7. Product and Process Comparisons 7.4. Comparisons based on data from more than two processes 7.4.7. How can we make multiple comparisons? 7.4.7.3. Bonferroni's method 

Simple method  The Bonferroni method is a simple method that allows many comparison statements to be made (or confidence intervals to be constructed) while still assuring an overall confidence coefficient is maintained. 
Applies for a finite number of contrasts  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 pairwise comparisons specified in the Tukey procedure. 
Valid for both equal and unequal sample sizes  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 \(g\). 
Bonferroni general inequality 
Formally, the Bonferroni general inequality is presented by:
$$ P\left( \bigcap_{i=1}^g A_i \right) \ge 1  \sum_{i=1}^g P[\bar{A_i}] \, , $$
where \(A_i\) and its complement \(\bar{A_i}\) are any events. 
Interpretation of Bonferroni inequality  In particular, if each \(A_i\) is the event that a calculated confidence interval for a particular linear combination of treatments includes the true value of that combination, then the lefthand side of the inequality is the probability that all the confidence intervals simultaneously cover their respective true values. The righthand side is one minus the sum of the probabilities of each of the intervals missing their true values. Therefore, if simultaneous multiple interval estimates are desired with an overall confidence coefficient \(1  \alpha\), one can construct each interval with confidence coefficient \((1  \alpha/g)\), and the Bonferroni inequality insures that the overall confidence coefficient is at least \(1  \alpha\). 
Formula for Bonferroni confidence interval 
In summary, the Bonferroni method states that the confidence
coefficient is at least \(1  \alpha\)
that simultaneously all the following confidence limits for the \(g\)
linear combinations \(C_i\)
are "correct" (or capture their respective true values):
$$ \hat{C}_{i} \pm t_{1\alpha/(2g), Nr} \,\, s_{\hat{C}_i} $$
where $$ s_{\hat{C}_i} = \hat{\sigma}_\epsilon \, \sqrt{\sum^{r}_{i=1} \frac{c^2_i}{n_i}} \, . $$ 
Example using Bonferroni method  
Contrasts to estimate  We wish to estimate, as we did using the Scheffe method, the following linear combinations (contrasts): $$ \begin{eqnarray} C_1 & = & \frac{\mu_1 + \mu_2}{2}  \frac{\mu_3 + \mu_4}{2} \\ & & \\ C_2 & = & \frac{\mu_1 + \mu_3}{2}  \frac{\mu_2 + \mu_4}{2} \, , \\ \end{eqnarray} $$ and construct 95 % confidence intervals around the estimates. 
Compute the point estimates of the individual contrasts  The point estimates are: $$ \begin{eqnarray} \hat{C}_1 & = & \frac{\bar{Y}_1 + \bar{Y}_2}{2}  \frac{\bar{Y}_3 + \bar{Y}_4}{2} = 0.5 \\ & & \\ \hat{C}_2 & = & \frac{\bar{Y}_1 + \bar{Y}_3}{2}  \frac{\bar{Y}_2 + \bar{Y}_4}{2} = 0.34 \, .\\ \end{eqnarray} $$ 
Compute the point estimate and variance of \(C\) 
As before, for both contrasts, we have
$$ \sum_{i=1}^4 \frac{c_i^2}{n_i} = \frac{4(1/2)^2}{5} = 0.2 $$
and
$$ s_{\hat{C}}^2 = \hat{\sigma}_\epsilon^2 \, \sum_{i=1}^4 \frac{c_i^2}{4} = 1.331(0.2) = 0.2661 \, , $$
where \(\sigma_\epsilon^2 \) = 1.331 was computed in our previous example. The standard error is 0.5158 (the square root of 0.2661). 
Compute the Bonferroni simultaneous confidence interval 
For a 95 % overall confidence coefficient using the Bonferroni
method, the \(t\)
value is \(t_{10.05/(2\cdot2), \, 16} = t_{0.9875, \, 16}\)
= 2.473
(from the t table in Chapter 1).
Now we can calculate the confidence intervals for the two contrasts. For \(C_1\)
we have confidence limits 0.5 ± 2.473 (0.5158) and for \(C_2\)
we have confidence limits
0.34 ± 2.473 (0.5158).
Thus, the confidence intervals are: $$ 1.776 \le C_1 \le 0.776 $$ and $$ 0.936 \le C_2 \le 1.616 \, .$$

Comparison to Scheffe interval 
Notice that the Scheffé interval
for \(C_1\)
is:
$$ 2.108 \le C_1 \le 1.108 \, , $$
which is wider and therefore less attractive. 
Comparison of Bonferroni Method with Scheffé and Tukey Methods  
No one comparison method is uniformly best  each has its uses 
