# Bonferroni Correction

Posted November 12, 2012

• Bonferroni Correction is also known as Bonferroni type adjustment
• Made for inflated Type I error (the higher the chance for a false positive; rejecting the null hypothesis when you should not)
• When conducting multiple analyses on the same dependent variable, the chance of committing a Type I error increases, thus increasing the likelihood of coming about a significant result by pure chance. To correct for this, or protect from Type I error, a Bonferroni correction is conducted.
• Bonferroni correction is a conservative test that, although protects from Type I Error, is vulnerable to Type II errors (failing to reject the null hypothesis when you should in fact reject the null hypothesis)
• Alter the p value to a more stringent value, thus making it less likely to commit Type I Error
• To get the Bonferroni corrected/adjusted p value, divide the original α-value by the number of analyses on the dependent variable. The researcher assigns a new alpha for the set of dependent variables (or analyses) that does not exceed some critical value: αcritical= 1 – (1 – αaltered)k, where k = the number of comparisons on the same dependent variable.
• However, when reporting the new p-value, the rounded version (of 3 decimal places) is typically reported. This rounded version is not technically correct; a rounding error. Example: 13 correlation analyses on the same dependent variable would indicate the need for a Bonferroni correction of (αaltered =.05/13) = .004 (rounded), but αcritical = 1 – (1-.004)13 = 0.051, which is not less than 0.05. But with the non-rounded version: (αaltered =.05/13) = .003846154, and αcritical = 1 – (1 – .003846154)13 = 0.048862271, which is in-fact less than 0.05! SPSS does not currently have the capability to set alpha levels beyond 3 decimal places, so the rounded version is presented and used.
• Another example: 9 correlations are to be conducted between SAT scores and 9 demographic variables. To protect from Type I Error, a Bonferroni correction should be conducted. The new p-value will be the alpha-value (αoriginal = .05) divided by the number of comparisons (9): (αaltered = .05/9) = .006. To determine if any of the 9 correlations is statistically significant, the p-value must be p < .006.

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