DesignandAnalysisafExperiments
withkFactorshavingpLevels
HenrikSpliid
LecturenotesintheDesignandAnalysisofExperiments
1stEnglishedition2002
InformaticsandMathematicalModelling
TechnicalUniversityofDenmark,DK{2800Lyngby,Denmark
0
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Foreword
Thesenoteshavebeenpreparedforuseinthecourse02411,StatisticalDesignofEx-
periments,attheTechnicalUniversityofDenmark.Thenotesareconcernedsolelywith
experimentsthathavekfactors,whichalloccuronplevelsandarebalanced.Suchex-
perimentsaregenerallycalledpk
factorialexperiments,andtheyareoftenusedinthe
laboratory,whereitiswantedtoinvestigatemanyfactorsinalimited-perhapsasfew
aspossible-numberofsingleexperiments.
Readersareexpectedtohaveabasicknowledgeofthetheoryandpracticeofthedesign
andanalysisoffactorialexperiments,or,inotherwords,tobefamiliarwithconcepts
andmethodsthatareusedinstatisticalexperimentalplanningingeneral,includingfor
example,analysisofvariancetechnique,factorialexperiments,blockexperiments,square
experiments,confounding,balancingandrandomisationaswellastechniquesforthecal-
culationofthesumsofsquaresandestimatesonthebasisofaveragevaluesandcontrasts.
ThepresentversionisarevisedEnglishedition,whichinrelationtotheDanishhasbeen
improvedasregardscontents,layout,notationand,inpart,organisation.Substantial
partsofthetexthavebeenrewrittentoimprovereadabilityandtomakethevarious
methodseasiertoapply.Finally,theexamplesonwhichthenotesarelargelybasedhave
beendrawnupwithagreaterdegreeofdetailing,andnewexampleshavebeenadded.
Sincethepresentversionisthe�rstinEnglish,errorsinformulationanspellingmay
occur.
HenrikSpliid
IMM,March2002
April2002:SincetheversionofMarch2002afewcorrectionshavebeenmadeonthe
pages21,25,26,40,68and82.
Lecturenotesforcourse02411.IMM-DTU.
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Contents
1
6
1.1
Introduction...................................
6
1.2
Literaturesuggestionsconcerningthedrawingupandanalysisoffactorial
experiments...................................
7
2
2k{factorialexperiment
9
2.1
Complete2kfactorialexperiments.......................
9
2.1.1
Factors..................................
9
2.1.2
Design..................................
9
2.1.3
Modelforresponse,parametrisation..................10
2.1.4
E�ectsin2k{factorexperiments....................11
2.1.5
Standardnotationforsingleexperiments
..............11
2.1.6
Parameterestimates..........................12
2.1.7
Sumsofsquares.............................13
2.1.8
Calculationmethodsforcontrasts
..................13
2.1.9
Yates'algorithm
............................14
2.1.10Replicationsorrepetitions.......................15
2.1.1123factorialdesign............................16
2.1.122kfactorialexperiment
........................19
2.2
Blockconfounded2kfactorialexperiment..................20
2.2.1
Constructionofaconfoundedblockexperiment...........25
2.2.2
Aone-factor-at-a-timeexperiment..................27
2.3
Partiallyconfounded2kfactorialexperiment.................28
2.3.1
Somegeneralisations..........................31
2.4
Fractional2kfactorialdesign
.........................34
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2.5
Factorson2and4levels............................43
3
Generalmethodsforpk-factorialdesigns
48
3.1
Completep
kfactorialexperiments.......................48
3.2
CalculationsbasedonKempthorne'smethod
................57
3.3
Generalformulationofinteractionsandarti�ciale�ects
..........60
3.4
Standardisationofgenerale�ects.......................62
3.5
Block-confoundedpkfactorialexperiment..................65
3.6
Generalisationofthedivisionintoblockswithseveralde�ningrelations..70
3.6.1
Constructionofblocksingeneral...................74
3.7
Partialconfounding...............................78
3.8
Constructionofafractionalfactorialdesign.................86
3.8.1
Resolutionforfractionalfactorialdesigns
..............90
3.8.2
Practicalandgeneralprocedure....................91
3.8.3
Aliasrelationswith1=pq�pkexperiments..............95
3.8.4
Estimationandtestingin1=pq�pkfactorialexperiments......101
3.8.5
Fractionalfactorialdesignlaidoutinblocks.............105
Index
.....
116
Myownnotes
.....
118
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Tabels
2.1
Asimpleweighingexperimentwith3items..................34
2.2
A1/4�25factorialexperiment.........................40
2.3
A2�4experimentin2blocks.........................44
2.4
Afractional2�2�4factorialdesign......................45
3.1
MakingaGraeco-Latinsquareina32factorialexperiment.........50
3.2
Latincubesin33experiments.........................53
3.3
EstimationandSSQinthe32-factorialexperiment..............58
3.4
Indexvariationwithinversionofthefactororder
..............61
3.5
Generalisedinteractionsandstandardisation.................62
3.6
Latinsquaresin23factorialexperimentsandYates'algorithm
.......63
3.7
23factorialexperimentin2blocksof4singleexperiments
.........65
3.8
32factorialexperimentin3blocks.......................66
3.9
Divisionofa23factorialexperimentinto22blocks..............69
3.10Dividinga33factorialexperimentinto9blocks...............71
3.11Divisionofa25experimentinto23blocks
..................72
3.12Divisionof3kexperimentsinto33blocks...................73
3.13Dividinga34factorialexperimentinto32blocks...............75
3.14Dividinga53factorialexperimentinto5blocks...............77
3.15Partiallyconfounded23factorialexperiment.................78
3.16Partiallyconfounded32factorialexperiment.................82
3.17FactorexperimentdoneasaLatinsquareexperiment............86
3.18Confoundingsina3�
1
�33factorialexperiment,aliasrelations......88
3.19A2�
2
�25factorialexperiment........................92
3.20Constructionof3�
2�35factorialexperiment................96
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3.21Estimationina3�
1
�33-factorialexperiment................101
3.22TwoSASexamples...............................104
3.23A3�
2
�35factorialexperimentin3blocksof9singleexperiments
....106
3.24A2�
4
�28factorialin2blocks........................110
3.25A2�
3
�27factorialexperimentin4blocks..................114
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1 1.1
Introduction
Theselecturenotesareconcerned
with
theconstructionofexperimentaldesignswhich
are
particularlysuitablewhenitiswantedtoexaminealargenumberoffactorsandoftenunder
laboratoryconditions.
Thecomplexityoftheproblem
canbeillustratedwiththefactthatthenumberofpossiblefactor
combinationsinamulti-factorexperimentistheproductofthelevelsofthesinglefactors.If,
forexample,oneconsiders10factors,eachononly2levels,thenumberofpossibledi�erent
experimentsis2�
2�
:::�
2=
2k=
1024.Ifitiswantedtoinvestigatethefactorson3levels,
thisnumberincreasesto310=
59049singleexperiments.Ascanbeseen,thenumberofsingle
experimentsrapidlyincreaseswiththenumberoffactorsandfactorlevels.
Forpracticalexperimentalwork,thisimpliestwomainproblems.
First,itquicklybecomes
impossibletoperform
allexperimentsinwhatiscalledacompletefactorstructure,andsecond,it
isdiÆculttokeeptheexperimentalconditionsunchangedduringalargenumberofexperiments.
Doingtheexperiments,forexample,necessarilytakesalongtime,useslargeamountsoftest
material,usesalargenumberofexperimentalanimals,orinvolvesmanypeople,allofwhich
tendtoincreasetheexperimentaluncertainty.
Thesenoteswillintroducegeneralmodelsforsuchmulti-factorexperimentswhereallfactors
areonplevels,andwewillconsiderfundamentalmethodstoreducetheexperimentalworkvery
considerablyinrelationtothecompletefactorialexperiment,andtogroupsuchexperimentsin
smallblocks.Inthisway,bothsavingsintheexperimentalworkandmoreaccurateestimates
areachieved.
Ane�orthasbeenmadetokeepthenotesas"non-mathematical"aspossible,forexampleby
showingthevarioustechniquesintypicalexamplesandgeneralisingonthebasisofthese.Onthe
otherhand,thishasthedisadvantagethatthetextisperhapssomewhatlongerthanapurely
mathematicalstatisticalrun-throughwouldneed.
Generally,extensivenumericalexamplesarenotgivennorexamplesofthedesignofexperiments
forspeci�cproblem
complexes,butthewholediscussioniskeptonsuchagenerallevelthat
experimentaldesignersfrom
di�erentdisciplinesshouldhavereasonablepossibilitiestobene�t
from
themethodsdescribed.Asmentionedintheforeword,itisassumedthatthereaderhasa
certainfundamentalknowledgeofexperimentalworkandstatisticalexperimentaldesign.
