Bonferroni

Bonferroni refers to two related concepts in probability and statistics named after the Italian mathematician Carlo Emilio Bonferroni (1892–1960). The term commonly denotes the Bonferroni inequality in probability theory and the Bonferroni correction used in multiple hypothesis testing.

The Bonferroni inequality provides a bound on the probability of the union of events. For events A1, A2, ..., Am, it states that P(∪i Ai) ≤ ∑i P(Ai). A simple corollary is that the probability that any of several events occurs cannot exceed the sum of their individual probabilities. The inequality becomes exact in certain limiting cases and is widely used as a tool to derive conservative bounds.

In statistics, the Bonferroni correction controls the family-wise error rate when performing multiple hypothesis tests. If m hypotheses are tested simultaneously and an overall significance level alpha is desired, each individual test uses alpha/m as its threshold. Equivalently, a p-value is compared to alpha/m. The method is valid under any dependency structure among tests but is conservative, especially when many tests are performed, potentially reducing statistical power.

Variants and related methods include Holm-Bonferroni, a stepwise procedure that often provides greater power while preserving control of the family-wise error rate. The Dunn-Sidak correction is another alternative. In modern practice, false discovery rate procedures such as the Benjamini–Hochberg method are sometimes preferred when tolerating some false positives in exchange for higher power.

Bonferroni's ideas have had a lasting impact on statistical methodology and applied testing.