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Wuchap_03a_09fall - Unit 3 Experiments with More Than One...

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Unit3:ExperimentswithMoreThanOneFactor Sources:Chapter3. Pairedcomparisondesign(Section3.1). Randomizedblockdesign(Section3.2). Two-wayandmulti-waylayoutwithfixedeffects(Sections3.3and3.5). LatinandGraeco-Latinsquaredesign(Sections3.6and3.7). Balancedincompleteblockdesign(Section3.8). Split-plotdesign(Section3.9). Analysisofcovariance(ANCOVA)(Section3.10). Transformationofresponse(Section3.11). 1 SewageExperiment Objective: TocomparetwomethodsMSIandSIBfordeterminingchlorine contentinsewageeffluents; y =residualchlorinereading. Table1:ResidualChlorineReadings,SewageExperiment Method Sample MSI SIB d i 1 0.39 0.36 - 0.03 2 0.84 1.35 0.51 3 1.76 2.56 0.80 4 3.35 3.92 0.57 5 4.69 5.35 0.66 6 7.70 8.33 0.63 7 10.52 10.70 0.18 8 10.92 10.91 - 0.01 ExperimentalDesign: Eightsampleswerecollectedatdifferentdosesand contacttimes.Twomethodswereappliedtoeachoftheeightsamples.Itis a pairedcomparison designbecausethepairoftreatmentsareappliedtothe samesamples(orunits). 2
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PairedComparisonDesignvs.UnpairedDesign PairedComparisonDesign :Twotreatmentsarerandomlyassignedtoeach block of two units.Caneliminateblock-to-blockvariationandiseffectiveif suchvariationislarge. Examples:pairsoftwins,eyes,kidneys,leftandrightfeet. (Subject-to-subjectvariationmuchlargerthanwithin-subjectvariation). UnpairedDesign :Eachtreatmentisappliedtoa separate setofunits,or calledthe two-sample problem.Usefulifpairingisunnecessary;alsoithas moredegreesoffreedomforerrorestimation(seepage5). 3 Paired t tests Paired t test: Let y i 1 , y i 2 betheresponsesoftreatments1and2forunit i, i = 1 , . . . N .Let d i = y i 2 - y i 1 , ¯ d and s 2 d thesamplemeanandvariance of d i . t paired = N ¯ d/s d Thetwotreatmentsaredeclaredsignificantlydifferentatlevel α if | t paired | > t N - 1 ,α/ 2 . (1) 4
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Unpaired t tests Unpaired t test: Theunpaired t testisappropriateifwerandomlychoose N ofthe 2 N unitstoreceiveonetreatmentandassigntheremaining N unitstothesecondtreatment.Let ¯ y i and s 2 i bethesamplemeanandsample varianceforthe i thtreatment, i =1and2.Define t unpaired = ( ¯ y 2 - ¯ y 1 ) / radicalBig ( s 2 2 /N ) + ( s 2 1 /N ) . Thetwotreatmentsaredeclaredsignificantlydifferentatlevel α if | t unpaired | > t 2 N - 2 ,α/ 2 . (2) Notethatthedegreesoffreedomin(1)and(2)are N - 1 and 2 N - 2 respectively.Theunpaired t testhasmoredf’sbutmakesurethatthe unit-to-unitvariationisundercontrol(ifthismethodistobeused). 5 AnalysisResults: t tests t paired = 0 . 4138 0 . 321 / 8 = 0 . 4138 0 . 1135 = 3 . 645 , t unpaired = 5 . 435 - 5 . 0212 radicalbig (17 . 811 + 17 . 012) / 8 = 0 . 4138 2 . 0863 = 0 . 198 . The p valuesare Prob ( | t 7 | > 3 . 645) = 0 . 008 , Prob ( | t 14 | > 0 . 198) = 0 . 848 . Unpaired t testfailstodeclaresignificantdifferencebecauseits denominator2.0863istoolarge.Why?Becausethedenominatorcontains thesample-to-samplevariationcomponent. 6
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AnalysisResults:ANOVAand F tests Wrongtoanalyzebyignoringpairing.Abetterexplanationisgivenby ANOVA. F statisticinANOVAforpaireddesignequals t 2 paired ;similarly, F statistic inANOVAforunpaireddesignequals t 2 unpaired .Datacanbeanalyzedin twoequivalentways.
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