Finally,Ithinkthat,onthebasisofthesenotes,apersonwouldbeabletounderstandtheidea
intheexperimentaldesignsshown,andwouldalsobeabletodraw
upandanalyseexperimen-
taldesignsthataresuitableingivenproblem
complexes.However,thismustnotpreventthe
designerofexperimentsfrom
consultingtherelevantspecialistliteratureonthesubject.Here
canbefoundmanynumericalexamples,bothdetailedandrelevant,andinmanycases,alter-
nativeanalysismethodsaresuggested,whichcanbeveryusefulintheinterpretationofspeci�c
experimentresults.Below,afew
examplesof"classical"literatureinthe�eldarementioned.
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DesignofExperiments,Course02411,IMM,DTU
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1.2
Literaturesuggestionsconcerningthedrawingupandana-
lysisoffactorialexperiments
. Box,G.E.P.,Hunter,W.G.andHunter,J.S.:StatisticsforExperimenters,Wiley,1978.
Chapter10introduces2k
factorialexperiments.Chapter11showsexamplesoftheiruseand
analysis.Inparticular,section10.9showsamethodofanalysingexperimentswithmanye�ects,
whereonedoesnothaveanexplicitestimateofuncertainty.Themethodusesthetechnique
from
thequantilediagram
(Q-Q
plot)andisbothsimpleandillustrativefortheuser.Anumber
ofstandardblockexperimentsaregiven.
Chapter12introducesfractionalfactorialdesigns
andchapter13givesexamplesofapplications.
Thebookcontainsmanyexamplesthatare
completelycalculated-althoughonthebasisofquitemodestamountofdata.Ingenerala
highlyrecommendablebookforexperimenters.
Davies,O.L.andothers:TheDesignandAnalysisofExperiments,OliverandBoyd,1960(1st
edition1954).
Chapters7,8,9and10dealwithfactorialexperimentswithspecialemphasison2k
and3k
factorialexperiments.A
largenumberofpracticalexamplesaregivenbasedonrealproblems
withachemical/technicalbackground.Eventhoughthebookisalittleold,itishighlyrecom-
mendableasabasisforconductinglaboratoryexperiments.Italsocontainsagoodchapter(11)
aboutexperimentaldeterminationofoptimalconditionswherefactorialexperimentsareused.
Fisher,R.A.:TheDesignofExperiments,OliverandBoyd,1960(1stedition1935)
A
classic(perhaps"theclassic"),writtenbyoneofthefoundersofstatistics.Chapters6,7and
8introducenotationandmethodsfor2kand3kfactorialexperiments.Veryinterestingbook.
Johnson,N.L.andLeone,F.C,:StatisticsandExperimentalDesign,VolumeII,Wiley1977.
Chapter15givesapracticallyorientatedandquitecondensedpresentationof2k
factorialex-
perimentsforuseinengineering.WithVolumeI,thisisagoodgeneralbookaboutengineering
statisticalmethods.
Kempthorne,O.:TheDesignandAnalysisofExperiments,Wiley1973(1stedition1952).
Thiscontainsthemathematicalandstatisticalbasisforpk
factorialexperimentswithwhich
thesenotesareconcerned(chapter17).Inadditionitdealswithanumberofspeci�cproblems
relevantformulti-factorialexperiments,forexampleexperimentswithfactorsonboth2and
3levels(chapter18).
Itisbasedon
agriculturalexperimentsinparticular,butisactually
completelygeneralandhighlyrecommended.
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Montgomery,D.C.:DesignandAnalysisofExperiments,Wiley1997(1stedition1976).
Thelatestedition(5th)isconsiderablyimprovedinrelationtothe�rsteditions.Thebook
givesagood,thoroughandrelevantrun-throughofmanyexperimentaldesignsandmethods
foranalysingexperimentalresults.Chapters7,8and9dealwith2kfactorialexperimentsand
chapter10dealswith3kfactorialexperiments.Anexcellentmanualand,uptoapoint,suitable
forself-tuition.
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2
2k{factorialexperiment
Chapter2discussessomefundamentalexperimentalstructuresformulti-factorexperi-
ments.Here,forthesakeofsimplicity,weconsideronlyexperimentswhereallfactors
occurononly2levels.Theselevelsforexamplecanbe\low"/"high"foranamountof
additiveor\notpresent"/"present"foracatalyst.
Aspecialnotationisintroducedandanumberoftermsandmethods,whicharegenerally
applicableinplanningexperimentswithmanyfactors.Thischaptershouldthusbeseen
asanintroductiontothemoregeneraltreatmentofthesubjectthatfollowslater.
2.1
Complete2k
factorialexperiments
2.1.1
Factors
Thename,2kfactorialexperiments,referstoexperimentsinwhichitiswishedtostudyk
factorsandwhereeachfactorcanoccurononly2levels.Thenumberofpossibledi�erent
factorcombinationsisprecisely2k,andifonechoosestodotheexperimentsothatall
thesecombinationsaregonethroughinarandomiseddesign,theexperimentiscalleda
complete2kfactorialexperiment.
Inthissection,themainpurposeistointroduceageneralnotation,sowewillonly
consideranexperimentwithtwofactors,eachhavingtwolevels.Thisexperimentisthus
calleda22factorialexperiment.
ThefactorsintheexperimentarecalledAandB,anditispractical,nottosayrequired,
alwaystousethesenames,evenifitcouldperhapsbewishedtouse,forexample,Tfor
temperatureorVforvolumeformnemonicreasons.
Inaddition,thefactorsareorganisedsothatAisalwaysthe�rstfactorandBisthe
secondfactor.
2.1.2
Design
Foreachcombinationofthetwofactors,weimaginethatanumber(r)ofmeasurements
aremade.Therandomerroriscalled(generally)E.Theresultofasingleexperiment
withacertainfactorcombinationisoftencalledtheresponse,andthisterminologyis
alsousedforthesumoftheresultsobtainedforthegivenfactorcombination.
Thisdesignisasfollowswheretherearerrepetitionsperfactorcombinationinacom-
pletelyrandomisedsetup:
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B=0
B=1
Y001
Y011
A=0
:
:
Y00r
Y01r
Y101
Y111
A=1
:
:
Y10r
Y11r
Ifforexampleweinvestigatehowtheoutputfromaprocessdependsonpressureand
temperature,thetwolevelsoffactorAcanrepresenttwovaluesofpressurewhilethetwo
levelsoffactorBrepresenttwotemperatures.Themeasuredvalue,Yij�,thengivesthe
resultofthe�'thmeasurementwiththefactorcombination(Ai,Bj).
2.1.3
Modelforresponse,parametrisation
Itisassumed,asmentioned,thattheexperimentisdoneasacompletelyrandomised
experiment,thatis,thatthe2�2�robservationsaremade,forexample,incompletely
randomorderorrandomlydistributedovertheexperimentalmaterialwhichmaybeused
intheexperiment.
Themathematicalmodelfortheyieldofthisexperiment(theresponse)is,inthatfactor
Aisstillthe�rstfactorandfactorBisthesecondfactor:
Yij�=�+Ai+Bj+ABij+Eij�,wherei=(0;1);j=(0;1);�=(1;2;::;r)
wheretheususalrestrictionsapply
1 X i=0
Ai=0
;
1 X j=0
Bj=0
;
1 X i=0
ABij=0
;
1 X j=0
ABij=0
Theserestrictionsimplythat
A0=�A1
;
B0=�B1
;
AB00=�AB10=�AB01=+AB11
Therefore,inreality,thereareonly4parametersinthismodel,namelytheexperiment
level�andthefactorparametersA1,B1andAB11,ifone,(asusual)referstothe\high"
levelsofthefactors.
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2.1.4
E�ectsin2k{factorexperiments
Ina2-levelfactorialexperiment,oneoftenspeaksofthe"e�ects"ofthefactors.Bythis
isunderstoodinthisspecialcasethemeanchangeoftheresponsethatisobtainedby
changingafactorfromits"low"toits"high"level.
Thee�ectsinanexperimentwherethefactorshaveprecisely2levelsarethereforede�ned
inthefollowingmanner:
A=A1�A0=2A1
,andlikewiseB=2B1
;
AB=2AB11
Inotherfactorialexperiments,oneoftenspeaksmoregenerallyaboutfactore�ectsas
expressionsoftheactionofthefactorsontheresponse,withouttherebyreferringtoa
de�niteparameterform.
