JMP - Statistics and Actuarial Science

104 Pages·2015·2.53 MB·English

Chapter 6
Single factor - pairing and blocking
Contents
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
6.2 Randomizationprotocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
6.2.1 Someexamplesofseveraltypesofblockdesigns . . . . . . . . . . . . . . . . 373
6.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
6.3.1 Doestheanalysismatchthedesign? . . . . . . . . . . . . . . . . . . . . . . 376
6.3.2 Additivitybetweenblocksandtreatments. . . . . . . . . . . . . . . . . . . . 377
6.3.3 Nooutliersshouldbepresent . . . . . . . . . . . . . . . . . . . . . . . . . . 378
6.3.4 Equaltreatmentgroupstandarddeviations? . . . . . . . . . . . . . . . . . . . 379
6.3.5 Aretheerrorsnormallydistributed?. . . . . . . . . . . . . . . . . . . . . . . 380
6.3.6 Aretheerrorsindependent? . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
6.4 Comparingtwomeansinapaireddesign-thePairedt-test . . . . . . . . . . . . . 381
6.5 Example-effectofstreamslopeuponfishabundance . . . . . . . . . . . . . . . . 381
6.5.1 Introductionandsurveyprotocol . . . . . . . . . . . . . . . . . . . . . . . . 381
6.5.2 UsingaDifferencesanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . 383
6.5.3 UsingaMatchedpairedanalysis . . . . . . . . . . . . . . . . . . . . . . . . 386
6.5.4 UsingaSingleFactorRCBanalysis . . . . . . . . . . . . . . . . . . . . . . 388
6.5.5 UsingaGeneralModelinganalysis . . . . . . . . . . . . . . . . . . . . . . . 390
6.5.6 Whichanalysistochoose? . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
6.5.7 Commentsabouttheoriginalpaper . . . . . . . . . . . . . . . . . . . . . . . 393
6.6 Example-Qualitycheckontwolaboratories. . . . . . . . . . . . . . . . . . . . . 393
6.7 Example-Comparingtwovarietiesofbarley . . . . . . . . . . . . . . . . . . . . 396
6.8 Example-Comparingprepofmosaicvirus . . . . . . . . . . . . . . . . . . . . . 399
6.9 Example-Comparingturbidityattwosites . . . . . . . . . . . . . . . . . . . . . 401
6.9.1 Introductionandsurveyprotocol . . . . . . . . . . . . . . . . . . . . . . . . 401
6.9.2 UsingaDifferencesanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . 403
6.9.3 UsingaMatchedpairedanalysis . . . . . . . . . . . . . . . . . . . . . . . . 406
6.9.4 UsingaGeneralModelinganalysis . . . . . . . . . . . . . . . . . . . . . . . 408
6.9.5 Whichanalysistochoose? . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
6.10 Powerandsamplesizedetermination . . . . . . . . . . . . . . . . . . . . . . . . . 412
6.11 SingleFactor-RandomizedCompleteBlock(RCB)Design . . . . . . . . . . . . . 416
6.11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
6.11.2 Thepotato-peelingexperiment-revisited . . . . . . . . . . . . . . . . . . . . 417
6.11.3 Anagriculturalexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
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6.11.4 Basicideaoftheanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
6.12 Example-Comparingeffectsofsalinityinsoil . . . . . . . . . . . . . . . . . . . . 420
6.12.1 “AdjustingforBlocks”-UsingtheAnalyze->FitY-by-Xplatform . . . . . . . 423
6.12.2 Modelbuilding-fittingalinearmodel . . . . . . . . . . . . . . . . . . . . . 430
6.13 Example-Comparingdifferentherbicides . . . . . . . . . . . . . . . . . . . . . . 436
6.14 Example-Comparingturbidityatseveralsites . . . . . . . . . . . . . . . . . . . 440
6.15 PowerandSampleSizeinRCBs . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
6.16 Example-BPK:Bloodpressureatpresyncope . . . . . . . . . . . . . . . . . . . . 450
6.16.1 Experimentalprotocol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450
6.16.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
6.16.3 Powerandsamplesize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460
6.17 Finalnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
6.18 FrequentlyAskedQuestions(FAQ) . . . . . . . . . . . . . . . . . . . . . . . . . . 466
6.18.1 Differencebetweenpairingandconfounding . . . . . . . . . . . . . . . . . . 466
6.18.2 WhatisthedifferencebetweenapaireddesignandanRCBdesign? . . . . . . 467
6.18.3 Whatisthedifferencebetweenapairedt-testandatwo-samplet-test? . . . . 467
6.18.4 Howtointerpretthegraphofamatchedpairsanalysis . . . . . . . . . . . . . 468
6.18.5 WhentouseAnalyze->FitModelandAnalyze->FitY-by-X(BlockCentered)
platformsforanRCB? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
6.18.6 HowtousetheAnalyze->FitModelplatformtoadjustforblocksinanRCB? . 469
6.18.7 CanyouusemeandiamondswithanRCBdesign? . . . . . . . . . . . . . . . 470
6.18.8 ThelogicbehindtheAnalyze->FitModelplatform . . . . . . . . . . . . . . . 470
6.18.9 PowerinRCB/matchedpairdesign-whatisrootMSE? . . . . . . . . . . . . 470
6.18.10Testingforblockeffects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
6.18.11Presentingresultsforblockedexperiment. . . . . . . . . . . . . . . . . . . . 471
6.18.12Whatisamarginalmean? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
6.18.13Multipleexperimentalunitswithinablock? . . . . . . . . . . . . . . . . . . 472
6.18.14Howdoesablockdifferfromacluster?. . . . . . . . . . . . . . . . . . . . . 472
Thesuggestedcitationforthischapterofnotesis:
Schwarz,C.J.(2015). Singlefactor-pairingandblocking.
