6.3 Simultaneous Confidence Statements
A significant \(T^{2}\) says the mean vectors differ but not in which variables. Two devices answer that, and they differ in what they guarantee.
Theorem 6.8 (\(T^{2}\) intervals). With probability \(1-\alpha \), every linear combination \(\underline {a}'\left (\underline {\mu }_1-\underline {\mu }_2\right )\) simultaneously satisfies \[\underline {a}'\left (\overline {\underline {X}}_1-\overline {\underline {X}}_2\right ) \pm \sqrt {c^{2}}\, \sqrt {\underline {a}'\left (\tfrac {1}{n_1}+\tfrac {1}{n_2}\right )S_{\text {pooled}}\,\underline {a}},\] where \(c^{2} = \frac {(n_1+n_2-2)p}{n_1+n_2-p-1}F_{p,n_1+n_2-p-1,\alpha }\).
Note 6.9. The guarantee covers infinitely many linear combinations at once, including those chosen after seeing the data, which is what makes the intervals safe for exploration. The price is width. If only the \(p\) individual components are of interest, Bonferroni intervals using \(t_{n_1+n_2-2,\,\alpha /2p}\) are shorter and still control the overall error rate, but they cover only the \(p\) comparisons named in advance.
A common and instructive outcome is a significant \(T^{2}\) with no individual component significant. Nothing is wrong: the difference lies in a linear combination of the variables rather than in any one of them, and finding which combination is precisely what the discriminant function of Section 7 computes.
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