2.1.5
Standardnotationforsingleexperiments
Inthetheoreticaltreatmentofthisexperiment,itispracticaltointroduceastandard
notationfortheexperimentalresultsinthesamewayasforthee�ectsinthemathematical
model.
Fortheexperimentsthataredoneforexamplewiththefactorcombination(A1;B0),the
sumoftheresultsoftheexperimentisneeded.Thissumiscalleda,thatis
a=
r X �=1
Y10�
wherethissumisthesumofalldatawithfactorAonthehighlevelandtheotherfactors
onthelowlevel.Asmentioned,aisalsocalledtheresponseofthefactorcombinationin
question.
Inthesameway,thesumfortheexperimentswiththefactorcombination(A0;B1)is
calledb,whilethesumfor(A1;B1)iscalled"ab".Finally,thesumfor(A0;B0)iscalled
"(1)".
Inthedesignabove,cellsumsarethusfoundasinthefollowingtable
B=0
B=1
A=0
(1)
b
A=1
a
ab
Somepresentationsusenamesthatdirectlyrefertothefactorlevelsasforexample:
B=0
B=1
A=0
00
01
A=1
10
11
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Whenoneworkswiththesecellsums,theyaremostpracticallyshownintheso-called
standardorderforthe22experiment:
(1);a;b;ab
Itisimportanttokeepstrictlytotheintroducednotation,i.e.upper-caseletterforparam-
etersinthemodelandlower-caselettersforcellsums,andthattheorderofparameters
aswellasdata,iskeptasshown.Ifnot,thereisaconsiderableriskofmakingamessof
it.
2.1.6
Parameterestimates
Wecannowformulatetheanalysisoftheexperimentinmoregeneralterms.
We�ndthefollowingestimatesfortheparametersofthemodel:
^�=[(1)+a+b+ab]=(4�r)=[(1)+a+b+ab]=(2
k�r)
wherek=2,asmentioned,givesthenumberoffactorsinthedesignandristhenumber
ofrepetitionsofthesingleexperiments.
Furtherwe�nd:
b A 1=�b A 0=[�(1)+a�b+ab]=(2
k�r)
b B 1=�b B 0=[�(1)�a+b+ab]=(2
k�r)
d AB11=�d AB10=�d AB01=d AB00=[(1)�a�b+ab]=(2
k�r)
IfwealsowanttoestimateforexampletheA-e�ect,i.e.thechangeinresponsewhen
factorAischangedfromlow(i=0)tohigh(i=1)level,we�nd
b A=b A 1�b A 0=2b A 1=[�(1)+a�b+ab]=(2
k�
1�r)
Theparenthesis[�(1)+a�b+ab]givesthetotalincreaseinresponse,whichwasfound
bychangingthefactorAfromitslowleveltoitshighlevel.Thisamountiscalledthe
A-contrast,andiscalled[A].Therefore,inthecaseofthefactorA,wehaveinsummary
theequations:
[A]=[�(1)+a�b+ab]
;
b A 1=�b A 0=[A]=(2
k�r)
;
b A=2b A 1
andcorrespondinglyfortheothertermsinthemodel.Speci�callyforthetotalsumof
observations,thenotation[I]=[(1)+a+b+ab]isused.Thisquantitycanbecalledthe
pseudo-contrast.
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2.1.7
Sumsofsquares
Further,wecanderivethesumsofsquaresforalltermsinthemodel.Thiscanbedone
withordinaryanalysisofvariancetechnique.Forexample,thisgivesinthecaseoffactor
A:
SSQA
=[A]2=(2
k�r)
Correspondingexpressionsapplyforalltheotherfactore�ectsinthemodel.
Thesumsofsquaresforthesefactore�ectsallhave1degreeoffreedom.
Iftherearerepeatedmeasurementsforthesinglefactorcombinations,i.e.r>1,wecan
�ndtheresidualvariationasthevariationwithinthesinglecellsinthedesignintheusual
manner:
SSQresid=
1 X i=0
1 X j=0
([r X �
=1
Y2 ij�]�T
2 ij�=r)
;
where
Tij�
=
r X �=1
Yij�
isthesum(thetotal)incell(i;j).
Wecansummarisetheseconsiderationsinananalysisofvariancetable:
Sourceof
Sumofsquares
Degreesof
S2
F-value
variation
=SSQ
freedom=f
=SSQ/f
A
[A]2=(2
k�r)
1
S2 A
FA
=S
2 A=S
2 resid
B
[B]2=(2k�r)
1
S2 B
FB
=S
2 B=S
2 resid
AB
[AB]2=(2k�r)
1
S2 AB
FAB
=S
2 AB=S
2 resid
Residual
SSQresid
2k�(r�1)
S2 resid
Totalt
SSQtot
r�2k�1
Inthetable,forexample,FA
iscomparedwithanFdistributionwith(1;2k�(r�1))
degreesoffreedom.
2.1.8
Calculationmethodsforcontrasts
Thesalientpointintheaboveanalysisisthecalculationofthecontrasts.Variousmethods,
somemorepracticalthanothers,canbegiventosolvethisproblem.
Mathematically,thecontrastscanbecalculatedbythefollowingmatrixequation:
2 6 6 6 4I A B A
B
3 7 7 7 5=2 6 6 6 41
1
1
1
�1
1
�1
1
�1
�1
1
1
1
�1
�1
13 7 7 7 52 6 6 6 4(1)
a b ab
3 7 7 7 5
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Onenotesthatbothcontrastsandcellsumsaregiveninstandardorder.Inadditionit
canbeseenthattherowforexamplefortheA-contrastcontains+1foraandab,where
factorAisatitshighlevel,but-1for(1)andb,wherefactorAisatitslowlevel.Finally,
itisnoticedthattherowforAB
foundbymultiplyingtherowsforAandB
byeach
other.
Insomepresentations,thematrixexpresssionshownisgivenjustas+and-signsina
table:
(1)
a
b
ab
I
+
+
+
+
A
�
+
�+
B
�
�+
+
AB
+
��+
2.1.9
Yates'algorithm
Finallywegiveacalculationalgorithm
whichisnamedaftertheEnglishstatistician
FrankYatesandiscalledYates'algorithm.Data,i.e.thecellsums,arearrangedin
standardorderinacolumn.Thenthesearetakeninpairsandsummed,andafterthat
thesamevaluesaresubtractedfromeachother.Thesumsareputatthetopofthenext
columnfollowedbythedi�erences.Whenformingthedi�erences,theuppermostvalue
issubtractedfromthebottomone(mnemonicrule:Ascomplicatedaspossible).The
operationisrepeatedasmanytimesastherearefactors.Herethiswouldbek=2times:
Cellsums
1sttime
2ndtime
=
Contrasts
SumofSq.
(1)
(1)+a
(1)+a+b+ab
=
[I]
[I]2=(2
k�r)
a
b+ab
�(1)+a�b+ab
=
[A]
[A]2=(2k�r)
b
�(1)+a
�(1)�a+b+ab
=
[B]
[B]2=(2k�r)
ab
�b+ab
(1)�a�b+ab
=
[AB]
[AB]2=(2k�r)
Wegiveanumericalexamplewherethedataareshowninthefollowingtable:
B=0
B=1
A=0
12.1
19.8
14.3
21.0
A=1
17.9
24.3
19.1
23.4
One�nds(1)=12:1+14:3=26:4,a=17:9+19:1=37:0,b=19:8+21:0=40:8and
ab=24:3+23:4=47:7.
Yates=algorithmnowgives
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Cellsums
1sttime
2ndtime
=
Contrasts
SumsofSquares
(1)=26.4
63.4
151.9
=
[I]
[I]2=(2k�r)=2884.20
a=37.0
88.5
17.5
=
[A]
[A]2=(2k�r)=38.28
b=40.8
10.6
25.1
=
[B]
[B]2=(2k�r)=78.75
ab=47.7
6.9
-3.7
=
[AB]
[AB]2=(2
k�r)=1.71
Inthisexperimentr=2,andSSQresid
canbefoundasthesumofsquareswithinthe
singlefactorcombinations.
SSQresid=(12:12+14:3
2�(12:1+14:3)2=2)
+(17:92+19:1
2�(17:9+19:1)2=2)
+(19:82+21:0
2�(19:8+21:0)2=2)
+(24:32+23:4
2�(24:3+23:4)2=2)=2:42+0:72+0:72+0:41=4:27
ANOVA
Sourceofvariation
SSQ
df
s2
F-value
Amaine�ect
38.28
2�1=1
38.28
35.75
Bmaine�ect
78.75
2�1=1
78.75
73.60
ABinteraction
1.71
(2�1)(2�1)=1
1.71
1.60
Residualvariation
4.27
4(2�1)=4
1.07
Totalt
123.01
8�1=7
Asweshallsee,Yates'algorithmisgenerallyapplicabletoall2kfactorialexperiments
andforexamplecanbeeasilyprogrammedonacalculator.Thealgorithmalsoappears
insignalanalysisunderthename\fastFouriertransform".