InCourseNotesforBeginningandIntermediateStatistics.
Available at http://www.stat.sfu.ca/~cschwarz/CourseNotes. Retrieved
2015-08-20.
6.1 Introduction
Inthecompletelyrandomizeddesign,acompleterandomizationofexperimentalunitstotreatmentswas
performed. Thisrandomizationensuredthattheeffectsofallpossibleothervariablesthatmightaffect
theresponseare,onaverage,equalinalltreatmentgroups.Consequently,differencesinthegroupmeans
canbeattributedtothetreatments.
In some cases, it is known or suspected in advance, that a variable, not of primary interest to the
experimenter,willaffecttheresultsanditispossibletogroupexperimentalunitsintoclusters(orblocks
orstrata)whereunitswithinaclusterhavesimilarvaluesofthisothervariable. Bychangingtheexperi-
mentaldesignslightly,itispossibletodesignamorepowerfulexperimentthatadjustsforthepotential
effectsofthisadditionalexplanatoryvariable.
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Forexample, supposethatanexperimentwastobeperformedtoinvestigatetheeffectofadrugin
loweringbloodpressure. Agroupoftestsubjectsisavailable.
In a completely randomized design, 1/2 of the test subjects would be assigned at random to the
controlgrouptoreceiveaplacebo,and1/2ofthetestsubjectswouldbeassignedtothedruggroup. By
randomizing, theeffectsofother, uncontrolledvariablessuchasamountofexercise, metabolism, diet,
etc.,wouldbeequal,onaverage,betweenthetwogroups. However,theseotheruncontrolledvariables
would result in a large variation in blood pressure within each group making it harder to detect any
changes. [Recallthatpowerforthistypeofexperimentisrelatedtotheratioofthedifferenceinmeans
tothestandarddeviationwithingroups.]
Thedesigncanbeimprovedbytreatingeachsubjectwithboththeplaceboandthedrug(inrandom
order). Now each subject serves as a “control” for these other variables and the difference in blood
pressurereadingswillbefree(wehope)oftheeffectsoftheseothervariables.Thisisknownasapaired
design,ormoregenerally,asablockeddesign. Thisdesignisnotperfect–onestillhastoworryabout
carry-overeffects(e.g.theresponseforthesecondtreatmentmightbeaffectedbywhathappenedinthe
firsttreatment),andabouttheinteractionoftheblockingfactorwiththetreatment(i.e.,perhapspeople
withhighbloodpressurereactdifferentlytothedrugthanpeoplewithlowbloodpressure). Itispossible
toblockbymorethanonevariable,e.g.thesubjectscouldbefurthergroupedbyinitialbloodpressure
levels–thisisbeyondthescopeofthiscourse.
Thisexamplewithtwotreatmentlevelscanbeextendedtoarandomizedcompleteblockexperiment
where2ormorelevelsarerandomizedwithineachblock.
Infieldbiology,commonblockingvariablesaresiteoftheexperimentorbiogeoclimaticzone.
Someotherexamplesofblockeddesignsare:
• Honeybeecoloniesarestackedonpallets,threeperpallet. Investigatorswishtodeterminewhich
of three brands of a chemical treatment is most effective in killing a bee mite. They randomly
assignthethreetreatmentswithineachpalletensuringthateachpalletreceivesallthreetreatments.
