Bonferroni Correction Calculator

This calculator adjusts p-values or alpha thresholds using the Bonferroni correction for multiple comparisons.

Original Significance Level (α):

Number of Comparisons (m):

P-value (optional, for adjustment):

Understanding the Bonferroni Correction

The Bonferroni correction is a statistical adjustment used to address the issue of multiple comparisons. When conducting multiple hypothesis tests, the probability of incorrectly rejecting a true null hypothesis (Type I error) increases. The Bonferroni correction controls for this by adjusting the significance level (\(\alpha\)) or the p-values to maintain the overall error rate.

Formula for the Bonferroni Correction

The Bonferroni-adjusted significance level is calculated as:

\[ \alpha_{adjusted} = \frac{\alpha}{m} \]

\(\alpha\): Original significance level (e.g., 0.05).
\(m\): Number of comparisons or tests conducted.

💡 Explanation: By dividing the original significance level (\(\alpha\)) by the number of comparisons, the Bonferroni correction ensures that the overall probability of making one or more Type I errors remains at or below the original \(\alpha\).

Adjusting the p-value

Alternatively, instead of adjusting \(\alpha\), the p-value for each hypothesis test can be multiplied by the number of comparisons. This yields the Bonferroni-adjusted p-value:

\[ p_{adjusted} = p \cdot m \]

\(p\): The original p-value from the hypothesis test.
\(m\): Number of comparisons or tests conducted.

💡 Explanation: Adjusting the p-value allows direct comparison with the original significance level (\(\alpha\)). If the adjusted p-value is less than or equal to \(\alpha\), the null hypothesis is rejected.

Comparison of Adjustments

Adjusting \(\alpha\): The threshold for significance is lowered, making it harder to reject the null hypothesis.
Adjusting p-values: The p-values are inflated by the number of comparisons, ensuring they are evaluated against the original significance level.

Example of Adjusting the p-value

Suppose you conduct five hypothesis tests (\(m = 5\)) and obtain a p-value of 0.01 for one of the tests. The Bonferroni-adjusted p-value is calculated as:

\[ p_{adjusted} = 0.01 \cdot 5 = 0.05 \]

If your original significance level (\(\alpha\)) is 0.05, this test would still be considered significant after adjustment.

Applications of Adjusting the p-value

Adjusting the p-value is particularly useful in scenarios where the significance level cannot be changed, such as pre-registered analyses. It is applied in various fields:

Medical Studies: To assess the effectiveness of multiple treatments in a single study.
Education Research: To evaluate the impact of different teaching strategies simultaneously.
Genomics: To control false positives when testing associations for thousands of genes.

Limitations of the Bonferroni Correction

While the Bonferroni correction is simple and widely used, it has limitations:

Conservativeness: The correction can be overly stringent when the number of comparisons is large, increasing the likelihood of Type II errors (failing to detect a true effect).
Independence Assumption: It assumes that all tests are independent. When this assumption is violated (e.g., in correlated data), the correction may be too strict.

Because of these limitations, other methods are sometimes preferred in certain scenarios:

Holm-Bonferroni Correction: A step-down procedure that is less conservative than the Bonferroni correction but still controls the family-wise error rate.
False Discovery Rate (FDR): Methods like the Benjamini-Hochberg procedure control the expected proportion of false positives, making them more appropriate for high-dimensional data (e.g., genomics).

💡 Takeaway: While the Bonferroni correction is robust and straightforward, researchers should consider alternative methods in studies with many comparisons or dependent tests to strike a balance between Type I and Type II error control.

Programmatically Calculating Adjusted p-values

Below are examples of how to calculate Bonferroni-adjusted p-values in Python, R, and JavaScript:

Python Implementation

Python (Adjusting p-value and Alpha)

# Input values
p = 0.01  # Original p-value
m = 5     # Number of comparisons
alpha = 0.05  # Significance level

# Calculate Bonferroni-adjusted values
p_adjusted = p * m
alpha_adjusted = alpha / m

print(f"Bonferroni-adjusted p-value: {p_adjusted:.5f}")
print(f"Bonferroni-adjusted significance level (Alpha): {alpha_adjusted:.5f}")

R Implementation

R (Adjusting p-value and Alpha)

# Input values
p <- 0.01  # Original p-value
m <- 5     # Number of comparisons
alpha <- 0.05  # Significance level

# Calculate Bonferroni-adjusted values
p_adjusted <- p * m
alpha_adjusted <- alpha / m

cat("Bonferroni-adjusted p-value:", round(p_adjusted, 5), "\n")
cat("Bonferroni-adjusted significance level (Alpha):", round(alpha_adjusted, 5), "\n")

JavaScript Implementation

JavaScript (Adjusting p-value and Alpha)

// Input values
const p = 0.01;  // Original p-value
const m = 5;     // Number of comparisons
const alpha = 0.05;  // Significance level

// Calculate Bonferroni-adjusted values
const pAdjusted = p * m;
const alphaAdjusted = alpha / m;

console.log(`Bonferroni-adjusted p-value: ${pAdjusted.toFixed(5)}`);
console.log(`Bonferroni-adjusted significance level (Alpha): ${alphaAdjusted.toFixed(5)}`);