8.2 General Strategies

Consider a balanced random model with Q sources of variation represented by the ANOVA shown in Table 8.1. Assuming all random errors are independent and normally distributed, the are jointly independent and each has a chi-squared distribution with n _q degrees of freedom for q = 1 , , Q (see, e.g., Hocking [31, p. 460]). These results are used to construct confidence intervals for all four parameters shown in Table 8.2, where c _q, d _r, and e _t ? 0. Note that ? ₃ and ? ₄ differ in two respects. First, ? ₃ has nonnegative linear combinations in both the numerator and the denominator, whereas ? ₄ allows negative linear combinations in the numerator. Second, the same expected mean square cannot appear in both the numerator and the denominator of ? ₃, but it can in ? ₄. Although ? ₃ is a special case of ? ₄, it is useful to consider the two cases separately. Note that neither ? ₃ nor ? ₄ allows negative coefficients in the denominator.