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10PowerFTest - POWER OF THE F-TEST Suppose C is a q p...

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POWER OF THE F-TEST A very common consulting question: “I’m planning a study to do .... ”. How many replicates (per treatment) should I use? I know 5 ways that can be used to determine an appropriate sample size As many as you can afford (time, money) n = 3 per treatment n so that s.e. (or s.e. of diff.) is ... n so that width of 1- α c.i. is ... n so that power of test (F or T) is ... for specified true difference Last is most common. e.g. NIH requires “proof that study is not underpowered” in grant applications. Often, need 80% power to detect an a-priori specified set of interesting differences 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 Suppose C is a q × p matrix such that C β is testable. Earlier, we established that the quadratic form incorporating C ˆ β d has a non-central F distribution F = ( C ˆ β d ) [ C ( X X ) C ] 1 ˆ σ 2 ( C ˆ β d ) / q F ( δ 2 ) q , n k where δ 2 = ( C β d ) [ C ( X X ) C ] 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 The non-centrality parameter δ 2 = ( C β d ) [ C ( X X ) C ] 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 X ) C . All else being equal, the noncentrality parameter increases as: the distance between C β and d increases, σ 2 gets smaller, the design, as defined by X , improves. (includes samplesize.) Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 3 / 21 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
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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 Let F α q , n k denote the 1 α quantile of the central F distribution with q and n k d.f. 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 . Consider a study with 4 trts and 8 reps per trt. Design study around test of Ho: μ i = μ 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
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