Bonferroni Correction Calculator

With our Bonferroni correction calculator, we aim to help you adjust p-values to account for multiple comparisons in statistical analyses. To understand more on this topic, check out our confidence interval calculator and F-statistic calculator.

We designed this tool to enhance your understanding of:

• What the Bonferroni correction, or Bonferroni adjustment, is;
• How to calculate the Bonferroni correction;
• When to use Bonferroni correction​; and
• The metric's significance in reducing false-positive results.

We have also included practical Bonferroni correction examples to help you understand how to apply the Bonferroni correction to your analyses.

The Bonferroni correction is a statistical method used to address the problem of multiple comparisons in hypothesis testing. This metric is especially useful when performing multiple statistical tests simultaneously. This is because the likelihood of obtaining at least one false-positive result, which can be interpreted as Type I error, increases during these situations. The Bonferroni correction helps control this error rate by adjusting the threshold for statistical significance.

The Bonferroni correction adjusts the significance level (α), typically set at 0.05, or 5%, by dividing it by the number of comparisons made. This adjusted significance level is then used to evaluate the p-values of individual tests.

Now that we understand what Bonferroni adjustment is, let's look at the example below to understand more about its calculation.

• Number of tests: 4
• Significance level (α): 5%
• p-value: 0.05

1. Choose the calculation method.

   The first step is to choose whether you want to use the classic Bonferroni correction​ or the Šidák correction.

   The classic Bonferroni correction simply divides the significance level by the number of tests, while the Šidák correction uses a multiplicative adjustment that assumes independence among tests, often making it slightly less conservative.

2. Determine the number of tests.

   Now, you need to compute the total number of hypotheses or tests being performed. In our example, we are conducting 4 tests.

3. Compute the p-value or significance level (α).

   If you choose the classic Bonferroni correction​, you need to determine the significance level (α).

   If you choose the Šidák correction, the next step is to determine the p-value for each individual hypothesis. The purpose of this step is to check whether it meets the Bonferroni-corrected threshold. If the p-value is smaller than the corrected α, the result is statistically significant.

   You can check out our p-value calculator to help you with this computation.

4. Calculate the Bonferroni correction.

   The last step is to calculate the Bonferroni correction.

   If you choose the classic Bonferroni correction, the Bonferroni correction formula will be:

   $\text{Bonferroni correction} = \frac{α}{\text{Number of tests}}$

   Hence, for this example, our Bonferroni correction is:

   $\text{Bonferroni correction} = \frac{5\%}{\text{Number of tests}} = 0.0125$

   This means that each individual test must have a p-value less than 0.0125 to be considered statistically significant.

   If you choose the Šidák correction, you can calculate it by using the following formula:

   $\text{Bonferroni correction} = (1 - \text{p-value}) ^ \frac{1}{\text{Number of tests}}$

   Thus, for this instance, our Bonferroni correction is:

   $\text{Bonferroni correction} = (1 - \text{0.05}) ^ \frac{1}{\text{4}} = 0.0127$

   This means that each individual test must have a p-value less than 0.0127 to be considered statistically significant.

Now that we have discussed when to use Bonferroni correction​ and looked at the Bonferroni correction example, let's take a deeper dive into its significance.

• Controlling false positives

  The Bonferroni correction ensures that the overall Type I error rate remains at or below the specified significance level. Hence, it can help reduce the risk of falsely declaring a result significant.

• Interpreting multiple comparisons

  When multiple hypotheses are tested, the Bonferroni correction is a great metric that ensures the significant results are genuinely meaningful rather than merely due to random chance.

While the Bonferroni correction is straightforward and effective, it does have a few limitations.

1. The metric can sometimes be overly conservative, particularly when dealing with multiple comparisons.

2. Although Bonferroni correction effectively controls false positives, it sometimes does so at the expense of increasing the likelihood of false negatives.

3. Hence, it is sometimes wise to consider alternative methods, such as the Holm-Bonferroni or Benjamini-Hochberg procedures.