Linear Models by Shayle R. Searle

Author:Shayle R. Searle
Language: eng
Format: epub, pdf
ISBN: 9781118491768
Publisher: Wiley
Published: 2012-09-02T16:00:00+00:00

so that Q of (21) is

Thus the F-value is = 5.4 as in (75) and Table 6.7.

Consider the hypothesis

(78)

Here we have

which is the estimable function typified in the last line of Table 6.8, with wij = nij/ni.. From (78), (72) and (73)

so that by (21) the numerator sum of squares for testing the hypothesis in (78) is

(79)

This is no accident. Although R(α | μ) is, as indicated in (74), the numerator sum of squares for testing the fit of α after μ, it is also the numerator sum of squares for testing

(80)

Furthermore, this hypothesis is orthogonal to

(81)

orthogonal in the sense of (62); e.g., k′ of (78) and K′ of (77) are examples of (80) and (81) respectively, and k′ and every row of K′ satisfy (62). And, in testing (80) by using (21) it will be found that F(H) given there reduces to F(α | μ), as exemplified in (79). Hence F(α | μ) tests (80), the numerator sum of squares being R(α | μ); and F(β: α | μ, α) tests (81), its numerator being R(β: α | μ, α). The two numerator sums of squares R(α | μ) and R(β: α | μ, α) are statistically independent, as can be established by expressing each of them as quadratics in y and applying Theorem 4 of Chapter 2 (see Exercise 9).

The equivalence of the F-statistic for testing (80) and F(α | μ) can also be appreciated by noting in (80) that if the βij did not exist then (80) would represent

H: all α’s equal (in the absence of β’s)

which is indeed the context of earlier interpreting F(α | μ) as testing a after μ.

g. Models that include restrictions

The general effect of having restrictions as part of the model has been discussed in Sec. 5.6 and illustrated in detail in Sec. 2h of this chapter. The points made there apply equally as well here: restrictions that involve non-estimable functions of the parameters affect the form of functions that are estimable and hypotheses that are testable. Of particular interest here are restrictions with for all i, because then from Table 6.8 we see that μ + αi and αi − αi, are estimable, and hypotheses about them are testable. If, further, the wij of the restrictions are nij/ni, so that the restrictions are for all i, then (80) becomes H: all αi’s equal and (80) is, as we have just shown, tested by F(α | μ), which is independent of F(β:α | μ, α) that tests H: all β’s equal within each α-level. However, if the wij of the restrictions are not nij/ni, but some other form satisfying for all i, e.g., wij = 1/bi, the hypothesis H: all αi’s equal can still be tested, but the F-statistic will not equal F(α | μ), nor will its numerator be independent of that of F(β : α | μ, α).

h. Balanced data

The position with balanced data (nij = n for all i and j, and bi = b for all i) is akin to that of the 1-way classification, discussed in Sec.

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