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designexper

# designexper - Design and Analysis of Experiments Pair-wise...

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1 ©Mike Pore - Mar 2005 Design and Analysis Design and Analysis of Experiments of Experiments Pair-wise t-tests 1-way ANOVAs 2-way ANOVAs Replications Factorial Designs Multiple Comparisons Response Surfaces 2 K Designs Fractional Factorial Designs Randomized Block Designs Fixed and Random Effects Crossed vs. Nested Factors

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2 ©Mike Pore - Mar 2005 A Test of Two “Treatments” A Test of Two “Treatments” How do we test for differences in means? H O : μ 1 = μ 2 H A : μ 1 μ 2 T 1 Control T 2 Treatment X 11 X 21 X 12 X 22 : : X 1n X 2n S 1 S 2 1 X 2 X
3 ©Mike Pore - Mar 2005 Multiple Treatments Multiple Treatments How do we test for differences in means? H O : μ i = μ j for all i, j H A : μ i μ j for some i, j T 1 Control T 2 T 3 . . . T K X 11 X 21 X 31 . . . X K1 X 12 X 22 X 32 . . . X K2 : : : : : X 1n X 2n X 3n . . . X Kn . . . S 1 S 2 S 3 . . . S K 1 X 2 X 3 X K X

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4 ©Mike Pore - Mar 2005 Multiple t-tests Multiple t-tests H O : μ 1 = μ 2 H A : μ 1 μ 2 H O : μ 2 = μ 3 H A : μ 2 μ 3 …………. …………. …………. H O : μ K-1 = μ K There are tests. Probs of Type I and Type II errors effected by multiple tests. If all H O true, then the prob of falsely rejecting at least one of the tests is ( 29 2 1 2 - = K K K ( 29 ( 29 2 1 1 - α - K K
5 ©Mike Pore - Mar 2005 Multiple t-tests Multiple t-tests H O : μ i = μ j for all i, j H A : μ i μ j for some i, j How to test with a given level of significance, α O ? For multiple t-test, we would have to set and solve for the α for each test. . . . Or, test all the means simultaneously. Do this! ( 29 ( 29 ( 29 2 1 1 1 - α - - = = α K K O I Type P

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6 ©Mike Pore - Mar 2005 A One-Way Design A One-Way Design T 1 Control T 2 T 3 . . . T K X 11 X 21 X 31 . . . X K1 X 12 X 22 X 32 . . . X K2 : : : : : X 1n X 2n X 3n . . . X Kn . . . S 1 S 2 S 3 . . . S K 1 X 2 X 3 X K X ( 29 ( 29 ( 29 n x x k s s n s n s x n k x k x p x k i i x i k i k i i p k i n j ij k i i 2 2 2 1 2 2 1 1 2 2 1 1 1 1 1 1 1 1 1 σ = σ - - = = - - = = = ∑∑ = = = = = = in the case n i = n
7 ©Mike Pore - Mar 2005 A One-Way Design A One-Way Design ? s n ˆ for s assumption the are What ? s ˆ for s assumption the are What ? better is , ˆ , estimator Which s ˆ also and s n ˆ or n so n X p p p p p p X p X p p X 2 2 2 2 2 2 2 2 2 2 2 2 2 = σ = σ σ = σ = σ σ = σ σ = σ

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8 ©Mike Pore - Mar 2005 A One-Way Design A One-Way Design j , i some for : H j , i all for : H j i A j i O μ μ μ = μ 2 2 2 2 X p A X p O n : H n : H σ < σ σ = σ A test on means A test on variances }
9 ©Mike Pore - Mar 2005 A One-Way Design A One-Way Design Source of Variation Degrees of Freedom Sum of Squares MS F p Between Treatments K-1 Within Treatments K (n-1) Total nK - 1 ( 29 2 1 1 ∑∑ = = - K i n j ij X X ( 29 2 1 1 ∑∑ = = - K i n j i ij X X ( 29 2 1 = - K i i X X n Analysis of Variance Table

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10 ©Mike Pore - Mar 2005 A One-Way Design A One-Way Design Source of Variation df SS Mean Squares F statistic P value Between K-1 SS B Within K (n-1) SS W Total nK - 1 SS T Analysis of Variance Table ( 29 1 - K SS B ( 29 1 - n K SS W W B MS MS W B MS MS F P
11 ©Mike Pore - Mar 2005 A One-Way Design

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