ABSTRACT
The test for H 0 : cr1 = 0 against H a : cr1 > 0 shown in Table 5. 2. 3 is not exact under model (5.4.1). This is because in the unbalanced design neither n 1St/61 nor n2S~/62 are in general chi-squared random variables under the null hypothesis H0 : cr~ = 0. Tietjen (1974) recommended that one still use the test shown in Table 5.2.3 and noted that it is exact for the special case where Ku = K. For this special case
n2S~/62 has an exact chi-squared distribution and n1Srte 1 has an exact chi-squared distribution under the null hypothesis. Cummings and Gaylor (1974) recommended two alternative tests based on the Type I sums of squares and the Satterthwaite approximation. Their results suggest that non-chi-squaredness and dependence have canceling effects on the size of the test. Thus, they concluded such an approach wi11 perform similarly to a Satterthwaite approximation in a balanced design. Tan and Cheng (1984) compared these three tests with a fourth test statistic based on a Satterthwaite approximation involving the Type I sums of squares. Their results suggest that the test in Table 5.2.3 cannot be recommended for extremely unbalanced designs, but that the other three tests can generally be recommended. Khuri (1987) proposed an exact test for H0 : cr~ = 0 and compared the power of his test to the four tests examined by Tan and Cheng. When 1; = }, Khuri's test statistic reduces to STKIS~K where STK and S~K are the means squares defined earlier in this section. Although Khuri's exact test appears to have greater power than the approximate tests recommended by Tan and Cheng, it has the disadvantage that computation of the test statistic requires a non-unique partitioning of the error sums of squares. Given this disadvantage, we recommend testing H
~ = 0 with the lower bound defined in (5.4.5). In particular, one rejects H0 : a1 = 0 against Ha: ~ > 0 if the lower bound in (5.4.5) is greater than zero. Hernandez et al. (1992) showed this test has comparable power to the approximate tests studied by Tan and Cheng and is better at maintaining the stated test size. The power of the test is also comparable to Kburi' s exact test. Hussein and Milliken (1978b) provide an exact test of H 0 : cr1 = 0 where Var(A;) = d;al, Var(B;) = a~, and Kij = K. If d; = d, the test reduces to the one reported in Table 5.2.3.
