Two Factor Designs - Statistics and Actuarial Science - Simon Fraser

78 Pages·2015·1.12 MB·English

Chapter 10
Two Factor Designs - Single-sized
Experimental units - CR and RCB
designs
Contents
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
10.1.1 Treatmentstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
10.1.2 Experimentalunitstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . 536
10.1.3 Randomizationstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
10.1.4 Puttingthethreestructurestogether . . . . . . . . . . . . . . . . . . . . . . . 539
10.1.5 Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
10.1.6 Fixedorrandomeffects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540
10.1.7 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
10.1.8 Generalcomments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542
10.2 Example-Effectofphoto-periodandtemperatureongonadosomaticindex-CRD 543
10.2.1 Designissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543
10.2.2 Preliminarysummarystatistics . . . . . . . . . . . . . . . . . . . . . . . . . 544
10.2.3 Thestatisticalmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548
10.2.4 Fittingthemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
10.2.5 Hypothesistestingandestimation . . . . . . . . . . . . . . . . . . . . . . . . 549
10.2.6 Modelassessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554
10.2.7 Unbalanceddata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
10.3 Example-Effectofsexandspeciesuponchemicaluptake-CRD. . . . . . . . . . 555
10.3.1 Designissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
10.3.2 Preliminarysummarystatistics . . . . . . . . . . . . . . . . . . . . . . . . . 557
10.3.3 Thestatisticalmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
10.3.4 Fittingthemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
10.4 Powerandsamplesizefortwo-factorCRD . . . . . . . . . . . . . . . . . . . . . . 564
10.5 Unbalanceddata-Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567
10.6 Example-Streamresidencetime-UnbalanceddatainaCRD . . . . . . . . . . . 569
10.6.1 Preliminarysummarystatistics . . . . . . . . . . . . . . . . . . . . . . . . . 571
10.6.2 TheStatisticalModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572
10.6.3 Hypothesistestingandestimation . . . . . . . . . . . . . . . . . . . . . . . . 573
526
CHAPTER10. TWOFACTORDESIGNS-SINGLE-SIZEDEXPERIMENTALUNITS-CRAND
RCBDESIGNS
10.6.4 Powerandsamplesize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
10.7 Example-Energyconsumptioninpocketmice-UnbalanceddatainaCRD . . . 579
10.7.1 Designissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580
10.7.2 Preliminarysummarystatistics . . . . . . . . . . . . . . . . . . . . . . . . . 580
10.7.3 Thestatisticalmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582
10.7.4 Fittingthemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582
10.7.5 Hypothesistestingandestimation . . . . . . . . . . . . . . . . . . . . . . . . 583
10.7.6 Adjustingforunequalvariances? . . . . . . . . . . . . . . . . . . . . . . . . 587
10.8 Example:Use-DependentInactivationinSodiumChannelBetaSubunitMutation
-BPK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
10.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
10.8.2 Experimentalprotocol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
10.8.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
10.9 Blockingintwo-factorCRDdesigns . . . . . . . . . . . . . . . . . . . . . . . . . 598
10.10FAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598
10.10.1Howtodeterminesamplesizeintwo-factordesigns . . . . . . . . . . . . . . 598
10.10.2Whatisthedifferencebetweena‘block’anda‘factor’? . . . . . . . . . . . . 599
10.10.3Ifthereisevidenceofaninteraction,doestheanalysisstopthere? . . . . . . . 599
10.10.4WhenshouldyouuserawmeansorLSmeans? . . . . . . . . . . . . . . . . . 601
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).
527
(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.
528
(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.
529
(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:
530
(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:
531
(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
532
(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.
533
(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?).
534
(cid:13)c2015CarlJamesSchwarz 2015-08-20
CHAPTER10. TWOFACTORDESIGNS-SINGLE-SIZEDEXPERIMENTALUNITS-CRAND
RCBDESIGNS
• NomaineffectofFactorA;largemaineffectofFactorB;largeinteractionbetweenFactorAand
B.Asbefore,theremaybeproblemsininterpretingmaineffectsinthepresenceofaninteraction
(why?).
535
(cid:13)c2015CarlJamesSchwarz 2015-08-20