Thelastcolumninthealgorithmgivesthecontraststhatareusedfortheestimationas
wellasthecalculationofthesumsofsquaresforthefactore�ects.
2.1.10
Replicationsorrepetitions
Beforewemoveontoexperimentswith3ormorefactors,letuslookatthefollowing
experiment
B=0
B=1
A=0
Y001
Y011
A=1
Y101
Y111
Dayno.1
,
B=0
B=1
A=0
Y002
Y012
A=1
Y102
Y112
Dayno.2
,���,
B=0
B=1
A=0
Y00R
Y01R
A=1
Y10R
Y11R
Dayno.R
c hs.
DesignofExperiments,Course02411,IMM,DTU
15
thatis,a2�2,replicatedR
times.Themathematicalmodelforthisexperimentis
notidenticalwiththemodelpresentedonpage10thebeginningofthischapter.The
experimentisnotcompletelyrandomisedinthatrandomisationisdonewithindays.
Anexperimentalcollectionofsingleexperimentsthatcanberegardedashomogeneous
withrespecttouncertainty,suchasthedaysintheexample,isgenerallycalledablock.
Ifitisassumedthatthecontributionfromthedayscanbedescribedbyanadditivee�ect,
correspondingtoageneralincreaseorreductionoftheresponseonthesingledays(block
e�ect),areasonablemathematicalmodelwouldbe:
Yij�=�+Ai+Bj+ABij+D�+Fij�
;
i=(0;1);j=(0;1);�=(1;2;:::;R);
whereD�givesthecontributionfromthe�'thdag,andFij�givesthepurelyrandomerror
withindays.
Wewillsaythatthe22experimentisreplicatedRtimes.
Thisisessentiallydi�erentfromthecasewhereforexample2�2�rmeasurementsare
madeinacompletelyrandomiseddesignasonpage10.
Ifoneisinthepracticalsituationofhavingtochoosebetweenthetwodesigns,anditis
assumedthatbothexperiments(becauseofthetimeneeded)mustextendoverseveral
days,thelatterdesignispreferable.Inthe�rstdesigntherandomisationisdoneacross
dayswithrrepetitions,andtheexperimentaluncertainty,Eij�
willalsocontainthe
variationbetweendays.
OnecanregardD�,i.e.thee�ectfromthe�'thday,asarandomlyvaryingamountwith
thevariance�
2 D,whileFij�,i.e.theexperimentalerrorwithinoneday,isassumedtohave
thevariance�
2 F.FromthiscanbederivedthatEij�,i.e.thetotalexperimentalerrorin
acompletelyrandomiseddesignoverseveraldays,hasthevariance
�2 E
=�
2 D
+�
2 F
Theexampleillustratestheadvantageofdividingone'sexperimentintosmallerhomo-
geneousblocksasdistinctfromcompleterandomisation.Italsoshowsthatthereisa
fundamentaldi�erencebetweentheanalysisofanexperimentwithrrepetitionsina
completelyrandomiseddesignandarandomiseddesignreplicatedRtimes.
2.1.11
23
factorialdesign
Wenowstatethedescribedtermsforthe23factorialexperimentwithaminimumof
comments.
ThefactorsarenowA,B,andCwithindicesi,jandk,respectively.Thefactorsare
againorderedsoAisthe�rstfactor,BthesecondandCthethirdfactor.
c hs.
DesignofExperiments,Course02411,IMM,DTU
16
Themathematicalmodelwithrrepetitionspercellinacompletelyrandomiseddesignis:
Yijk�=�+Ai+Bj+ABij+Ck+ACik+BCjk+ABCijk+Eijk�
wherei;j;k=(0;1)and�=(1;::;r).
Theusualrestrictionsare:
1 X i=0
Ai=
1 X j=0
Bj=
1 X i=0
ABij=
1 X j=0
ABij=
1 X k=0
Ck=���=
1 X k=0
ABCijk=0
whichimpliesthat
A1=
�
A0
;
B1=
�
B0
;
AB11=
�
AB10=
�
AB01=
AB00
;
C1=
�
C0
;
�
�
�
;
(andfurtheronuntil)
AB
C000=
�
AB
C100=
�
AB
C010=
AB
C110=
�
AB
C001=
AB
C101=
AB
C011=
�
AB
C111
Thee�ectsoftheexperiment(whichgivethedi�erenceinresponsewhenafactoris
changedfrom\low"levelto\high"level,cf.page11)are
A=2A1
;
B=2B1
;
AB=2AB11
;
C=2C1
;���;ABC=2ABC111
Thestandardorderforthe23=8di�erentexperimentalconditions(factorcombinations)
is:
(1)
,
a
,
b
,
ab
,
c
,
ac
,
bc
,
abc
wheretheintroductionofthefactorCisdonebymultiplyingcontothetermsforthe22
experimentandaddingtheresultingtermstothesequence:(1);a;b;ab;((1);a;b;ab)c=
(1);a;b;ab;c;ac;bc;abc.
c hs.
DesignofExperiments,Course02411,IMM,DTU
17
[I]
=
[+(1)+a+b+ab+c+ac+bc+abc]
[A]
=
[�(1)+a�b+ab�c+ac�bc+abc]
[B]
=
[�(1)�a+b+ab�c�ac+bc+abc]
[AB]
=
[+(1)�a�b+ab+c�ac�bc+abc]
[C]
=
[�(1)�a�b�ab+c+ac+bc+abc]
[AC]
=
[+(1)�a+b�ab�c+ac�bc+abc]
[BC]
=
[+(1)+a�b�ab�c�ac+bc+abc]
[ABC]
=
[�(1)+a+b�ab+c�ac�bc+abc]
orinmatrixformulation
2 6 6 6 6 6 6 6 6 6 6 6 6 6 4I A B A
B C AC
BC
ABC
3 7 7 7 7 7 7 7 7 7 7 7 7 7 5=2 6 6 6 6 6 6 6 6 6 6 6 6 6 41
1
1
1
1
1
1
1
�1
1
�1
1
�1
1
�1
1
�1
�1
1
1
�1
�1
1
1
1
�1
�1
1
1
�1
�1
1
�1
�1
�1
�1
1
1
1
1
1
�1
1
�1
�1
1
�1
1
1
1
�1
�1
�1
�1
1
1
�1
1
1
�1
1
�1
�1
13 7 7 7 7 7 7 7 7 7 7 7 7 7 52 6 6 6 6 6 6 6 6 6 6 6 6 6 4(1)
a b ab c a
c bc abc
3 7 7 7 7 7 7 7 7 7 7 7 7 7 5
Yates'algorithmisperformedasabove,buttheoperationonthecolumnsshouldnowbe
done3timesasthereare3factors.Ifonewritesindetailwhathappens,onegets:
response
1sttime
2ndtime
3rdtime
contrasts
(1)
(1)+a
(1)+a+b+ab
(1)+a+b+ab+c+ac+bc+abc
[I]
a
b+ab
c+ac+bc+abc
�(1)+a�
b+ab�
c+ac�
bc+abc
[A]
b
c+ac
�(1)+a�
b+ab
�(1)�
a+b+ab�
c�
ac+bc+abc
[B]
ab
bc+abc
�c+ac�
bc+abc
(1)�
a�
b+ab+c�
ac�
bc+abc
[AB]
c
�(1)+a
�(1)�
a+b+ab
�(1)�
a�
b�
ab+c+ac+ab+abc
[C]
ac
�b+ab
�c�
ac+bc+abc
(1)�
a+b�
ab�
c+ac�
bc+abc
[AC]
bc
�c+ac
(1)�
a�
b+ab
(1)+a�
b�
ab�
c�
ac+bc+abc
[BC]
abc
�bc+abc
c�
ac�
bc+abc
�(1)+a+b�
ab+c�
ac�
bc+abc
[ABC]
Parameterestimatesare,withk=3:
^�=
[I]
2k�r
;
b A 1=
[A]
2k�r
;b B 1=
[B]
2k�r
;:::;
dABC111=
[ABC]
2k�r
c hs.
