Bonferroni Correction Calculator
Adjust alpha levels and p-values for multiple statistical hypothesis tests using the Bonferroni correction method.
Understanding the Bonferroni Correction
When running multiple statistical hypothesis tests simultaneously, the probability of encountering a false positive (Type I error) increases significantly. The Bonferroni correction is a classic method in statistics designed to control the Family-Wise Error Rate (FWER) by adjusting the significance threshold.
Formula for Adjusted Significance Level
If the desired overall significance level for a family of tests is $\alpha$ and you are performing $n$ independent tests, the Bonferroni adjusted alpha level $\alpha_{adj}$ is given by:
$$\alpha_{adj} = \frac{\alpha}{n}$$
Alternatively, an individual test p-value $p_i$ can be adjusted directly:
$$p_{adj} = \min(1, p_i \times n)$$
Why Bonferroni Correction Matters
Without adjustment, running 20 independent tests at $\alpha = 0.05$ yields approximately a $64\%$ chance ($1 - (1 - 0.05)^{20}$) of getting at least one false positive. Applying the Bonferroni correction sets each individual test threshold to $0.05 / 20 = 0.0025$, keeping the total false positive probability below $5\%$.
Frequently Asked Questions
What is the main limitation of the Bonferroni correction?
The Bonferroni correction is quite conservative, especially when the number of tests $n$ is very large or when tests are highly correlated. This can increase the risk of Type II errors (false negatives).
When should you use the Bonferroni correction?
Use it when conducting a small to moderate number of planned post-hoc pairwise comparisons or multiple hypothesis tests where controlling false positives is critical.
How do you interpret adjusted p-values?
Adjusted p-values can be directly compared to your original significance level $\alpha$. If $p_{adj} \le \alpha$, the result remains statistically significant after accounting for multiple testing.
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