7. Product and Process Comparisons
7.4. Comparisons based on data from more than two processes
7.4.6. How can we make multiple comparisons?

7.4.6.3.

Bonferroni's method

This method applies to an ANOVA situation when the analyst has picked out a particular set of pairwise comparisons or contrasts or linear combinations in advance. This set is not infinite, as in the Scheffé case, but may exceed the set of paiwise comparisons specified in the Tukey procedure. 

The Bonferroni method is valid for equal and unequal sample sizes. We restrict ourselves to only linear combinations or comparisons of treatment level means (pairwise comparisons and contrasts are special cases of linear combinations). We denote the number of statements or comparisons in the finite set by g. 

Formally, the Bonferroni general inequality is presented by: 

where A_i   and its complement  are any events, such as the event that a calculated confidence interval for a particular linear combination of treatments includes the true value of that combination. Therefore, if multiple interval estimates are desired with an overall confidence coefficient 1-a , one constructs each interval with confidence coefficient (1-a/g), and the Bonferroni inequality insures that the overall confidence coefficient is at least 1-a. 

In summary, the Bonferroni method states that the confidence coefficient is at least 1-a that simultaneously all the following confidence limits for the g linear combinations C_i are correct: 

Example

We wish to estimate, as we did using the Scheffe method, the following linear combinations (contrasts): 

and construct 95 percent confidence intervals around the estimates.. 

The point estimates are: 

As before, for both contrasts, we have 

and  so that the standard error is .5158 (the square root of .2661). 

For a 95 percent overall confidence coefficient using the Bonferroni method the t-value is t_.05/4;16 = t_.0125;16 = 2.473. Now we can calculate the confidence intervals for the two contrasts.For C₁ we have confidence limits -.5 ± 2.473 (.5158) and for C₂ we have confidence limits .34 ± 2.473 (.5158). 

Thus, the confidence intervals are: 

Notice that the Scheffé interval for C₁ is:  which is wider and therefore less attractive. 
 

Comparison of Bonferroni Method with Scheffé and Tukey Methods.

1. If all pairwise comparisons are of interest, Tukey has the edge. If only a subset of pairwise comparisons are required, Bonferroni may sometimes be better. 

2. When the number of contrasts to be estimated is small, (about as many as that there are factors) Bonferroni is better than Scheffé. Actually, unless the number of desired contrasts is at least twice the number of factors Scheffé  will always show wider confidence bands than Bonferroni. 

3. Many computer packages calculate all three methods. So, study the output and select the method with the smallest confidence band.