Theblocksarethepallets;thefactoristhechemicaltreatment;thelevelsarethethreebrands.
• Restingheartratevariesconsiderablyamongpeople. Consequently,youmaydecidetomeasurea
personbeforeandafterexercisetoseethechangeinheartrate. Theblocksarepeople;thefactor
is time; the levels are before and after exercise. Notice in this experiment, you can’t randomize
time-thiscanintroducesubtleproblemsintotheanalysis.1
• Driving habits vary considerably among drivers. Consequently, you may decide to compare the
durability of different brands of tires by mounting all brands on the same car and doing a direct
comparison under the same driving conditions, rather than using different cars for each grand
withdifferentdriversand(presumably)differentdrivingconditions. Theblocksaredriver/car;the
factorisbrandoftire;thelevelsaretheparticularbrandschosenintheexperiment.
Therearetwoseeminglydifferentexperimentalproceduresandanalyses.
1. Paireddesign. Therearetwotreatmentlevels. Apairedt-testisusedtoanalyzethedata.
2. Blockeddeign. Therearetwoormoretreatmentlevels. AnANOVAisusedtoanalyzethedata.
1Forthetechnicalmasochistsintheaudience,thesubtleproblemisthattheerrortermslikelynolongerhaveacompound
symmetriccovariancestructure. Measurementsthatareclosertogetherintimewillbemorerelatedthanmeasurementsthatare
distantintime.
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Thepairedt-testisaspecialcaseofamoregeneralANOVAapproachandthetwoapproacheswillgive
identicalresultsfordesignswithexactly2treatmentlevels. Incaseswithmorethan2levels,thepaired
approachcannotbeused.
ThenullhypothesisisexactlythesameregardlessifthedesignisaCRDoranRCB!Moregen-
erally,forfixedeffects,thehypothesesarealwaysaboutthemeanresponsesatdifferentlevels.Typically
thenullhypothesisisH:noeffectoffactorXuponthemeanresponseorH:themeanresponseforeach
leveloffactorXisthesame. ThealternatehypothesisisthatthereissomeeffectoffactorXuponthe
meanresponseorthatthemeanresponsediffersamongthelevelsoffactorX.
Thereasonthatthehypothesesarethesameisthathypothesesareconcernedabouttreatmenteffects.
Thetreatmentstructureisquiteindependentoftheexperimentalunitstructure(i.e.,anRCB,CRD,sub-
sampling,orsplit-plotdesign)andtherandomizationstructure(i.e.,wasitcompleterandomization).
AdvantagesandDisadvantagesofanRCB
WhychooseaRCBoveraCRDorvice-versa? Herearetheadvantagesanddisadvantagesofeach
design.
Design Advantages Disadvantages
CRD Easytoconstructthedesign. Experimental units are presumed to
Easy to analyze even if the number be homogeneous so that complete
ofreplicationsdiffersineachgroup. randomizationisappropriate.
Canbeusedforanynumberoftreat-
ments.
RCB Onesourceofheterogeneityamong Maybedifficulttogetlargeenough
experimentalunitscanbeaccounted blocksifyouhavealargenumberof
for. treatments.
More complex analysis if doing
by hand - otherwise must properly
specifydesigntocomputerpackage.
Can be a complicated analysis if
many values go missing - but in
many cases modern software han-
dles missing values without prob-
lems.
In general, it is almost always advantageous to block. If the blocking is successful, you can
substantiallyincreasethepowerofyourdesigntodetectdifferences;ifblockingisunimportant,thereis
verylittlelossofefficiencyinthedesign.
Whatisthedifferencebetweentreatingblocksassimplyblocksortreatingblocksasanother
factor? Insomecases,thecreationofblocksisdependantuponavariablethatlookslikeafactor. For
example,blockscouldbeformedbasedupon(hypothesized)fertilitydifferencesamongfields. Thekey
differencebetweentreatingblocksassimplynusiancevariablesortreatingthemasafactorishowyou
goaboutmeasuringtheblockvariable. Ifblocksareformedupon(hypothesized)differencesinfertility
amongplots,itisNOTnecessarytomeasuretheactualfertilitylevels,norisitnecessarytorestrictthe
number of fertility levels to a small number of levels. If you wanted to treat fertility as a factor, you
would normally only have a few levels (e.g. low, medium, high), you would be forced to measure the
fertilityofeachplotofland,andyouwouldliketohavereplicatesofeachleveloffertility. Youarenow
intotherealmoftwofactordesignswhicharediscussedinotherchapters.