DesignofExperiments,Course02411,IMM,DTU
18
Correspondingly,thee�ectestimatesare:
b A=2b A 1;
b B=2b B 1;
���;
dABC=2dABC111
Thesumsofsquaresare,forexample:
SSQA
=
[A]2
2k�r
;SSQB
=
[B]2
2k�r
;SSQABC
=[ABC]2
2k�r
Thevariancesofthecontrastsarefound,with[A]asexample,as
Varf[A]g=Varf�(1)+a�b+ab�c+ac�bc+abcg=2k�r��
2
;
wherek=3here.
Theresultisseenbynotingthatthereare2kterms,whichallhavethesamevariance,
whichforexampleis
Varfabg=Varf
r X �=1
Y110�g=r��
2
Further,itisnowfound,that
Varfb A 1g=Varf[A]=(2
k�r)g=�
2=(2
k�r)
Varfb Ag=Varf2b A 1g=�
2=(2
k�
2�r)
2.1.12
2k
factorialexperiment
Thestatedequationsaregeneraliseddirectlytofactorialexperimentswithkfactors,each
on2levels,withrrepetitionsinarandomiseddesign.Writingupthemathematical
model,namesforcellsums,calculationofcontrastsetc.aredoneinexactlythesameway
asdescribedabove.Forestimatesandsumsofsquares,thengenerally
Parameterestimate=(Contrast)/(2k�r)
E�ectestimate=2�Parameterestimate
Sumofsquares(SSQ)=(Contrast)
2=(2
k�r)
c hs.
DesignofExperiments,Course02411,IMM,DTU
19
Regardingtheconstructionofcon�denceintervalsfortheparametersande�ects,the
varianceoftheestimatescanbederived.One�nds
VarfContrastg=�
2�2k�r
VarfParameterestimateg=VarfContrastg/(2k�r)2=�
2=(2k�r)
VarfE�ectestimateg=22�
2=(2k�r)=�
2=(2k�
2�r)
Thecon�denceintervalsforparametersore�ectscanbeconstructedifonehasanestimate
of�
2.Supposethatonehassuchanestimate,^�
2
=s2,andthatithasfdegreesof
freedom.If(1��)con�denceintervalsarewanted,onetherebygets
I 1�
�(parameter)=Parameterestimate�s�t(f) (1�
�=2)=
p 2k�r
I 1�
�(e�ekt)=E�ectestimate�2�s�t(f) (1�
�=2)=
p2k�r
wheret(f) (1�
�=2)
denotesthe(1��=2)-fractileinthet-distributionwithfdegreesof
freedom.
2.2
Blockconfounded2k
factorialexperiment
Inexperimentswithmanyfactors,thenumberofsingleexperimentsquicklybecomes
verylarge.Forpracticalexperimentalwork,thismeansthatitcanbediÆculttoensure
homogeneousexperimentalconditionsforallthesingleexperiments.
Agenerallyoccurringproblemisthatinaseriesofexperiments,rawmaterialisused
thattypicallycomesintheformofbatches,i.e.homogeneousshipments.Aslongaswe
performtheexperimentsonrawmaterialfromthesamebatch,theexperimentswillgive
homogeneousresults,whileresultsofexperimentsdoneonmaterialfromdi�erentbatches
willbemorenon-homogeneous.Thebatchesofrawmaterialinthiswayconstituteblocks.
Inthesameway,itwilloftenbethecasethatexperimentsdoneclosetogetherintime
aremoreuniformthanexperimentsdonewithalongtimebetweenthem.
Inaseriesofexperimentsonewilltrytodoexperimentsthataretobecomparedonthe
mostuniformbasispossible,sincethatgivesthemostexactevaluationofthetreatments
thatarebeingstudied.Forexample,onewilltrytodotheexperimentonthesamebatch
andwithinasshortaspaceoftimeaspossible.Butthisofcourseisaproblemwhenthe
numberofsingleexperimentsislarge.
Letusimaginethatwewanttodoa23factorialexperiment,i.e.anexperimentwith8
singleexperiments,correspondingtothe8di�erentfactorcombinations.Supposefurther
c hs.
DesignofExperiments,Course02411,IMM,DTU
20
thatitisnotpossibletodoallthese8singleexperimentsonthesameday,butperhaps
onlyfourperday.
Anobviouswaytodistributethe8singleexperimentsoverthetwodayscouldbetodraw
lots.Weimaginethatthisdrawinglotsresultsinthefollowingdesign:
day1
day2
(1)
c
abc
a
bc
ac
ab
b
Forthisdesign,wegetforexampletheA-contrast:
[A]=[�(1)+a�b+ab�c+ac�bc+abc]
Aslongasthetwodaysgiveresultswithexactlythesamemeanresponse,thisestimate
will,inprinciple,bejustasgoodasiftheexperimentshadbeendoneonthesameday.
(howeverthevarianceisgenerallyincreasedwhenexperimentsaredoneovertwodays
insteadofononeday).
Butifontheotherhandthereisacertainunavoidabledi�erenceinthemeanresponse
onthetwodays,weobviouslyhaveariskthatthisa�ectstheestimates.Asasimple
modelforsuchadi�erenceinthedays,wecanassumethattheresponseonday1is1g
undertheideal,whileitis2govertheidealonday2.Ane�ectofthistypeisablock
e�ect,andthedaysconstitutetheblocks.Onesaysthattheexperimentislaidoutintwo
blockseachwith4singleexperiments.
FortheA-contrast,itisshownbelowhowtheseunintentional,butunavoidable,e�ects
ontheexperimentalresultsfromthedayswilla�ecttheestimation,as1gissubtracted
fromalltheresultsfromday1and2gisaddedtoalltheresultsfromday2:
[A]=[�((1)�1g)+(a�1g)�(b+2g)+(ab+2g)�(c�1g)+(ac+2g)�(bc+2g)+(abc�1g)]
=[�(1)+a�b+ab�c+ac�bc+abc]+[1�1�2+2+1+2�2�1]g
=[�(1)+a�b+ab�c+ac�bc+abc]
Thus,adi�erenceinlevelontheresultsfromthetwodays(blocks)willnothaveanye�ect
ontheestimateforthemaine�ectoffactorA.Inotherwords,factorAisinbalance
withtheblocks(thedays).
Ifwerepeattheprocedureforthemaine�ectoffactorB,weget
[B]=[�((1)�1g)�(a�1g)+(b+2g)+(ab+2g)�(c�1g)�(ac+2g)+(bc+2g)+(abc�1g)]
c hs.
DesignofExperiments,Course02411,IMM,DTU
21
=[�(1)�a+b+ab�c�ac+bc+abc]+[1+1+2+2+1�2+2�1]g
=[�(1)+a�b+ab�c+ac�bc+abc]+6g
TheestimatefortheBe�ect(i.e.thedi�erenceinresponsewhenBischangedfromlow
tohighlevel)istherebyonaverage(6g=4)=1:5ghigherthantheidealestimate.
Ifwelookbackatthedesign,thisisbecausefactorBwasmainlyat"highlevel"onday
2,wheretheresponseonaverageisalittleabovetheideal.
ThesamedoesnotapplyinthecaseoffactorA.Thishasbeenat\highlevel"twotimes
eachdayandlikewiseat\lowlevel"twotimeseachday.ThesameappliesforfactorC.
ThusfactorsAandCareinbalanceinrelationtotheblocks(thedays),whilefactorB
isnotinbalance.
Anoverallevaluationofthee�ectoftheblocks(thedays)ontheexperimentcanbeseen
fromthefollowingmatrixequation
2 6 6 6 6 6 6 6 6 6 6 6 6 6 41
1
1
1
1
1
1
1
�1
1
�1
1
�1
1
�1
1
�1
�1
1
1
�1
�1
1
1
1
�1
�1
1
1
�1
�1
1
�1
�1
�1
�1
1
1
1
1
1
�1
1
�1
�1
1
�1
1
1
1
�1
�1
�1
�1
1
1
�1
1
1
�1
1
�1
�1
13 7 7 7 7 7 7 7 7 7 7 7 7 7 52 6 6 6 6 6 6 6 6 6 6 6 6 6 4(1)�1g
a�1g
b+2g
ab+2g
c�1g
ac+2g
bc+2g
abc�1g
3 7 7 7 7 7 7 7 7 7 7 7 7 7 5=2 6 6 6 6 6 6 6 6 6 6 6 6 6 4I+4g
A B+6g
AB�6g
C AC
BC�6g
ABC�6g
3 7 7 7 7 7 7 7 7 7 7 7 7 7 5
ItcanbeseenthatallcontraststhatonlyconcernfactorsAandCarefoundcorrectly,
becausethetwofactorsareinbalanceinrelationtotheblocksinthedesign,whileall
contraststhatalsoconcernBarea�ectedbythe(unintentional,butunavoidable)e�ect
fromtheblocks.
Whatwenowcanaskiswhetheritispossibleto�ndadistributionoverthetwodaysso
thatthein uencefromtheseiseliminatedtothegreatestpossibleextent.
Wecannotethatitisthedi�erencebetweenthedaysthatisimportantfortheestimates
ofthee�ectsofthefactors,whilegenerallevelofthedaysisabsorbedinthecommon
averageforalldata.
Ifweoncemoreregardthecalculationofthecontrast[A],wecandrawupthefollowing
table,whichshowshowthein uenceofthedaysisweightedintheestimate:
Contrast[A]
Response
(1)
a
b
ab
c
ac
bc
abc
Weight
�
+
�
+
�+
�
+
Day
1
1
2
2
1
2
2
1
c hs.
DesignofExperiments,Course02411,IMM,DTU
22
Wenotethatday1entersanequalnumberoftimeswith+andwith�,andday2as
well.Ifwelookatoneofthecontrastswherethedaysdonotcancel,e.g.[B],wegeta
tablelikethefollowing:
Contrast[B]
Response
(1)
a
b
ab
c
ac
bc
abc
Weight
�
�+
+
��
+
+
Day
1
1
2
2
1
2
2
1
wherethebalanceisobviouslynotpresent.
Theconditionthatisnecessarysothatane�ectisnotin uencedbythedaysisobviously
thatthereisabalanceasdescribed.Thepossibilitiesforcreatingsuchabalanceare
linkedtothematrixofonesintheestimation:
2 6 6 6 6 6 6 6 6 6 6 6 6 6 4I A B A
B C AC
BC
ABC
3 7 7 7 7 7 7 7 7 7 7 7 7 7 5=2 6 6 6 6 6 6 6 6 6 6 6 6 6 41
1
1
1
1
1
1
1
�1
1
�1
1
�1
1
�1
1
�1
�1
1
1
�1
�1
1
1
1
�1
�1
1
1
�1
�1
1
�1
�1
�1
�1
1
1
1
1
1
�1
1
�1
�1
1
�1
1
1
1
�1
�1
�1
�1
1
1
�1
1
1
�1
1
�1
�1
13 7 7 7 7 7 7 7 7 7 7 7 7 7 52 6 6 6 6 6 6 6 6 6 6 6 6 6 4(1)
a b ab c a
cbc a
bc3 7 7 7 7 7 7 7 7 7 7 7 7 7 5
Thismatrixhasthespecialcharacteristicthattheproductsumofanytworowsiszero.
Ifoneforexampletakestherowsfor[A]and[B],onegets(-1)(-1)+(+1)(-1)+...+
(+1)(+1)=0.Thetwocontrasts[A]and[B]arethusorthogonalcontrasts(linearly
independent).
IfonethereforechoosesforexampleadesignwherethedaysfollowfactorB,itisabsolutely
certainthatinanycasefactorAwillbeinbalanceinrelationtothedays.Thisdesign
wouldbe:
day1
day2
(1)
a
c
ac
b
ab
bc
abc
Thein uencefromthedayscannowbecalculatedbyadding�1gtoalldatafromday1
andadding+2gtoalldatafromday2:
c hs.
DesignofExperiments,Course02411,IMM,DTU
23
2 6 6 6 6 6 6 6 6 6 6 6 6 6 41
1
1
1
1
1
1
1
�1
1
�1
1
�1
1
�1
1
�1
�1
1
1
�1
�1
1
1
1
�1
�1
1
1
�1
�1
1
�1
�1
�1
�1
1
1
1
1
1
�1
1
�1
�1
1
�1
1
1
1
�1
�1
�1
�1
1
1
�1
1
1
�1
1
�1
�1
13 7 7 7 7 7 7 7 7 7 7 7 7 7 52 6 6 6 6 6 6 6 6 6 6 6 6 6 4(1)�1g
a�1g
b+2g
ab+2g
c�1g
ac�1g
bc+2g
abc+2g
3 7 7 7 7 7 7 7 7 7 7 7 7 7 5=2 6 6 6 6 6 6 6 6 6 6 6 6 6 4I+4g
A B+12g
AB
C AC
BC
ABC
3 7 7 7 7 7 7 7 7 7 7 7 7 7 5
Onecanseethatnow,becauseofthedescribedattributeofthematrix,itisonlytheB
contrastandtheaveragethatarea�ectedbythedistributionoverthetwodays.
Ofcoursethisdesignisnotveryusefulifwealsowanttoestimatethee�ectoffactor
B,aswecannotunequivocallyconcludewhetheraB-e�ectfoundcomesfromfactorB
orfromdi�erencesintheblocks(thedays).Ontheotherhand,alltheothere�ectsare
clearlyfreefromtheblocke�ect(thee�ectofthedays).
Onesaysthatmaine�ectoffactorBisconfoundedwiththee�ectoftheblocks(the
word\confound"isfromLatinandmeansto\mixup").
Thelastexampleshowshowwe(byfollowingthe+1and�1variationforthecorrespond-
ingcontrast)candistributethe8singleexperimentsoverthetwodayssothatprecisely
oneofthee�ectsofthemodelisconfoundedwithblocks,andnomorethantheone
chosen.Onecanshowthatthiscanalwaysjustbedone.
If,forexample,wechoosetodistributeaccordingtothethree-factorinteractionABC,it
canbeseenthattherowfor[ABC]has+1fora,b,cogabc,but�1for(1),ab,acogbc.
Onecanalsofollowthe+and�signsinthefollowingtable:
(1)
a
b
ab
c
ac
bc
abc
I
+
+
+
+
+
+
+
+
A
�
+
�+
�+
�
+
B
�
�+
+
��
+
+
AB
+
��+
+
�
�
+
C
�
���+
+
+
+
AC
+
�+
��+
�
+
BC
+
+
����
+
+
ABC
�
+
+
�+
�
�
+
Thisgivesthefollowingdistribution,aswenowingeneraldesignatethedaysasblocks
andletthesehavethenumbers0and1:
block0
block1
(1)
ab
ac
bc
a
b
c
abc
c hs.
DesignofExperiments,Course02411,IMM,DTU
24
Theblockthatcontainsthesingleexperiment(1)iscalledtheprincipalblock.The
practicalmeaningofthisisthatonecanmakeastartinthisblockwhenconstructingthe
design.
2.2.1
Constructionofaconfoundedblockexperiment
Theexperimentdescribedaboveiscalledablockconfounded(orjustconfounded)23
factorialexperiment.Thechosenconfoundingisgivenwiththeexperiment's
de�ningrelation:I=
ABC
AndinthisconnectionABCiscalledthede�ningcontrast.
Aneasywaytocarryoutthedesignconstructionistoseeifthesingleexperimentshavean
evenoranunevennumberoflettersincommonwiththede�ningcontrast.Experiments
withanevennumberincommonshouldbeplacedintheoneblockandexperimentswith
anunevennumberincommonshouldgointheotherblock.
Alternativelyonemayusethefollowingtabularmethodwherethecolumnfor'Block'
isfoundbymultiplyingtheA,BandCcolumns:
A
B
C
code
Block=ABC
�1
�1
�1
(1)
�1
+1
�1
�1
a
+1
�1
+1
�1
b
+1
+1
+1
�1
ab
�1
�1
�1
+1
c
+1
+1
�1
+1
ac
�1
�1
+1
+1
bc
�1
+1
+1
+1
abc
+1
Theexperimentisanalysedexactlyasanordinary23factorialexperiment,butwiththe
exceptionthatthecontrast[ABC]cannotunambiguouslybeattributedtothefactorsin
themodel,butisconfoundedwiththeblocke�ect.
Onecanaskwhetheritispossibletodotheexperimentin4blocksof2singleexperiments
inareasonableway.Thishasgeneralrelevance,sincepreciselytheblocksize2(which
naturallyisthesmallestimaginable)occursfrequentlyinpracticalinvestigations.
Onecouldimaginethatthe8observationswereputintoblocksaccordingtotwocriteria,
i.e.bychoosingtwode�ningrelationsthatforexamplecouldbe:
c hs.
DesignofExperiments,Course02411,IMM,DTU
25
I 1=
(1)
ab
ac
bc
ABC
c
abc
a
b
I 2=AB
block(0,0)�1
block(0,1)�2
block(1,0)�3
block(1,1)�4
Onenoticesforexamplethattheexperimentsinblock(0,1)haveanevennumberof
lettersincommonwithABCandanunevennumberoflettersincommonwithAB.
