By Virginia Valian

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Here the setup is the same as in the case of a one-way layout except that in addition to the treatment factor, we have a nonrandom quantitative factor X,called a covariate or concomitant. )’. i , hi ri. and X i . and are the-sample means of the observations from the ith treatment. The vector 6 N(8, uZV)where V = { uii} is given by - i, 1 + (Xi. f. where N = C;”=, ni. 3 (B&nced Incomplete Block (BIB) Design). In a BIB design k treatments are arranged in 6 blocks each of size p < k such that 26 MULTIPLE COMPARISON PROCEDURES FOR RXED-EFFECTS LINEAR MODELS each treatment is replicated I times (at most once in each block) and every pair of treatments occurs together in A blocks.

Note that this model assumes no treatment X block interaction. The 0,’s and p,’s are not estimable but under the side conditions b Zrz1 0, = C,=, /3, = 0, the unique LS estimates are given by = f , p, = Y ,- F. ( 1 S j S b), and 1 - * PQi 8. =- ‘ Ak ( 1 5 i S k) . i is the mean of the Y,,’s from the jth block, and Qi is the ith “adjusted” treatment total given by the sum of all observations on the ith treatment -(l/p) X the sum of block totals for all blocks containing the ith treatment (1 S i S k ) .

1981) studied several conservative approximations to the critical constant [ and found that this Bonferroni approximation is generally the best choice. Recently Bofinger (1985) has proposed an MCP for ordering all pairs of treatment means (Ot, q ) in a balanced one-way layout that controls the Type 111 FWE (the probability that any pair (O,, 0,) is incorrectly ordered) at a designated level a. 2, the T-procedure can be used for ordering the treatment means (declare 0, > 0, or