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6.2 Randomization protocol
Asinglefactor,randomizedcompleteblockdesign(RCB)hasthefollowingattributes:
1. Thereisasinglesizeofexperimentalunit.
2. Experimentalunitscanbegroupedintoclusters(orblocksorstrata)andwithineachblock,exper-
imentalunitsareassimilaraspossible.
3. Within each block, experimental units are completely randomized (independently within each
block) to treatments such that every treatment occurs once and only once in each block and all
treatmentsoccurineachblock.
Therestrictedrandomizationprocedureisakeypointofblockeddesigns-therandomizationtakes
placeindependentlywithineachblock.
The most common violation of this protocol is incomplete randomization within each block. For
example,inadrugstudytocomparethebloodpressureofadrugvs.thebloodpressurewhentakinga
placebo, theplacebomayalwaysbegivenbeforethedrug. Inthiscase, itisimpossibletoknowifthe
drugcausedanychangeinmeanbloodpressure; perhapsthedifferentamountofsunlightbetweenthe
twooccasionscausedthechange.
6.2.1 Someexamplesofseveraltypesofblockdesigns
SupposethattheMinistryofForestsisinterestedinstudyingtheeffectsofsupplementalfertilizationon
thegrowthofseedlingsafterplanting. Threelevelsofsupplementalfertilizationwillbeused(none,low,
orhighamountsoffertilization). Theexperimentwillbeconductedatsixdifferenttestsitesaroundthe
province. Ateachlocation,threerecentlyreplantedforestplotsareavailableforuseintheexperiment.
Hereareseveralpossibledesigns.
Completelyrandomizeddesign-noblocking
Plot number at each site
Site 1 2 3
---------------------
1 0 high 0
2 low low high
3 high low 0
4 low 0 high
5 low 0 low
6 0 high high
---------------------
The experimental units were completely randomized to the treatments ignoring the site groupings.
Thereisnoguaranteethateachsitereceiveseachtreatmentlevel. Consequentlycomparisonsbetween
treatment levels (e.g. high vs. low) include extra variation because different locations are used for the
twotreatments.
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Randomizedcompleteblockdesign-RCBdesign
Plot number at each site
Site 1 2 3
---------------------
1 0 high low
2 low 0 high
3 high low 0
4 low 0 high
5 high 0 low
6 0 high low
---------------------
Within each block, all treatments occur once and only once and within each block randomization
wasperformedindependentlyoftherandomizationinotherblocks.
Randomizedcompleteblockdesign-RCBdesign-missingvalues
Plot number at each site
Site 1 2 3
---------------------
1 0 high low
2 low 0 *
3 high low 0
4 low 0 high
5 high 0 low
6 0 high low
---------------------
Within each block, all treatments occur once and only once and within each block randomization was
performedindependentlyoftherandomizationinotherblocks. However, duringtheexperiment, some
plots were rendered unusable (e.g. damage by deer) and some treatments could not be measured (e.g.
plot 3 in site 2). In the past (when computations were done by hand), this made the analysis more
difficult.Withmodernsoftware(e.g.NOTExcel!),thisusuallydoesn’tcausemanyproblemunlessthere
are substantial number of missing values. You may wish to seek advice on the analysis of blocked
designswithmissingvalues.
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Incompleteblockdesign-notanRCB
Plot number at each site
Site 1 2
---------------
1 0 high
2 low 0
3 high low
4 low 0
5 high 0
6 0 high
---------------------
Insomebases,blocksmaynotbelargeenoughtocontainalltreatments. Theblocksareincompletebut
randomizationstilltakesplacewithineachblockandindependentlyofotherblocks. Seekadviceonthe
designandanalysisofsuchdesigns. Forexample,theabovedesignisnotbalancedasnotallpairsof
treatmentsoccurequallyoften(the0vs.highpairoccursmoreoftenthanotherpairs).
Generalizedrandomizedcompleteblockdesign
Plot number at each site
Site 1 2 3 4
---------------------------
1 0 high low low
2 low 0 high 0
3 high low 0 0
4 low 0 high high
5 high 0 low high
6 0 high low low
---------------------------
Insomecases,theblocksarelargeenoughtohavesomeorallofthetreatmentsreplicatedwithineach
block.Thisprovidesadditionalinformationaboutthevariabilityoftreatmentswithinblocks.Inaddition,
thefactthatyounowknowthevariabilityofresponseswithinblocksforthesametreatment,allowsthe
experimentertoconductaformalstatisticaltesttoexamineiftheblock-treatmentadditivityassumption
holdsinthisexperiment.