Thetabularmethodgives
A
B
C
code
B1=ABC
B2=AB
Blockno.
�1
�1
�1
(1)
�1
+1
1�(0,0)
+1
�1
�1
a
+1
�1
4�(1,1)
�1
+1
�1
b
+1
�1
4�(1,1)
+1
+1
�1
ab
�1
+1
1�(0,0)
�1
�1
+1
c
+1
+1
3�(1,0)
+1
�1
+1
ac
�1
�1
2�(0,1)
�1
+1
+1
bc
�1
�1
2�(0,1)
+1
+1
+1
abc
+1
+1
3�(1,0)
Inthe�gure,thereisa2�2blocksystem,correspondingtothegroupingaccordingto
ABCandAB.OnecannotethatthefactorsAandBarebothon\high"aswellas\low"
levelinall4blocks.Thesefactorsareobviouslyinbalanceinrelationtotheblocks.
However,thisdoesnotapplytofactorC.Itisat\high"levelintwooftheblocksandat
\low"levelintheothertwo.Ifitissounfortunatethatthetwoblocksdesignated(0,0)
and(1,1)togetherresultinahigherresponsethantheothertwoblocks,wewillgetan
undervaluationofthee�ectoffactorC.ThusfactorCisconfoundedwithblocks.
Tobeabletoforeseethis,onecanperceiveABC
andAB
asfactorsandthenwitha
formalcalculation�ndtheinteractionbetweenthem:
Blocke�ect=Blocklevel+ABC+AB+(ABC�AB)
Forthee�ectthuscalculated(ABC�AB)=A
2B
2C,thearithmeticruleisintroduced
thatinthe2kexperiment,theexponentsarereducedmodulo2.Thus(ABC�AB)=
A2B
2C
�!A
0B
0C
�!C
.Therebyonegetstheformalexpressionfortheblock
confounding:
Blocke�ect=Blocklevel+ABC+AB+C
whichtellsusthatitispreciselythethreee�ectsABC,ABandCthatbecomeconfounded
withtheblocksinthegivendesign.
c hs.
DesignofExperiments,Course02411,IMM,DTU
26
Ifonewantstoestimatethemaine�ectofC,thisdesignisthereforeunfortunate.A
betterdesignwouldbe:
I 1=
(1)
abc
ac
b
AC
c
ab
a
bc
I 2=AB
block0,0
block0,1
block1,0
block1,1
Since(AC�AB)=A
2BC=BC,thein uenceoftheblocksinthedesignisformally
givenby
Blocks=Blocklevel+AC+AB+BC
andthede�ningrelation:I=AB=AC=BC.
Thethreee�ectsAB,ACandBCareconfoundedwithblocks.Allothere�ectscanbe
estimatedwithoutin uencefromtheblocks.Takespecialnotethatthemaine�ectsA,
BandCallappearatbothhighandlowlevelsinall4blocks.Thethreefactorsarethus
allinbalanceinrelationtotheblocks.
Thedesignshownisthebestexistingdesignforestimatingthemaine�ectsof3factors
inminimalblocks,thatis,with2experimentsineach.Sinceminimalblocksatthesame
timeresultinthemostaccurateexperiments,thedesignisparticularlyimportant.
Thedesigndoesnotgivethepossibilityofestimatingthetwo-factorinteractionsAB,AC
andBC.
2.2.2
A
one-factor-at-a-timeexperiment
Itcouldbeinterestingtocomparethedesignshownwiththefollowingone-factor-at-a-
timeexperiment,whichisalsocarriedoutinblocksofsizeof2:
(1)a
(1)b
(1)c
thatis3blocks,wherethefactorsareinvestigatedeachinoneblock.
Theexperimentcouldbeaweighingexperiment,whereonewantstodeterminetheweight
ofthreeitems,A,BandC.Themeasurement(1)correspondstothezeropointreading,
whileagivesthereadingwhenitemAis(alone)ontheweightandcorrespondinglyforb
andc.
Inthisdesign,anestimateforexampleoftheAe�ectisfoundas
c hs.
DesignofExperiments,Course02411,IMM,DTU
27
b A=[�(1)+a]withvariance2�
2
whereitishereassumedthatr=1.Intheprevious23designin2�2blocks,itwasfound
b A=[�(1)+a�:::+abc]=(2
3�
1)withvariance�
2=2
Ifoneistoachieveanaccuracyasgoodasthe\optimal"designwithrepeateduseofthe
one-factor-at-a-timedesign,ithastoberepeated(2��
2=(�
2=2)=4times.Thus,there
willbeatotalof4�6=24singleexperimentsincontrasttothe8thatareusedinthe
\optimal"design.
Anotherone-factor-at-a-timein2blocksof2singleexperimentsisthefollowingexperi-
ment:
(1)
a
b
c
block0
block1
Whyisthisahopelessexperiment?Whatcanoneestimatefromtheexperiment?
2.3
Partiallyconfounded2k
factorialexperiment
Wewillagainconsiderthe2�2experimentwiththetwofactorsAandB:
B=0
B=1
A=0
(1)
b
A=1
a
ab
Supposethatthisexperimentistobedoneinblocksofthesize2.Theblockscan
correspondforexampletobatchesofrawmaterialthatarenolargerthanatmost2
experimentsperbatchcanbedone.Bychoosingthede�ningcontrastasI=AB,the
followingblockgroupingisobtained:
Experiment1:
(1) 1
ab 1
a2
b 2
batch1
batch2
I=AB
Themathematicalmodeloftheexperimentisthefollowing:
Yij�=�+Ai+Bj+ABij+Eij�,wherei=(0;1);j=(0;1);�=1
c hs.
DesignofExperiments,Course02411,IMM,DTU
28
buttheABinteractione�ectisconfoundedwithblocks.
Supposenowthatwefurtherwanttoestimateand/ortesttheinteractioncontribution
ABij.Thisofcoursecanonlybedonebydoingyetanotherexperiment,inwhichABis
notconfoundedwithblocks.Therearetwosuchexperiments,onewhereAisconfounded
withblocksandonewhereBisconfoundedwithblocks.Asexamplewechoosethelatter:
Experiment2:
(1) 3
a3
b 4
ab 4
batch3
batch4
I=B
Fromthetwoexperimentsshown(eachwithtwoblocksandtwosingleexperimentsin
eachblock)wewillnowestimatethevariouse�ects.Themaine�ectforfactorAcanbe
estimatedbothinthetwo�rstblocksandinthetwolastblocksandatotalAcontrastis
foundas:
[A] total=[A] 1+[A] 2
;
thatis,thesumoftheAcontrastsinboththetwoexperimentalparts:
[A] 1=�(1) 1+a2�b 2+ab 1
(from
experiment1)
[A] 2=�(1) 3+a3�b 4+ab 4
(from
experiment2)
astheindexofthe8singleexperimentscorrespondstotheblock(batch)inwhichthe
singleexperimentsweremade.Theindexofthecontrastgiveswhetheritisthe�rstor
thesecondexperimentalpartitiscalculatedin.
Further,wecannow�ndacontrastforthemaine�ectB,butonlyfromthe�rstexperi-
ment:
[B] 1=�(1) 1�a2+b 2+ab 1
Finallyacontrastfortheinteractione�ectABisfound,butnowfromtheotherexperiment
whereitisnotconfoundedwithblocks:
[AB] 2=+(1) 3�a3�b 4+ab 4
SincethetwoAcontrastsarebothfreeofblocke�ects,inadditiontotheirsumwecan
�ndtheirdi�erence:
[A] di�erence=[A] 1�[A] 2
c hs.
DesignofExperiments,Course02411,IMM,DTU
29
Thisamountmeasuresthedi�erencebetweentheAestimatesinthetwopartsofthe
experiment.Thisdi�erence,astheexperimentislaidout,canonlybeduetoexperimental
uncertainty,andcanthusbeinterpretedasanexpressionoftheexperimentaluncertainty,
thatis,theresidualvariation.
Thetwocontrasts
[B] 2=�(1) 3�a3+b 4+ab 4
(from
experiment2)
[AB] 1=+(1) 1�a2�b 2+ab 1
(from
experiment1)
arebothconfoundedwithblocks.
Asexpressionsoftheexperimentallevelsinthetwopartsoftheexperimentwe�nd
[I] 1=+(1) 1+a2+b 2+ab 1
(from
experiment1)
[I] 2=+(1) 3+a3+b 4+ab 4
(from
experiment2)
Thisresultsin
[I] total=[I] 1+[I] 2
[I] di�erence=[I] 1�[I] 2
Thequantity[I] totalandthecontrast[I] di�erencemeasurethelevelofthewholeex-
perimentandthedi�erenceinlevelbetweenthe�rstandsecondpartoftheexperiment,
respectively.
