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03_MultComp_1r

03_MultComp_1r - Multiple Comparisons Contrasts among means...

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Multiple Comparisons Contrasts among means Notation: ψ i …. i th contrast among population means. i ψ ˆ …. The estimate of ψ i . Example: For 3 groups, 12 1124 3 13 21 35 2 23 3236 1 ˆˆ 2 2 2 YY Y Y Y •• + ψ= − ψ= + + (4.1-1) • Coefficients for the linear combinations 3 3 2 2 1 1 ˆ Y c Y c Y c i + + = ψ 1 ˆ ψ 2 ˆ ψ 3 ˆ ψ 4 ˆ ψ 5 ˆ ψ 6 ˆ ψ 1- 1 0 10- 1 01- 1 0.5 0.5 - 1 0.5 0.5 -1 0.5 0.5 (4.1-2) . 0 1 = = p j j c • Any contrast has to satisfy = = = 3 2 Y + = 2 2 Y + = 1 2 Y + =
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Pairwise v.s. Non-pairwise comparisons • Pairwise comparisons Compares only two groups. (e.g.) ψ 1 , ψ 2 , ψ 3 in the three-group example. The number of pairwise comparisons . 2 ) 1 ( 2 = = p p p • Non-pairwise comparisons Compares any combinations of groups. (e.g.) ψ 4 , ψ 5 , ψ 6 in the three-group example. The number of non-pairwise comparisons are infinite when n 3. Orthogonal Contrasts • Conceptually, orthogonal contrasts are mutually non-redundant contrast. (Example) ψ 2 and ψ 3 are not orthogonal contrasts, because both of them can be expressed as linear combinations of ψ 1 and ψ 4 . () 4 1 3 3 1 3 2 1 2 1 3 2 1 2 1 4 1 2 ˆ ˆ 2 1 ˆ 2 1 2 1 2 1 2 1 2 2 1 ˆ ˆ 2 1 ˆ ψ + ψ = ψ = + + = + = ψ + ψ = ψ
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03_MultComp_1r - Multiple Comparisons Contrasts among means...

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