A Proof of the Conjecture that the TUKEY-KRAMER Multiple Comparisons Procedure is Conservative

where the eij are independent N(O, a2) random variables and gii is the mean of the ith treatment (1 c i c k). The gii and a2 are unknown parameters. Let Xi be the sample mean of the ith treatment based on ni observations (1 c i c k), and let S2 be an unbiased estimate of a2 which is distributed independently of the Xi as a a 2x / v random variables. Usually the ANOVA mean square error with v = ^=, nk degrees of freedom is used as the estimate S2. A commonly occurring inference problem in practice is that of making simultaneous pairwise comparisons between the treatment means jui. Tukey (1953) proposed his celebrated T-procedure to do this in the special case when all the ni are equal to a common sample size n (say). This procedure can be summarized by the following probability statement which gives exact (1 a)-level joint confidence intervals for all the differences ,u -j: