Suppose n 30 experimental units are available let ni

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Unformatted text preview: rality parameter δ2 = (Cβ − d) [C(X X)− C ] σ2 −1 (Cβ − d) increases as . . . the distance between Cβ and d increases, σ 2 decreases, the design, as defined by X, improves (e.g., sample size increases). Copyright c 2012 (Iowa State University) Statistics 511 4 / 10 Consider a completely randomized design (CRD) with three treatments. Suppose n = 30 experimental units are available. Let ni denote the number of experimental units assigned to treatment i for i = 1, 2, 3. Suppose we consider the model Yij = µi + ij Copyright c 2012 (Iowa State University) i = 1, 2, 3; j = 1, . . . , ni ; ∼ N (0, σ 2 I) Statistics 511 5 / 10 Consider the test of H0 : µ1 = µ2 = µ3 . This null hypothesis is equivalent to H0 : Cβ = 0, where µ1 1 −1 0 C= and β = µ2 . 0 1 −1 µ3 (Actually, C could be any 2 × 3 matrix with row space equal to the row space of the C matrix above.) Copyright c 2012 (Iowa State University) Statistics 511 6 / 10 n1 0 0 X X = 0 n2 0 0 0 n3 −1 Copyright c 2012 (Iowa State University) = 1 n2 + 1 n2 0 (X X)−1 = 0 0 1 n1 1 + n2 1 − n2 C(X X)−1 C = C(X X)−1 C 1 n1 1 n3 1 n1 1 n2 + 1 n2 0 0 0 1 n3 1 − n2 1 1 n2 + n3 1 1 n2 1 n1 n2 + 1 n1 n3 + 1 n2 n3 Statistics 511 7 / 10 = 1 n2 + 1 n2 1 n3 1 n1 1 n2 + n1 n2 n3 n 1 n2 Suppose µ1 = 3, µ2 = 2, µ3 = 1, and σ 2 = 1 Then Cβ − d = δ 2 1 1 = [ 1, 1] = Copyright c 2012 (Iowa State University) and 1 n2 + 1 n2 1 n3 1 n1 1 n2 + 1 n2 1 1 n1 n2 n3 n 4n1 n3 + n1 n2 + n2 n3 . n Statistics 511 8 / 10 Now suppose µ1 = 3, µ2 = µ3 = 1, σ 2 = 1 Then Cβ − d = 2 0 δ 2 = [ 2, 0] = Copyright c 2012 (Iowa State University) and 1 n2 + 1 n2 1 n3 1 n1 1 n2 + 1 n2 2 0 n1 n2 n3 n 4n1 (n2 + n3 ) . n Statistics 511 9 / 10 µ1 µ2 µ3 321 321 321 311 311 311 Copyright c 2012 (Iowa State University) n1 5 10 12 5 10 12 n2 20 10 6 20 10 6 n3 5 10 12 5 10 12 δ2 10.0 20.0 24.0 ¯ 16.6 ¯ 26.6 28.8 Statistics 511 10 / 10...
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