Multiple Comparison Confidence Intervals
The tukey multiple comparison C.Is for all simple contrasts \(\beta _j-\beta _{j'}\quad j<j'\) with confidence coefficients of \(1-\alpha \) given by \[\widehat {\beta }_j-\widehat {\beta }_{j'}\pm \frac {1}{\sqrt {2}}\, l_{(1-\alpha ,k,v)} \,\sqrt {\text {MSE}\,\Big (\frac {1}{n_j}+\frac {1}{n_{j'}}\Big )}\]
- \(n_j=\) number of observations for the \(j^{\text {th}}\) \(+ve\).
- \(n_{j'}=\) number of observations for \(j'^{\text {th}}\) \(+ve\).
\[n_j=n\quad \widehat {\beta }_j-\widehat {\beta }_{j'} \pm l_{(1-\alpha ,k,v)}\,\sqrt {\frac {\text {MSE}}{n}}\]
What is \(\widehat {\beta }_j\) for one-way ANOVA? \(\widehat {\beta }_j=\overline {Y}._j\)
Example 3.7.8. Same data as in Example 3.7.7
| Inhibitors | \(A\) | \(B\) | \(C\) | \(D\) |
| Means | 43 | 89 | 67 | 40 |
\[\text {SSE} = 72\, ,\quad v=16\,, \quad X^tX=5I_4\] \(95\%\) Simultaneous C.I for all simple contrasts.
\begin {align*} \beta _j-\beta _{j'} :\quad & \widehat {\beta }_j-\widehat {\beta }_{j'}\pm l(0.95,4,16)\,\sqrt {\frac {4.5}{5}}\\ & \overline {Y}._j-\overline {Y}._{j'}\pm 4.05\,\sqrt {\frac {4.5}{5}}\\ \end {align*}