Chapter 10 Two Factor Designs - Single-sized Experimental units - CR and RCB designs Contents 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 10.1.1 Treatmentstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 10.1.2 Experimentalunitstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 10.1.3 Randomizationstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 10.1.4 Puttingthethreestructurestogether . . . . . . . . . . . . . . . . . . . . . . . 516 10.1.5 Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 10.1.6 Fixedorrandomeffects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 10.1.7 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 10.1.8 Generalcomments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 10.2 Example-Effectofphoto-periodandtemperatureongonadosomaticindex-CRD 520 10.2.1 Designissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 10.2.2 Preliminarysummarystatistics . . . . . . . . . . . . . . . . . . . . . . . . . 521 10.2.3 Thestatisticalmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 10.2.4 Fittingthemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 10.2.5 Hypothesistestingandestimation . . . . . . . . . . . . . . . . . . . . . . . . 527 10.2.6 Modelassessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 10.2.7 Unbalanceddata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 10.3 Example-Effectofsexandspeciesuponchemicaluptake-CRD. . . . . . . . . . 535 10.3.1 Designissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 10.3.2 Preliminarysummarystatistics . . . . . . . . . . . . . . . . . . . . . . . . . 537 10.3.3 Thestatisticalmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 10.3.4 Fittingthemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 10.4 Powerandsamplesizefortwo-factorCRD . . . . . . . . . . . . . . . . . . . . . . 548 10.5 Unbalanceddata-Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 10.6 Example-Streamresidencetime-UnbalanceddatainaCRD . . . . . . . . . . . 554 10.6.1 Preliminarysummarystatistics . . . . . . . . . . . . . . . . . . . . . . . . . 556 10.6.2 TheStatisticalModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 10.6.3 Hypothesistestingandestimation . . . . . . . . . . . . . . . . . . . . . . . . 558 503 CHAPTER10. TWOFACTORDESIGNS-SINGLE-SIZEDEXPERIMENTALUNITS-CRAND RCBDESIGNS 10.6.4 Powerandsamplesize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 10.7 Example-Energyconsumptioninpocketmice-UnbalanceddatainaCRD . . . 567 10.7.1 Designissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 10.7.2 Preliminarysummarystatistics . . . . . . . . . . . . . . . . . . . . . . . . . 568 10.7.3 Thestatisticalmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 10.7.4 Fittingthemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571 10.7.5 Hypothesistestingandestimation . . . . . . . . . . . . . . . . . . . . . . . . 572 10.7.6 Adjustingforunequalvariances? . . . . . . . . . . . . . . . . . . . . . . . . 578 10.8 Example:Use-DependentInactivationinSodiumChannelBetaSubunitMutation -BPK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 10.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 10.8.2 Experimentalprotocol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 10.8.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 10.9 Blockingintwo-factorCRDdesigns . . . . . . . . . . . . . . . . . . . . . . . . . 589 10.10FAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 10.10.1Howtodeterminesamplesizeintwo-factordesigns . . . . . . . . . . . . . . 590 10.10.2Whatisthedifferencebetweena‘block’anda‘factor’? . . . . . . . . . . . . 591 10.10.3Ifthereisevidenceofaninteraction,doestheanalysisstopthere? . . . . . . . 591 10.10.4WhenshouldyouuserawmeansorLSmeans? . . . . . . . . . . . . . . . . . 592 Thesuggestedcitationforthischapterofnotesis: Schwarz,C.J.(2015). TwoFactorDesigns-Single-sizedExperimentalunits-CRandRCB designs. InCourseNotesforBeginningandIntermediateStatistics. Availableathttp://www.stat.sfu.ca/~cschwarz/CourseNotes. Retrieved2015-08-20. 