Boole, Bonferroni, Tukey, Scheffe´: Seneta (1992) surveyed probability inequalities presented by Boole and Bonferroni and related results of Frechet. For an overview of ´

Tukey’s contributions to multiple comparisons, see Benjamini and Braun (2002) and Tukey (1994). With unbalanced data, Kramer (1956) suggested replacing s∕

√n in the Tukey interval by s

√1 2

[

(1∕na) + (1∕nb)

]

for groups a and

b. Hayter (1984) showed this is slightly conservative. For the normal linear model, Scheffe (1959) proposed a method ´

that applies simultaneously to all contrasts of c means. For estimating a contrast ∑

i ai

????i in the one-way layout (possibly unbalanced), it multiplies the usual estimated standard error s

√∑

i

(a2 i ∕ni

) for ∑

i ai ȳi by √(c − 1)F1−????,c−1,n−c to obtain the margin of error. For simple differences between means, these are wider than the Tukey intervals, because they apply to a much larger family of contrasts. Hochberg and Tamhane (1987) and Hsu

(1996) surveyed multiple comparison methods.