ABSTRACT
Hypothesis tests for the fixed effect parameters in (7.5.1) are shown in Table 7.5.2. All of these tests are exact and have optimal properties. Confidence intervals on the fixed effect parameters are obtained in the following manner. The best linear unbiased estimator for the contrast wa = I jcjaj with 'J.{j = 0 is wa = Ijci *j*· The variance of wa is Var(wa) = (Ijcj)(cr£ + K<T?v)I(IK) = (IjcJ)63 /(/K). Using Sj to estimate e3, an exact 1 -a confidence interval on (l)a is
H0 : 13k = 0 for all k Ha : 13k =I= 0 for at least one k
(7.5.2)
In a similar manner, an exact 1 - a confidence interval on the contrast w13 = Ikckl3k with Ikck = 0 is
(7 .5.3)
situations where the (cx(3)jk interaction effect is significant. Ifj = j', an exact 1 - a confidence interval on the expected value of this difference is
(7.5 .4)
If j =I= j', Var(Y *jk - Y *j'k') = 2(cr?v + a~)! I . Examination of Table 7.5.1 shows that no single mean square can be used to provide an unbiased esimator of this variance. Thus, as was the case in Section 7.3, no exact confidence interval based on the ANOVA mean squares is available. Using the results of Burdick and Sielken (1978), an exact confidence interval on the expected difference of the cell means can be obtained. However, like the exact method of Khuri (1984), it suffers a loss of power in comparison to competing approximate methods. The Satterthwaite approximation is one such method. Here we approximate m'YI-y as a chi-squared random variable with m degrees of freedom where
'Y = cr?v + cr~ S~ + (K - l)S~ 'Y = K
and
(7 .5.5)
Using (7.5 .5), an approximate 1 - a confidence interval on the expected value of Y *jk - Y */ k' for j =I= j' is
(7.5.6)
Table 7 .5.3 Analysis of Variance for Seed Data
In practice , the greatest integer less than m is used in determining F cx:t.m· Unlike the Satterthwaite approximation employed in (7.3.2), 'Y has positive coefficients on both S~ and S~ and is always positive. Thus , the approximation is expected to work well unless n3 is small and n6 - n3 is large. Milliken and Johnson (1984, p. 303) recommend using F* in place ofF ex: t.m in (7 .5.6) where
Example 7.5.1 Steel and Torrie (1960, pp. 236-240) report the results of a split plot design used to study the yields of J = 4 lots of oats forK = 4 chemical seed treatments. The seed lots, factor A, were assigned at random to whole plots and the chemical seed treatments, factor B, were assigned at random to the subplots within each whole plot. Each seed lot was applied in I= 4 blocks . Tables 7.5.3 and 7.5.4 report the ANOVA table and the sample means for the 16 seed lot-seed treatment combinations.