10.1 Introduction So far we’ve looked at two different experimental designs, the single-factor completely randomized design(1-factorCRD),andthesingle-factorrandomizedcompeteblockdesign(1-factorRCB). Bothdesignsinvestigatedifdifferencesinthemeanresponsecouldbeattributedtodifferentlevels ofasinglefactor. However, inmanyexperiments, interestliesnotonlyintheeffectofasinglefactor, butinthejointeffectsof2ormorefactors. Forexample: • Yieldofwheat. Theyieldofwheatdependsuponmanyfactors-twoofwhichmaybethevariety andtheamountoffertilizerapplied. Thishastwofactors-(1)varietywhichmayhavethreelevels representingthreepopulartypesofseeds,and(2)theamountoffertilizerwhichmaybesetattwo levels. • Pesticidelevels. Thepesticidelevelsmaybemeasuredinbirdswhichmaydependuponsex(two levels)anddistanceofthewinteringgroundsfromagriculturalfields(threelevels). • Performanceofaproduct. Thestrengthofpapermaydependupontheamountofwateradded (twolevels)andthetypeofwoodfiberusedinthemix(threelevels). 504 (cid:13)c2015CarlJamesSchwarz 2015-08-20 CHAPTER10. TWOFACTORDESIGNS-SINGLE-SIZEDEXPERIMENTALUNITS-CRAND RCBDESIGNS There are many ways to design experiments with multiple factors - we will examine three of the mostcommondesignsusedinecologicalresearch-thecompletelyrandomizeddesign(thischapter),the randomizedblockdesign(thischapter),andthesplit-plotdesign(nextchapter). Asnotedmanytimesinthiscourse,itisimportanttomatchtheanalysisofthedatawiththewaythe datawascollected. Beforeattemptingtoanalyzeanyexperiment,thefeaturesoftheexperimentshould beexaminedcarefully. Inparticular,caremustbetakentoexamine • thetreatmentstructure; • theexperimentalunitstructure; • therandomizationstructures; • thepresenceorabsenceofbalance; • ifthelevelsoffactorsarefixedorrandomeffects;and • theassumptionsimplicitlymadeforthedesign. Ifthesefeaturesarenotidentifiedproperly,thenanincorrectdesignandanalysisofanexperimentwill bemade. 10.1.1 Treatmentstructure Thetreatmentstructurereferstohowthevariouslevelsofthefactorsarecombinedintheexperiment. The first step in any design or analysis is to start by identifying the factors in the experiment,their associatedlevels,andthetreatmentsintheexperiment. Treatmentsarethecombinationsoffactorlevels thatare‘applied’1toexperimentalunits. Thetwo-factordesignhas,asthenameimplies,twofactors. WegenericallycalltheseFactorAand FactorBwithaandblevelsrespectively. Wewillexamineonlyfactorialtreatmentstructures,i.e. every treatmentcombinationappearssomewhereintheexperiment. Forexample,ifFactorAhas2levels,and FactorBhas3levels,thenall6treatmentcombinationsappearintheexperiment. Whyfactorialdesigns? Why do we insist on factorial treatment structures? There is a temptation to investigate multi-factor effects using a ‘change-one-at-time’ structure. For example, suppose you are investigating the effects of process temperature (at two levels, H & L), fiber type (at two levels - deciduous and coniferous) and initial pulping method (at two levels - mechanical or chemical) upon the strength of paper. In the ‘change-one-at-a-time’treatmentstructure,thefollowingtreatmentcombinationswouldbetested: 1. L deciduous mechanical 2. H deciduous mechanical 3. L coniferous mechanical 4. L deciduous chemical 1Recallthatinanalyticalsurveys,thefactorlevelscannotbeassignedtounits(e.g. youcan’tassignsextoananimal)andso thekeypointisthatunitsarerandomlyselectedfromtherelevantpopulation. 505 (cid:13)c2015CarlJamesSchwarz 2015-08-20 CHAPTER10. TWOFACTORDESIGNS-SINGLE-SIZEDEXPERIMENTALUNITS-CRAND RCBDESIGNS The researcher then argues that the effect of fiber type could be found by examining the difference in strengthbetweentreatments(1)and(3);theeffectofpulpingmethodcouldbefoundbyexaminingthe difference in strength between treatments (4) and (1); and the effect of process temperature could be foundbyexaminingthedifferenceinstrengthbetweentreatments(1)and(2). Thisisvalidprovidedthattheresearcheriswillingtoassumethetreatmenteffectsareadditive, i.e.,thattheeffectofprocesstemperatureisthesameatalllevelsoftheotherfactors;thattheeffectof