bonferroni correction modifies probability (p) values to account for the increased likelihood of a “type I” error while doing numerous statistical tests.
• It employs Bonferroni inequalities, made famous by Italian mathematician Carlo Emilio Bonferroni, and hence has that name.
• Here, the critical P value (α) should be divided by the number of comparisons in the test.
• The test minimizes the chance of “Type I” error. However, it is often done at the expense of an increased probability of Type II error.

The Bonferroni test method finds its use of the Bonferroni inequalities, which were known after Carlo Emilio Bonferroni, an Italian mathematician. The Bonferroni correction is applied to P values. When performing repeated statistical tests, there is a higher chance of making a type I error, which is why the Bonferroni correction modifies probability (p) results. One can run numerous dependent or independent statistical tests concurrently on a single data set. 

Bonferroni test ANOVA is a test where the mean of a numerical outcome can be compared between two or more independent groups. The Bonferroni test ANOVA can be helpful when there are more than two groups. On sample data, to carry out a Bonferroni adjustment, the number of comparisons should divide the crucial P value (α). One can evaluate the statistical power of the data based on the modified P value. For example, suppose we are testing 20 hypotheses. The modified new critical value of P would be (α)/20. This value is what is useful to calculate the statistical power.

The Bonferroni correction helps in lowering the likelihood of receiving false-positive findings (type I errors). Several pair-wise tests are run on a single data set. As the number of studies and hypotheses increases, the likelihood of finding at least one significant result occurring from the chance of the hypothesis also increases. However, the routine application of this test has drawn criticism for testing the incorrect hypothesis, resulting in a detrimental effect on the strong statistical judgment. It will also increase the likelihood of a type II error while decreasing the likelihood of a type I error.

A single statistical test determines if two group means are equal. Assuming the p-value to test the alpha level is 0.05, if the p-values determined are less than 0.05, one can reject the null hypothesis and conclude that the group means are different. If one conducts multiple tests at the same time, there will be several groups' means. When the amount of data is greater, there exists a high chance of committing a type I error and rejecting the null hypothesis when it is true. This is where the adjustment level comes into play. The null hypothesis of individual tests is then eliminated when the p-value is less than the adjusted alpha (α) level. 

Bonferroni test calculator helps with easy achievement of results.

Bonferroni adjustment illustration:

α' = 1 - ( 1 - α ) 1/k

Where “α'” represents Bonferroni Correction, “α” denotes Critical P Value and “k” denotes the Number of Tests

α' = 1 - ( 1 - α ) 1/k

= 1 - (1 - 0.05)1/20

Adjusted α: 0.00250

Above is an example from the Bonferroni test calculator. Here, we have conducted 20 comparisons. In cases where the null hypotheses of comparisons have a p-value less than 0.00250, we can eliminate them. Using the Bonferroni test calculator can thus save a lot and effort.

The Bonferroni adjustment typically controls for false positives; as the number of tests rises, it might become overly conservative. It, in turn, raises the possibility of getting erroneous negative results (type II errors).

The specifics of the study will determine whether or not to apply the Bonferroni correction. We can use the method in certain circumstances, such as if

(1) A test of the "universal null hypothesis" (denoted as Ho) showing that all tests examined are not significant is necessary,

(2) Avoiding a “type I” error is critical, and

(3) There are no predetermined hypotheses while performing most tests.

In conclusion, testing several hypotheses on a single data set increases the chance of drawing false-positive findings that aren't accurate. The Bonferroni correction is a straightforward statistical technique for reducing this risk, and when done properly, it can guarantee the objectivity of research that employs several significance tests.

This has been a guide to what is Bonferroni Test and its definition. Here we discuss how Bonferroni test works, illustration, when to use it and correction example. You can learn more from the following articles -

• One-Tailed Test
• T-Test
• Anova vs T-test