ch03 - Chapter 3 Supplemental Text Material S3-1. The...

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Chapter 3 Supplemental Text Material S3-1. The Definition of Factor Effects As noted in Sections 3-2 and 3-3, there are two ways to write the model for a single- factor experiment, the means model and the effects model. We will generally use the effects model y ia jn ij i ij =++ = = R S T µτ ε 12 ,, , " " where, for simplicity, we are working with the balanced case (all factor levels or treatments are replicated the same number of times). Recall that in writing this model, the i th factor level mean µ i is broken up into two components, that is τ i =+ i , where i is the i th treatment effect and is an overall mean. We usually define = = i i a a 1 and this implies that i i a = = 1 0. This is actually an arbitrary definition, and there are other ways to define the overall “mean”. For example, we could define µµ == = = ww ii i i a i a where 1 1 1 This would result in the treatment effect defined such that w i a = = 0 1 Here the overall mean is a weighted average of the individual treatment means. When there are an unequal number of observations in each treatment, the weights w i could be taken as the fractions of the treatment sample sizes n i /N . S3-2. Expected Mean Squares In Section 3-3.1 we derived the expected value of the mean square for error in the single- factor analysis of variance. We gave the result for the expected value of the mean square for treatments, but the derivation was omitted. The derivation is straightforward. Consider EM S E SS a Treatments Treatments () = F H G I K J 1 Now for a balanced design SS n y an y Treatments i i a =− = 11 22 1 .. . and the model is
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y ia jn ij i ij =++ = = R S T µτ ε 12 ,, , " " In addition, we will find the following useful: EEE E En Ea n ij i ij i () () ,() ,() .. . . . . εε σ === = = = 0 222 22 2 Now ESS E n yE an y Treatments i i a ( ) ( =− = 11 1 ) . Consider the first term on the right hand side of the above expression: E n y n n i i a ii i a ( 2 1 2 1 =+ + == ∑∑ ) Squaring the expression in parentheses and taking expectation results in E n y n an n an n a i i a i i a i i a [ ( ) . 2 1 22 2 2 1 2 1 + =+ + = ] because the three cross-product terms are all zero. Now consider the second term on the right hand side of : Treatments E an y an Ean n an i i a 1 2 1 2 2 F H G I K J + = µε since Upon squaring the term in parentheses and taking expectation, we obtain τ i i a = = 1 0. E an y an an an an [( ) ] 2 F H G I K J µσ since the expected value of the cross-product is zero. Therefore, E n an y an n a an Treatments i i a i i a i i a ( ) ( ) . =+ +− + + = = = 1 1 1 1 µ στ 2
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Consequently the expected value of the mean square for treatments is EM S E SS a an a n a Treatments Treatments i i a i i a () = F H G I K J = −+ + = = 1 1 1 1 22 1 2 2 1 στ σ τ This is the result given in the textbook. S3-3. Confidence Interval for σ 2 In developing the analysis of variance (ANOVA) procedure we have observed that the error variance is estimated by the error mean square; that is, 2 ± 2 = SS Na E We now give a confidence interval for . Since we have assumed that the observations are normally distributed, the distribution of 2 SS E 2 is . Therefore, χ 2
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ch03 - Chapter 3 Supplemental Text Material S3-1. The...

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