# Bonferroni adjustment for multiplicity

Calculating a confidence interval always takes more or less the same form: it involves an estimate; a multiplier; and the standard error, as below.

Normally, the multiplier is collected from a table, based on the degrees of freedom, the significance level of the confidence interval (which is oftentimes a number like 95% of 98%), and whether the confidence interval is one-sided or two-sided. Generally, we will report a two-sided range in the format

The Bonferroni confidence interval does not drastically innovate on this approach. However, it does adjust for the effect of working with multiple variables that may be influencing one another. It is named after Carlo Emilio Bonferroni, and was developed by Olive Jean Dunn. Details are below,

Approach | Multiplier | Description |
---|---|---|

Bonferroni confidence interval | The "Bonferroni adjustment" simply involves changing the multiplier we choose. This is useful when the variables may undergo a linear combination. It essentially involves crafting a bespoke multiplier that results in a wider confidence interval which takes the multiple variables into consideration. It is generally not possible to look the multiplier up with a t-distribution table, but it is easily achievable using statistical software or spreadsheet software. The degree of freedom is the usual when calculating a t-multiplier. The probability is the usual ( for a one-sided test; for a two-sided test) but then divided by the number of variables.
For example, for a two-sided test with a 95% significance level and four variables, we will plug in for the t-multiplier with | |

Bonferroni corrected confidence interval | This involves changing the multiplier we choose, as well as processing the standard error. This is useful when the variables will not undergo a linear combination. We choose the t-multiplier based on a significance level of |