fibertypeisthesameatalllevelsoftheotherfactors;andthattheeffectofinitialpulpingmethodisthe sameatalllevelsoftheotherfactors. Unfortunately,thereisnomethodavailabletotestthisassumption withthesetoftreatmentslistedabove. Itisusuallynotagoodideatomakethisverystrongassumption-whathappensiftheassumptionis nottrue? Inthepreviousexample,itmeansthatyour‘effects’areonlyvalidfortheparticularlevelsof theotherfactorsthathappenedtobepresentinthecomparison. Forexample, theprocesstemperature effectwouldonlybevalidfordeciduousfibersourcesthataremechanicallypulped. Asuperiortreatmentstructureisthefactorialtreatmentstructure.Inthefactorialtreatmentstructure, every combination of levels appears in the experiment. For example, referring back to the previous experiment,allofthefollowingtreatmentswouldappearintheexperiment: 1. L deciduous mechanical 2. H deciduous mechanical 3. L coniferous mechanical 4. H coniferous mechanical 5. L deciduous chemical 6. H deciduous chemical 7. L coniferous chemical 8. H coniferous chemical Now,themaineffectsofeachfactorarefoundas: • maineffectoftemperature-treatments1,3,5,7vs.2,4,6,8 • maineffectofsource-treatments1,2,5,6vs.3,4,7and8 • maineffectofmethod-treatments1,2,3,4vs.5,6,7,8 Eachmaineffectwouldbeinterpretedatthe‘averagechange’overthelevelsoftheotherfactors. Inaddition,itispossibletoinvestigateifinteractionsexistbetweenthevariousfactors.Forexample, istheeffectofprocesstemperaturethesameformechanicalandchemicalpulpingmethods? Thiswould beexaminedbycomparingthechangein(1)+(3)vs.(2)+(4)[representingtheeffectoftemperaturefor mechanicallypulpedwood]andthechangein(5)+(7)vs.(6)+(8)[representingtheeffectoftemperature for chemically pulped wood]. Can you specify how you would investigated the interaction between temperatureandsource? Whataboutbetweensourceandmethodofpulping? Alloftheseareknownas twofactorinteractions. Theconceptofatwo-factorinteractioncanalsobegeneralizedtothree-factorandhigherinteraction termsinmuchthesameway. 506 (cid:13)c2015CarlJamesSchwarz 2015-08-20 CHAPTER10. TWOFACTORDESIGNS-SINGLE-SIZEDEXPERIMENTALUNITS-CRAND RCBDESIGNS Whynotfactorialdesigns? While a factorial treatment structure provides the maximal amount of information about the effects of factors and their interactions, there are some disadvantages. In general, the number of treatments that willappearintheexperimentisequaltotheproductofthelevelsfromallofthefactors.Inanexperiment withmanyfactors,thiscanbeenormous. Forexample,ina10factordesign,witheachfactorat2levels, thereare1024treatmentcombinations. Itturnsoutthatinsuchlargeexperiment,therearebetterways toproceedthatarebeyondthescopeofthiscourse-anexampleofwhichisafractionalfactorialdesign which selects a subset of the possible treatments to run with the understanding that the subset chosen losesinformationonsomeofthehigherorderinteractions.Ifyouarecontemplatingsuchanexperiment, pleaseseekcompetenthelp. As well, in some cases, interest lies in estimating a response surface, e.g. factors are continuous variables(suchatemperature)andtheexperimenterisinterestedinfindingtheoptimalconditions. This gives rise to a class of designs called response surface designs which are beyond the scope of this course. Again,seekcompetenthelp. Displayingandinterpretingtreatmenteffects-profileplots An important part of the design and analysis of experiment lies in predicting the type of response ex- pected - in particular, what do you expect for the size of the main effects and do you expect to see an interaction. During the design phase, these are useful to determining the power and needed sample sizes for an experiment. Duringthe analysis phase, these valuesand plots helpin interpreting theresults of the statisticalanalysis. Withtwofactors(AandB)eachattwolevels,youcanconstructaprofileplot. Theseprofileplots showtheapproximateeffectofbothfactorssimultaneously. Thekeythingtolookforisthe‘parallelism’ofthetwolines. Profileplotswithnointeractionbetweenfactors