ch13 - Chapter 13. Supplemental Text Material S13-1....

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Unformatted text preview: Chapter 13. Supplemental Text Material S13-1. Expected Mean Squares for the Random Model We consider the two-factor random effects balanced ANOVA model y i a j b k n ij i j ij ijk = + + + + = = = R S | T | µ τ β τβ ε ( ) , , , , , , , , , 1 2 1 2 1 2 " " " given as Equation (13-15) in the textbook. We list the expected mean squares for this model in Equation (13-17), but do not formally develop them. It is relatively easy to develop the expected mean squares from direct application of the expectation operator. For example, consider finding E MS E SS a a E SS A A A ( ) ( = − ) F H G I K J = − 1 1 1 where SS A is the sum of squares for the row factor. Recall that the model components τ β τβ i j i , ( and j ) β are normally and independently distributed with means zero and variances respectively. The sum of squares and its expectation are defined as σ σ σ τ β τ 2 2 2 , , and SS bn y y abn E SS bn E y E y abn A i i a A i i a = − = − F H G I K J = = ∑ ∑ 1 1 2 1 2 2 1 2 .. ... .. ... ( ) Now y y bn bn n n i ijk i i i k n j b .. . . .. ( ) = = + + + = = ∑ ∑ µ τ β τβ ε 1 1 + and 1 1 2 2 2 1 2 1 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 bn E y bn E bn bn bn bn bn bn a bn a bn ab n abn abn abn abn an an a i i a i i i i i i i a .. .. .. .. ( ) ( ) ( ) ( ) ( ) ( ) = = ∑ ∑ = + + + + + = + + + + = + + + + µ τ ε µτ µε τ µ σ σ σ σ µ σ σ σ σ τ β τβ τ β τβ ε Furthermore, we can show that y abn bn an n ... . . .. ... ( ) = + + + + µ τ β τβ ε so the second term in the expected value of SS A becomes 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 abn E y abn abn a bn b an abn abn abn bn an n ( ) ( ) ( ) ( ) ... = + + + + = + + + + µ σ σ σ µ σ σ σ σ τ β τβ τ β τβ σ We can now collect the components of the expected value of the sum of squares for factor A and find the expected mean square as follows: E MS E SS a a bn E y E y abn a a n a bn n bn A A i i a ( ) ( ) ( ) .. ... = − F H G I K J − − F H G I K J L N M O Q P = − − + − + = + + = ∑ 1 1 1 1 1 1 1 1 2 1 2 2 2 2 2 2 σ σ σ σ σ τβ τ τβ τ 2 σ This agrees with the first result in Equation (15-17). S13-2. Expected Mean Squares for the Mixed Model As noted in Section 13-3 of the textbook, there are several version of the mixed model, and the expected mean squares depend on which model assumptions are used. In this section, we assume that the restricted model is of interest. The next section considers the unrestricted model....
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This note was uploaded on 03/20/2011 for the course STATISTIC 101 taught by Professor Fandia during the Spring '10 term at UCLA.

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ch13 - Chapter 13. Supplemental Text Material S13-1....

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