Forthisreason,Irecommendtheuseofthelatterdesign(theGeneralizedRandomizedCom-
plete Block Design) whenever possible The design gives you the advantage of blocking - increased
precision. Aswell,itallowsyoutoempiricallytesttheblock-treatmentassumptionofaddivitivity.
The“balanced”caseofaGRCBdesignwouldhaveanequalnumberofreplicatesofeachtreatment
ineachblock(e.g. 2replicateswithineachblock). However,thisisnotessentialiftheblockscannotbe
made big enough. While there is no “formal” rule, it seems sensible to spread the replication over the
differenttreatmentsamongtheblocks, i.e.insomeblockreplicatetreatmentlevelsa1anda2, whilein
other blocks, replicate a2 and a3, and then a1 and a3 etc. Seek advice on the design and analysis of
suchdesigns.
Twonicereferencesare:
• Addelman,S.(1969).
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TheGeneralizedRandomizedBlockDesign.
AmericanStatistician,23,35-36.
http://dx.doi.org/10.2307/2681737.
• GatesC.E.(1999).
Whatreallyisexperimentalerrorinblockdesigns.
AmericanStatistician49,362-363.
http://dx.doi.org/10.2307/2684574.
6.3 Assumptions
As noted earlier, each and every statistical procedure makes a number assumptions about the data that
shouldbeverifiedastheanalysisproceeds. Someoftheseassumptionscanbeexaminedusingthedata
athand;other,oftenthemostimportantcanonlybeassessedusingthemeta-dataabouttheexperiment.
The set of assumptions for the single factor RCB are, for the most part, identical to those for the
single factor CRD. To make this chapter self-contained, they are repeated in detail below and the key
differencesforanRCBwillbehighlighted.
6.3.1 Doestheanalysismatchthedesign?
THISISTHEMOSTCRUCIALASSUMPTION!
Inthischapter,thedatawerecollectedunderablockeddesign.
Itisnotpossibletocheckthisassumptionbyexaminingthedataandyoumustspendsometimeex-
aminingexactlyhowthetreatmentswererandomizedtoexperimentalunits,andiftheobservationalunit
isthesameastheexperimentalunit(i.e.themeta-dataabouttheexperiment). Thiscomesdowntothe
RRR’sofstatistics-howweretheexperimentalunitsrandomized,whatarethenumbersofexperimental
units,andaretheregroupingsofexperimentalunits(blocks)?
The key features of a RCB design are the restricted randomization of the treatments within each
block,andthattheblockarecomplete(i.e.everytreatmentoccursexactlyonceineveryblock).2
Typicalproblemsarelackofrandomizationwithineachblock,pseudo-replication,and(ironically)a
lackofblocking.
Wasrandomizationcomplete? Ifyouaredealingwithanalyticalsurvey,thenverifythatthesam-
plesaretruerandomsamples(notmerelyhaphazardsamples).Ifyouaredealingwithatrueexperiments,
ensurethattherewasacompleterandomizationoftreatmentstoexperimentalunits.
Whatisthetruesamplesize? Aretheexperimentalunitsthesameastheobservationalunits? In
pseudo-replication(Hurlbert,1984),theexperimentalandobservationalunitsaredifferent. Anexample
of pseudo-replication are experiments with fish in tanks where the tank is the experimental unit (e.g.
chemicalsaddedtothetank)butthefisharetheobservationalunits.
Isblockingpresent? Theexperimentalunitsshouldbegroupedintomorehomogeneousunitswith
restrictedrandomizationswithineachgroup.NotehowthisdiffersfromaCRDwherethereisnoa-priori
2Theassumptionofacompleteblockcanberelaxedsomewhatthrougheitheradditionalreplicatesineachblockorincomplete
blocks.Ineithercase,pleaseseekadditionalhelpindesignandtheanalysisofsuchdesigns.
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groupingofexperimentalunitsandthereiscompleterandomizationoftreatmentstoexperimentalunits.
6.3.2 Additivitybetweenblocksandtreatments
THISISACRUCIALASSUMPTIONthatallowstheanalysistobeinterpreted!
Acrucialfeatureinthedesignandanalysisofblockeddesignsistheassumptionofadditivitybe-
tweentreatmenteffectsandblockeffectsupontheresponsevariable.
