10PowerFTest

10PowerFTest - POWEROFTHEF-TEST :“I’mplanningastudytodo.”.Howmanyreplicates(pertreatment)shouldIuse samplesize (time,money n =3pertreatment n

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Unformatted text preview: POWEROFTHEF-TEST Averycommonconsultingquestion:“I’mplanningastudytodo ....”.Howmanyreplicates(pertreatment)shouldIuse? Iknow5waysthatcanbeusedtodetermineanappropriate samplesize Asmanyasyoucanafford(time,money) n =3pertreatment n sothats.e.(ors.e.ofdiff.)is... n sothatwidthof1- α c.i.is... n sothatpoweroftest(ForT)is...forspecifiedtruedifference Lastismostcommon.e.g.NIHrequires“proofthatstudyisnot underpowered”ingrantapplications. Often,need80%powertodetectan a-priori specifiedsetof interestingdifferences Wefocusonthepowerapproach. 1 − α ci.widthapproachisthe powerapproachwithpower= 1 − α/ 2 forrank1tests. Copyright c 2011Dept.ofStatistics(IowaStateUniversity) Statistics511 1/21 Suppose C isa q × p matrixsuchthat C β istestable. Earlier,weestablishedthatthequadraticformincorporating C ˆ β − d hasanon-centralFdistribution F = ( C ˆ β − d ) [ C ( XX ) − C ] − 1 ˆ σ 2 ( C ˆ β − d ) / q ∼ F ( δ 2 ) q , n − k where δ 2 = ( C β − d ) [ C ( XX ) − C ] − 1 σ 2 ( C β − d ) Copyright c 2011Dept.ofStatistics(IowaStateUniversity) Statistics511 2/21 Non-centralityparameterofanFdistribution Thenon-centralityparameter δ 2 = ( C β − d ) [ C ( XX ) − C ] − 1 ˆ σ 2 ( C β − d ) quantifiesthediscrepancybetween C β and d withrespectto Var ( C ˆ β )= σ 2 C ( XX ) − C . Allelsebeingequal,thenoncentralityparameterincreasesas: thedistancebetween C β and d increases, σ 2 getssmaller, thedesign,asdefinedby X ,improves.(includessamplesize.) Copyright c 2011Dept.ofStatistics(IowaStateUniversity) Statistics511 3/21 1 2 3 4 5 7 . 5 1 1- 3- 2- 1 1 2 3- 3- 2- 1 1 2 3 + σ 2 = 1 Copyright c 2011Dept.ofStatistics(IowaStateUniversity) Statistics511 4/21 1 2 2 3 3 4 5-3-2-1 1 2 3- 3- 2- 1 1 2 3 + σ 2 = 2 Copyright c 2011Dept.ofStatistics(IowaStateUniversity) Statistics511 5/21 Let F α q , n − k denotethe 1 − α quantileofthecentral F distribution with q and n − k d.f. Thepowerofthe α − leveltestof H : C β = d isgivenby P ( F ≥ F α q , n − k ) andisanincreasingfunctionofthenoncentralityparameter δ 2 . Considerastudywith4trtsand8repspertrt.Designstudy aroundtestofHo: μ i = μ ∀ i .Consider3distributions:central F 3 ,...
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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.

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10PowerFTest - POWEROFTHEF-TEST :“I’mplanningastudytodo.”.Howmanyreplicates(pertreatment)shouldIuse samplesize (time,money n =3pertreatment n

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