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Unformatted text preview: POWER OF THE FTEST I A very common consulting question: Im 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 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 apriori specified set of interesting differences I We focus on the power approach. 1 ci. width approach is the power approach with power = 1 / 2 for rank 1 tests. Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 1 / 21 I Suppose C is a q p matrix such that C is testable. I Earlier, we established that the quadratic form incorporating C  d has a noncentral F distribution F = ( C  d ) [ C ( X X ) C ] 1 2 ( C  d ) / q F ( 2 ) q , n k where 2 = ( C  d ) [ C ( X X ) C ] 1 2 ( C  d ) Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 2 / 21 Noncentrality parameter of an F distribution I The noncentrality parameter 2 = ( C  d ) [ C ( X X ) C ] 1 2 ( C  d ) quantifies the discrepancy between C and d with respect to Var ( C ) = 2 C ( X X ) C . I All else being equal, the noncentrality parameter increases as: I the distance between C and d increases, I 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 1 2 3 4 5 7.5 10 10 321 1 2 3321 1 2 3 + 2 = 1 Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 4 / 21 1 2 2 3 3 4 5 321 1 2 3321 1 2 3 + 2 = 2 Copyright c 2011 Dept. of Statistics (Iowa State University) Statistics 511 5 / 21 I Let F q , n k denote the 1 quantile of the central F distribution with q and n k d.f. I The power of the  level test of H : C = d is given by P ( F F q , n k ) and is an increasing function of the noncentrality parameter 2 ....
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This note was uploaded on 02/11/2012 for the course STAT 511 taught by Professor Staff during the Spring '08 term at Iowa State.
 Spring '08
 Staff

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