to the set of all contrasts.

[Use the fact that the event |yi − y0| ≤ ∆ for i = 1,...,s is equivalent to the event |

s i=0 ciyi| ≤ ∆s i=1 |ci| for all (c0,...,cs) satisfying s i=0 ci = 0.]

Note. As is pointed out in Problems 9.26(iii) and 9.32, the intervals resulting from the extension of the Tukey (and Dunnett) methods to all contrasts are shorter than the Scheff´e intervals for the differences for which these methods were designed and for contrasts close to them, and longer for some other contrasts.

For details and generalizations, see for example Miller (1981), Richmond (1982), and Shaffer (1977a).