ABSTRACT
To illustrate Khuri's method, we first compute a 1 - 2« = .90 simultaneous set of two-sided confidence intervals on 6 1 ,62 ,63 , and 64 . Since Q = 4 , an overall two-sided confidence of at least 90% is obtained using (3.5.2) with «K = (1 - (.90) 114)/2 = .013 . Using the function CINV of SAS©, we obtain F.013,3 ,co = 3.592, F.013,24,oo = 1.749, F.OI3:72,oo = 1.407, F .OI3:1100.oo = 1.097, F .987:3.oo = .0458, F.987:24,oo = .4693, F.987:72,oo = .6661 , and F 987: IIOO.co = .9075. The computed confidence intervals on 61,92 ,63 , and 64 are [12,990; 1,018,756], [1854; 6910], [326; 689], and [211; 255], respectively. These four intervals form the confidence region A. For this model, the expected mean squares satisfy the constraints 91 2: 63 2: 64 and 92 ~ 63 2: 64 • This defines the constrained space, R. By looking at the computed intervals that form A, it is seen that A is completely contained in R. Thus, we can use (3.5.3) to compute confidence intervals on the variance components CJ~, CJ~, CJ~, and CJk. Since CJ~ = (6 1 - 63)/300, the lower bound on CJ~ is computed using the lower bound on 91 and the upper bound on 93 • That is, L = (12,990 - 689)/300 = 41. The upper bound on cr~ is U = (1,018,756-326)/300 = 3395. In a similar manner, the confidence intervals on cr~, CJ~ and CJ~ are computed as [24.3; 137], [5.92; 39.8], and [211; 255], respectively. Now suppose the investigator wants a set of 90% simultaneous twosided confidence intervals on only CJ~, CJj, and CJk. The Bonferroni inequality can be used with (3.3.1) and (3.3.2) to provide shorter intervals than those computed with Khuri 's method. To illustrate, we compute the confidence interval on cr~ using this approach. If we desire equal confidence coefficients on the individual intervals for CJl, CJj, and
CJ~8 , equation (2.4.4) requires that 1 - 6a = .90 and« = .10/6 = .01667 for each interval. That is, each individual two-sided interval has a confidence coefficient of 1 - 2(.01667) = .967. Since CJ~ (61 - 93)/300, we replace c2 , S~, and n2 with c3 , S5, and n3 , respectively , in (3.3.1) and (3.3 .2). Using CINV and FINV of SAS© we obtain G 1 = .7069, H3 = .4736, G13 = .03398, H 1 = 17.41 , G3 = .2788, and H 13 = - 1.239. Placing these values into (3.3.1) and (3 .3.2) with ST = 46,659, s5 = 459, n 1 = 3, n3 = 72, and c 1 = c3 = 1/300, we compute L = 44 and U = 2861. This interval is shorter than the interval on CJ~ obtained from (3 .5. 3). Similar results are obtained for CJ~ and CJ~ .
