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10PowerFTest(1)

# 10PowerFTest(1) - POWER OF THE F-TEST A very common...

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POWER OF THE F-TEST I A very common consulting question: “I’m planning a study to do .... ”. How many replicates (per treatment) should I use? I I know 5 ways that can be used to determine an appropriate sample size I As many as you can afford (time, money) I n = 3 per treatment I n so that s.e. (or s.e. of diff.) is ... I n so that width of 1- α c.i. is ... I n so that power of test (F or T) is ... for specified true difference I Last is most common. e.g. NIH requires “proof that study is not underpowered” in grant applications. I Often, need 80% power to detect an a-priori specified set of interesting differences I We focus on the power approach. 1 - α ci. width approach is the power approach with power = 1 - α/ 2 for rank 1 tests. Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 1 / 21

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I Suppose C is a q × p matrix such that C β is testable. I Earlier, we established that the quadratic form incorporating C ˆ β - d has a non-central F distribution F = ( C ˆ β - d ) 0 [ C ( X 0 X ) - C 0 ] - 1 ˆ σ 2 ( C ˆ β - d ) / q F ( δ 2 ) q , n - k where δ 2 = ( C β - d ) 0 [ C ( X 0 X ) - C 0 ] - 1 σ 2 ( C β - d ) Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 2 / 21
Non-centrality parameter of an F distribution I The non-centrality parameter δ 2 = ( C β - d ) 0 [ C ( X 0 X ) - C 0 ] - 1 ˆ σ 2 ( C β - d ) quantifies the discrepancy between C β and d with respect to Var ( C ˆ β ) = σ 2 C ( X 0 X ) - C 0 . I All else being equal, the noncentrality parameter increases as: I the distance between C β and d increases, I σ 2 gets smaller, I the design, as defined by X , improves. (includes samplesize.) Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 3 / 21

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1 2 3 4 5 7.5 10 10 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 + σ 2 = 1 Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 4 / 21
1 2 2 3 3 4 5 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 + σ 2 = 2 Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 5 / 21

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I Let F α q , n - k denote the 1 - α quantile of the central F distribution with q and n - k d.f. I The power of the α - level test of H 0 : C β = d is given by P ( F F α q , n - k ) and is an increasing function of the noncentrality parameter δ 2 . I Consider a study with 4 trts and 8 reps per trt. Design study around test of Ho: μ i = μ 2200 i . Consider 3 distributions: central F 3 , 28 (null), nc F 3 , 28 with δ 2 = 5 and nc F 3 , 28 with δ 2 = 10 Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 6 / 21
A COMPARISON OF CENTRAL AND NON-CENTRAL F DENSITIES 0

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10PowerFTest(1) - POWER OF THE F-TEST A very common...

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