Forexample,considerthetheoretical[itistheoreticalbecauseitshowsthepopulationmeanswhichare neverknownexactly]profileplotofthemeanresponsesbelow: 507 (cid:13)c2015CarlJamesSchwarz 2015-08-20 CHAPTER10. TWOFACTORDESIGNS-SINGLE-SIZEDEXPERIMENTALUNITS-CRAND RCBDESIGNS Inthisplot,theverticaldistancebetweenthetwoparallellinesegmentsistheeffectofFactorB,i.e., whathappenstothemeanresponsewhenyouchangethelevelofFactorB,butkeepthelevelofFactor Aconstant. ThemaineffectofFactorBistheAVERAGEverticaldistancebetweenthetwolineswhen averaged over all levels of Factor A. Notice that if the lines are parallel, the vertical distance between thetwolinesisconstant-thisimpliesthattheeffectofFactorB(theverticaldistancebetweenthetwo lines) is the same regardless of the level of Factor A and the effect of Factor B and the main effect of FactorBaresynonymous. Inthiscase,wesaythatthereisNOINTERACTIONbetweenFactorAand FactorB.Similarly, theeffectofFactorAisthechangeinthelinebetweenthetwolevelsofFactorA ataparticularvalueofFactorB,i.e.,theverticalchangeineacheachlinesegment. Themaineffectof FactorAistheAVERAGEchangewhenaveragedoveralllevelsofFactorB.Noticethatifthelinesare parallel,theverticalchangeisthesameforbothlines-thisimpliesthattheeffectofFactorAisthesame regardlessofthelevelofFactorBandthattheeffectofFactorAissynonymouswiththemaineffectof FactorA.Onceagain,thereisnointeractionbetweenAandB. Profileplotswithinteractionbetweenfactors Nowconsiderthefollowingtheoreticalprofileplot: 508 (cid:13)c2015CarlJamesSchwarz 2015-08-20 CHAPTER10. TWOFACTORDESIGNS-SINGLE-SIZEDEXPERIMENTALUNITS-CRAND RCBDESIGNS Inthisplot,theverticaldistancebetweenthelinesegmentsCHANGESdependingonwhereyouare inFactorA.ThisimpliesthattheeffectofFactorBchangesdependinguponthelevelofA,i.e.,there is INTERACTION between Factor A and B. The main effect of Factor A is the average effect when averaged over levels of B. In this case the main effect is not very interpretable (as will be seen in the plotsbelow). Similarly, theverticalchangeforeachlinesegmentisdifferentforeachsegment-again the effect of Factor A changes depending upon the level of Factor B - once again there is interaction betweenAandB. Theplotsfromanactualexperimentmustbeinterpretedwithagrainofsaltbecauseeveniftherewas nointeraction,thelinesmaynotbeexactlyparallelbecauseofsamplingvariationsinthesamplemeans. Thekeythingtolookforisthedegreeofparallelism. Anditdoesn’tmatterwhichfactorisplottedalong thebottom-theplotsmaylookdifferent,butyouwillcometothesameconclusions. Ifthereisinteraction,thelinesegmentsmayevencrossratherthanremainingseparate. Illustrationsofvarioustheoreticalprofileplots 509 (cid:13)c2015CarlJamesSchwarz 2015-08-20 CHAPTER10. TWOFACTORDESIGNS-SINGLE-SIZEDEXPERIMENTALUNITS-CRAND RCBDESIGNS • NomaineffectofFactorA(averageoflinesisflat);smallmaineffectofFactorB(iftherewasno maineffectofFactorBthelineswouldcoincide);andnointeractionofFactorsAandB. • LargemaineffectofFactorA;smallmaineffectofFactorB(averagedifferencebetweenlinesis small);andnointeractionbetweenFactorsAandB. • NomaineffectofFactorA;largemaineffectofFactorB;andnointeractionbetweenFactorsA andB. 510 (cid:13)c2015CarlJamesSchwarz 2015-08-20 CHAPTER10. TWOFACTORDESIGNS-SINGLE-SIZEDEXPERIMENTALUNITS-CRAND RCBDESIGNS • LargemaineffectofFactorA;largemaineffectofFactorB;andnointeractionbetweenFactors AandB. • NomaineffectofFactorA;nomaineffectofFactorB;butlargeinteractionbetweenFactorsA and B. This illustrates the dangers of investigating ‘main effects’ in the presence of interaction (why? -agoodexamquestion!). • LargemaineffectofFactorA;nomaineffectoffactorB:slightinteraction. Again,thisdiagram illustratesthefollyofdiscussingmaineffectsinthepresenceofaninteraction(why?). 511 (cid:13)c2015CarlJamesSchwarz 2015-08-20 CHAPTER10. TWOFACTORDESIGNS-SINGLE-SIZEDEXPERIMENTALUNITS-CRAND RCBDESIGNS • NomaineffectofFactorA;largemaineffectofFactorB;largeinteractionbetweenFactorAand B.Asbefore,theremaybeproblemsininterpretingmaineffectsinthepresenceofaninteraction (why?). 512 (cid:13)c2015CarlJamesSchwarz 2015-08-20