Simultaneous Confidence Intervals by Scheffé’s Method
Scheffé’s method gives the widest guarantee of the three: every contrast among the treatment means simultaneously, not merely a list fixed beforehand. That is what makes it the right method when the comparison of interest was suggested by looking at the data — the case in which Bonferroni’s count \(m\) is not known in advance and the guarantee it offers does not apply.
The price is width. When only pairwise differences are wanted, Tukey’s intervals are shorter; when a short list is named in advance, Bonferroni’s are usually shorter still.
Theorem 3.7.5. Let \(\widehat {B}\) be the least square estimator for \(B\) in the linear model \(Y=XB+\varepsilon \), \(\quad \varepsilon \thicksim N(0,I\sigma ^2)\), then the \(100(1-\alpha )\%\)
confidence interval for \(C^tB\) is given by
\[C^t\widehat {B}\pm \sqrt {Kf^{\alpha }_{K,N-K}}\sqrt {\text {MSE}\,C^t\big (X^tX\big )^{-1}C}\]
for all vectors \(C\).
Note. The length of the interval depend on the number of components of \(B\) and not on the number of intervals like the simultaneous confidence interval using the Bonferroni’s method.
\[K=5,\quad \beta _j-\beta _{j'},\quad i=j'\]
Data: \(\widehat {B}= \begin {pmatrix} 29.4 & 29.6 & 28.0\\ \end {pmatrix}^t \) \[\text {MSE} = 9.7\, ,\quad df=12\, ,\quad X^tX =5I_3\]
Find \(95\%\) simultaneous C.Is for \(\beta _1, \beta _2, \beta _3\) using S-method. \[K=3,\quad N-K=12\] \begin {align*} \beta _1:\quad & \widehat {\beta }_1 \pm \sqrt {Kf^{\alpha }_{K,N-K}}\sqrt {\text {MSE}\,\big (X^tX\big )^{-1}_{11}}\\ & 29.4 \pm \sqrt {3f^{0.05}_{3,12}}\sqrt {9.7\times \frac {1}{5}}\\ & 29.4\pm \sqrt {3\times 3.49}\sqrt {\frac {9.7}{5}} \end {align*}
\begin {align*} \beta _2:\quad & 29.6 \pm \sqrt {3\times 3.49}\sqrt {9.7/5} \end {align*}
\begin {align*} \beta _3:\quad & 28.0 \pm \sqrt {3\times 3.49}\sqrt {9.7/5}\\ \end {align*}
Example 3.7.7. In a study of effectiveness of different rust inhibitors, four brands \(A,B,C\) and \(D\) were tested, each on a different set of five units. The basic results were
| inhibitor | \(A\) | \(B\) | \(C\) | \(D\) |
| mean | 43 | 89 | 67 | 40 |
The higher the value the more effective the rust inhibitor.
The error sum of squares is 72. Find \(95\%\) simultaneous C.Is of the individual parameters together with all the simple contrasts, using Bonferroni and S-method
- 1.
- Bonferroni \[\alpha = 0.05,\quad \text {MSE} = \frac {72}{16},\quad 10\implies \frac {\alpha }{10}=0.005\] \[\beta _j : \hat {\beta }_j \pm t^{0.005}_{16}\sqrt {4.5\times \frac {1}{5}}\] \[\hat {\beta }_j\pm 3.252\sqrt {4.5\times \frac {1}{5}}\] \[\beta _j-\beta _{j'}: \hat {\beta }_j-\hat {\beta }_{j'}\pm 3.252\sqrt {4.5\times \frac {2}{5}}\]
- 2.
- S-Method \begin {align*} \beta _j:\quad & \widehat {\beta }_j \pm \sqrt {Kf^{0.05}_{K,N-K}}\,\sqrt {\text {MSE}\,\cdot \frac {1}{n}}\\ & \widehat {\beta }_j \pm \sqrt {4f^{0.05}_{4,16}}\,\sqrt {4.5\times \frac {1}{5}}\\ & \widehat {\beta }_j \pm \sqrt {4\times 3.06}\,\sqrt {4.5\times \frac {1}{5}} \end {align*}
\[\beta _j-\beta _{j'}:\quad \widehat {\beta }_j-\widehat {\beta }_{j'} \pm \sqrt {4\times 3.06}\,\sqrt {4.5\times \frac {2}{5}}\]
\(\bullet \) Bonferroni is better method for confidence interval than the S-method.