hw4 - 2 Y.j-Y.j Question 2 For the model in Question 1 we...

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MAT 3378 - Fall 2009 Assignment 5 : Deadline: Friday, July 16, 2009 (in class) Question 1: Consider a repeated measures design model: Y ij = μ .. + ρ i + τ j + ε ij , where the subject effects are ρ i independent N (0 , σ 2 ρ ), the treatment effects satisfy r i =1 τ j = 0, the random errors ε ij are independent N (0 , σ 2 ) and ρ i , ε ij are independent, for i = 1 , . . . , s and j = 1 , . . . , r . (a) Give the covariance structure associated to this model. That is give the different values taken by σ { Y ij , Y i 0 j 0 } . (b) Using the covariances from part (a), show that 1. σ 2 { Y .j } = σ 2 ρ + σ 2 s . 2. for j 6 = j 0 : σ { Y .j , Y .j 0 } = σ 2 ρ s 3. for j 6 = j 0 : σ 2 { Y .j - Y .j 0 } = 2 σ 2 s (c) Let j 6 = j 0 . Using the results from part (b), argue that we can use s 2 { Y .j - Y .j 0 } = 2 MSE s as a point estimate for
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Unformatted text preview: 2 { Y .j-Y .j } . Question 2: For the model in Question 1, we can show that E { SSTR } = σ 2 ( r-1) + s r X j =1 α 2 i . (a) Suppose that we are considered with the power for the test for A main ef-fects. Suppose that σ = 10 . 5 and that we would like to detect a range in the A 1 main effects of Δ = 20 units with high probability. Give the value of the non-centrality parameter for the corresponding non-central F distribution. (b) Refer to part (a), if α = 5%, the power of the test is the probability of which event? 2...
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