# sol4 - MAT 3378 Spring 2010 Assignment 4 Question 1...

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Assignment 4 : 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 2 ρ ), the treatment eﬀects satisfy r i =1 τ j = 0, the random errors ε ij are independent N (0 2 ) and ρ i 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 σ 2 { Y .j - Y .j 0 } . Solution: (a) σ { Y ij ,Y i 0 j 0 } = σ { μ .. + ρ i + τ j + ε ij .. + ρ i 0 + τ j 0 + ε i 0 j 0 } = σ { ρ i i 0 } + σ { ε ij i 0 j 0 } = σ 2 ρ + σ 2 , if

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sol4 - MAT 3378 Spring 2010 Assignment 4 Question 1...

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