Thisassumptionsstatesthatthedifferenceinthemeanresponsebetweenanytwotreatmentsisthe
sameinallblocks. Theoverallmeanoftheresponsesfromeachtreatmentmayvaryamongblocks,but
thedifferencesmustbeconstant.
Forexample,consideranexperimentwiththreetreatmentsconductedintwoblocks.Table6.1shows
actualpopulationmeansundertheassumptionofadditivity.
Table6.1: Populationmeansunderassumptionofadditivity
Treatment
Block a b c
1 10 20 15
2 35 45 40
Noticethattheresponseisgenerallyhigherinblock2thaninblock1–infactthedifferencebetween
thetwoblocksisaconstantvalueof25foreachtreatment. Themeanoftreatmentbisalways10units
higherinbothblocksthanthemeanfortreatmenta.Thedifferenceinthemeansforanypairoftreatment
isthesameinallblocks.
Note that the above table refers to POPULATION means - the sample means may not enjoy this
strictadditivityasanartifactofthesamplingprocess.
Anothernamefortheassumptionofadditivityisnointeractionbetweentreatmentsandblocks.
Therearetwowaysinwhichadditivitycanfail. First,theunitsmaybemeasuredonthewrongscale.
ConsiderTable6.2ofmeanswhereadditivitydoesnothold:
Table6.2: Populationmeanswhentheassumptionofadditivityisfalsebutcorrectable
Treatment
Block a b c
1 10 20 15
2 100 200 150
InTable6.2, thetreatmenteffectsarenotthesameinbothblocks(why?), butnoticethatthemeanfor
treatment b is always twice the mean for treatment a in both blocks. This suggests that the effects of
treatmentaremultiplicativeratherthanadditive, andthattheanalysisshouldproceedonthelog-scale.
Indeed,considerthesamevaluesinTable6.3afteralog-transformation:3
ThetreatmenteffectsinTable6.3arenowadditiveonthelog-scale(howcanyoutell?).
3Eitherthelnorlogtransformationcanbeused.Theresultswilldifferbyaconstant
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Table6.3: Populationmeanswhentheassumptionofadditivityisfalsebutcorrectable. Alog-transform
isapplied.
Treatment
Block a b c
1 2.30 3.00 2.71
2 4.61 5.30 5.01
Table6.4: Populationmeanswhentheassumptionofadditivityisfalseandnotcorrectable.
Treatment
Block a b c
1 10 20 15
2 35 30 25
Insomecases,notransformationwillcorrectnon-additivityasillustratedinTable6.4..
Notice that the response Table 6.4 is generally higher in block 2 than in block 1 – but there is no
simplepatterntothedifferenceamongmeans. Thedifferencebetweenthemeanresponsefortreatment
aandtreatmentbchangesinthetwoblocks. Inblock1, treatmentbhasameanthatis10unitslarger
thanthemeanfortreatmenta,whileinblock2,themeanforbis5unitslowerthanthemeanfora. This
isalsoknownasaninteractionbetweentreatmentsandblocks.
Thediscussionoftheconceptofinteractionandadditivityisresumedwhentwo-factordesignsare
examinedinalaterchapter..
Theassumptionofadditivitybetweenblocksandtreatmentscanbeassessed(roughly)intwoways
depending if the design is a simple paired experiment or a more complex design with more than two
treatments.4
Inapaireddesign,aplotofthedifferencesbetweenthetwotreatmentsisplottedagainstthesample
averageofthetwotreatmentsforeachpair. Thedatashouldshowaroughscatterinaparallelbandto
theX-axis(thiswillbedemonstratedintheexamples).
In blocked designs with more than two treatments, plot the data points against the treatments and
joinpointsfromthesameblock(againthiswillbedemonstratedintheexamples).5
6.3.3 Nooutliersshouldbepresent
ThisisthesameassumptionasintheCRD.
As seen in previous chapters, the idea behind the tests for equality of means is, ironically, to com-
paretherelativevariationamongmeanstothevariationwitheachgroup. Outlierscanseverelydistort
estimatesofthewithin-groupvariationandseverelydistorttheresultsofthestatisticaltest.
Construct side-by-side scatterplots of the individual observations for each group. Check for any
4A more formal test that compares complete additivity to a multiplicative effect is possible. It is called Tukey’s test for
additivity,butisrarelymoreusefulthanthesimpleplotsoutlinedinthissection
5TheMatchingColumnoptionsintheAnalyze->FitY-by-Xplatformgivesthis.
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(cid:13)c2015CarlJamesSchwarz 2015-08-20