ch05 - Chapter 5 Supplemental Text Material S5-1. Expected...

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Chapter 5 Supplemental Text Material S5-1. Expected Mean Squares in the Two-factor Factorial Consider the two-factor fixed effects model y ia jb kn ij i j ij ijk =++ + + = = = R S | T | µτ β τ ε () ,, , 12 " " " given as Equation (5-1) in the textbook. We list the expected mean squares for this model, but do not develop them. It is relatively easy to develop the expected mean squares from direct application of the expectation operator. Consider finding EM S E SS aa ESS A A A ( = ) F H G I K J = 1 1 1 where SS A is the sum of squares for the row factor. Since SS bn y y abn bn EyE y abn Ai i a i a =− F H G I K J = = 1 1 2 1 2 2 1 2 .. ... Recall that τβ . . . . , , , , == = = = 00 0 0 ji and 0 + , where the “dot” subscript implies summation over that subscript. Now y y bn bn n n bn bn i ijk i i i k n j b ii . . + + + =+ + = = 1 1 and 11 22 2 1 2 1 2 2 2 1 2 1 2 1 2 bn Ey bn Eb n b n b n b n b n bn a bn bn abn abn bn a i i a i i i i i a i i a i i a ()( ) ( ) ) = = ∑∑ =+ + + + + + L N M O Q P µ σ Furthermore, we can easily show that y abn = + so
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11 1 2 1 22 abn Ey abn E abn abn E abn abn abn abn abn abn () ( ) ... =+ + µε µ ε µσ ) Therefore EM S E SS a a ESS a abn bn a abn a ab n bn a A A A i i a i i a i i a = F H G I K J = ++ −+ L N M O Q P = L N M O Q P = = = 1 1 1 1 1 1 1 1 1 1 1 2 2 1 µτ σ στ τ 2 which is the result given in the textbook. The other expected mean squares are derived similarly. S5-2. The Definition of Interaction In Section 5-1 we introduced both the effects model and the means model for the two- factor factorial experiment. If there is no interaction in the two-factor model, then β ij i j = + + Define the row and column means as i ij j b j ij i a b a . . = = = = 1 1 Then if there is no interaction, ij i j = + ..
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where . It can also be shown that if there is no interaction, each cell mean can be expressed in terms of three other cell means: µµ µ == i j i a . / / j b . ij ij i j i j = + ′′ ′ This illustrates why a model with no interaction is sometimes called an additive model , or why we say the treatment effects are additive. When there is interaction, the above relationships do not hold. Thus the interaction term () τβ ij can be defined as ( ) τ β ij ij i j = + + or equivalently, ( ) ij ij ij i j i j ij ij i j i j = + =−−+ Therefore, we can determine whether there is interaction by determining whether all the cell means can be expressed as ij i j = + + . Sometimes interactions are a result of the scale on which the response has been measured. Suppose, for example, that factor effects act in a multiplicative fashion, µτβ ij i j = If we were to assume that the factors act in an additive manner, we would discover very quickly that there is interaction present. This interaction can be removed by applying a log transformation, since log log log log ij i j = + + This suggests that the original measurement scale for the response was not the best one to use if we want results that are easy to interpret (that is, no interaction). The log scale for the response variable would be more appropriate.
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ch05 - Chapter 5 Supplemental Text Material S5-1. Expected...

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