Onecaninvestigatewhetherthequantitiesdrawnupareorthogonalcontrastsbylooking
atthefollowingmatrixexpression:
2 6 6 6 6 6 6 6 6 6 6 6 6 6 4[I]
[A] total
[B] 1
[AB] 2
[A] 1�[A] 2
[B] 2
[AB] 1
[I] 1�[I] 2
3 7 7 7 7 7 7 7 7 7 7 7 7 7 5=2 6 6 6 6 6 6 6 6 6 6 6 6 6 41
1
1
1
1
1
1
1
�1
1
�1
1
�1
1
�1
1
�1
�1
1
1
0
0
0
0
0
0
0
0
1
�1
�1
1
�1
1
�1
1
1
�1
1
�1
0
0
0
0
�1
�1
1
1
1
�1
�1
1
0
0
0
0
1
1
1
1
�1
�1
�1
�1
3 7 7 7 7 7 7 7 7 7 7 7 7 7 52 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4(1) 1
a2 b 2 a
b 1(1) 3
a3 b 4 a
b 43 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
c hs.
DesignofExperiments,Course02411,IMM,DTU
30
Oneseesthatallthecontrastsaremutuallyorthogonalandthatthereareexactly7
contrastsandthesum(thepseudocontrast).Theycanthereforeasawholedescribeall
thevariationbetweenthe8singleexperimentsthathavebeencarriedout.
Btmeansofthegeneralformulaforthesumsofsquaresandforcontrastsinparticular,
wecanthen�ndallthesumsofsquares.
SSQA
=
[A]2 total
RA�2k�r
,
whereRA
=2andk=2
SSQB
=
[B]2 1
RB�2k�r
,
whereRB
=1andk=2
SSQAB
=
[AB]2 2
RAB�2k�r
,
whereRAB
=1andk=2
SSQA;uncertainty
=
[A]2 1+[A]2 2
2k�r
�[A]2 total
RA�2k�r
,
whereRA
=2
SSQB;blocks
=
[B]2 2
RB;blocks�
2k�r
,
whereRB;blocks=1
SSQAB;blocks
=
[AB]2 1
RAB;blocks�
2k�r
,
whereRAB;blocks=1
SSQleveldi�erence
=
[I]2 1+[I]2 2
2k�r
�[I]2 total
R�2k�r
,
whereR=2
Allthesesumsofsquareshave1degreeoffreedom,andwecannowdrawupananalysis
ofvariancetablebasedon:
SSQA
,
fA
=1
,
(precision=1)
SSQB
,
fB
=1
,
(precision=1/2)
SSQAB
,
fAB
=1
,
(precision=1/2)
SSQblocks=SSQB;blocks+SSQAB;blocks+SSQleveldi�erence;fblocks=3
SSQuncertainty=SSQA;uncertainty
;
funcertainty=1
2.3.1
Somegeneralisations
Asshown,allthedrawnupcontrastsaremutuallyorthogonal.
c hs.
DesignofExperiments,Course02411,IMM,DTU
31
Thismeansthatthesumofsquaresforthe7contrastsmakeupthetotalsumofsquares.
Thisisalsosynonymouswiththestatementthatthee�ectswehaveintheexperiment
areallmutuallybalanced.
Onecanalsonotethatthenumberofdegreesoffreedomisprecisely8�1=7,namely
oneforeachofthecontrasts,towhichcomes1forthetotalsum[I].
Onecanthentesttheindividuale�ectsofthemodelagainsttheestimateofuncertainty.
Intheexample,thisestimatehasonly1degreeoffreedom,andofcoursethisdoesnot
giveareasonablystrongtest.
ThefactthatoneseeminglycomestothesameFtestforthee�ectofA,Baswellas
ABisnotsynonymouswiththestatementthattheA,BandABe�ectsareestimated
equallyprecise.Forexamplethiscanbeseenbycalculatingthevariancesintheparameter
estimates: V
ar(b A 1)=Varf
[A]
RA
�2k�rg=�
2�
RA
�2k�r
(RA
�2k�r)2
=
�2
RA
�2k�r
;RA
=2
Var(b B 1)=Varf
[B] 1
RB
�2k�rg=�
2�
RB
�2k�r
(RB
�2k�r)2
=
�2
RB
�2k�r
;RB
=1
Var(d AB11)=Varf
[AB] 2
RAB
�2k�rg=�
2�
RAB
�2k�r
(RAB
�2k�r)2
=
�2
RAB
�2k�r
;RAB
=1
sothatthevariancesoftheBandABestimatesaredoublethevarianceoftheAestimate.
ThisofcourseisduetothefactthattheAestimateisbasedontwiceasmanyobservations
astheotherestimates(RA=RB
=2andRA=RAB
=2).
Thedi�erencebetweenthetestsofthethreee�ectsistheirpower.ThetestoftheA
e�ecthasgreaterpowerthantheothertwotests(forthesametestlevel�).
Onecangenerallywrite
VarfParameterestimateg=Varf[Contrast]=(R�2
k�r)g=�
2=(R�2
k�r)
;
VarfE�ectestimateg=Varf2�[Contrast]=(R�2
k�r)g=�
2�4=(R�2
k�r)
;
whereRgivesthenumberof2kfactorialexperimentsonwhichtheestimateisbased,
andrgivesthenumberofrepetitionsforthesinglefactorcombinationsinthesefactorial
c hs.
DesignofExperiments,Course02411,IMM,DTU
32
experiments.FortheAe�ectintheexample,R=2,whileR=1forboththeBandthe
ABe�ect.
Ifonehasrepeateda2kfactorialexperimentRtimes,whereane�ectcanbeestimated,
onecan�ndthevariationbetweentheseestimatesinasimilarwayasshownfortheA
e�ectintheexample.Ifwesuppose,forthesakeofsimplicity,thatitistheAe�ect
thatcanbeestimatedintheseRdi�erent2kexperimentswithrrepetitionsperfactor
combination,wecangenerally�ndanestimateofuncertaintyasthesquaresum:
SSQA;uncertainty=
[A]2 1+[A]2 2+���+[A]2 R
2k�r
�([A] 1+[A] 2+���+[A] R)2
R�2k�r
whichwillhaveR�1degreesoffreedom.Theamount[A] �givestheAcontrastinthe
�'thfactorialexperiment.
OnenotesthatthissumofsquaresisexactlythevariationbetweentheRestimatesfor
theAe�ect.Ifonehasseverale�ectswhichinthiswayareestimatedseveraltimes,all
theiruncertaintycontributionscanbesummedupinacommonuncertaintyestimate,
whichcanbeusedfortesting.
Estimationafblocke�ects
Insomeconnections,itcanbeofinteresttoestimatespeci�cblockdi�erences.Ifwe
againtakethetwoconfounded2kexperimentsthatformthebasisforthissection,we
couldforexamplebeinterestedinestimatingthedi�erencebetweenblock0andblock
1.Anestimateforthisdi�erencecanbederivedbyrememberingthatthedi�erence
betweenblocks0and1isconfoundedwiththeABe�ect,andthatthepureABe�ect
canbeestimatedinthesecondexperimentalhalf.Inotherwords,wecandrawupthe
contrast
[AB] 1�[AB] 2=[(1)1�a2�b 2+ab 1]�[(1)3�a3�b 4+ab 4]
Thisquantityhas2�(di�erencebetweenblock0andblock1)asitsexpectedvalue,and
onecanthereforeusetheestimateX1=([AB] 1�[AB] 2)=2astheestimatefortheblock
di�erenceBlock1-Block2.
Inthisestimationofblocke�ects,theprincipleisthesimpleonethatoneestimatesthe
e�ectsthattheblocksareconfoundedwithandthenbreakstheconfoundingwiththese
estimates.
Wewillnotgofurtherherewiththeseideas,butonlypointoutthegeneralpossibilities
thatlieinusingpartialconfounding.
c hs.
DesignofExperiments,Course02411,IMM,DTU
33
Itbecomespossibletotestandestimateallfactore�ectsinfactorialexperimentswith
smallblocks,justasitbecomespossibletoextractblocke�ectswiththeoutlinedestima-
tiontechnique.
2.4
Fractional2k
factorialdesign
Inthissection,wewillintroduceaspecialandveryimportanttypeofexperiment,which
undercertainassumptionscanhelptoreduceexperimentalworkgreatlyincomparison
withcompletefactorialexperiments.
Example2.1:
A
simpleweighingexperimentwith3items
Supposewewanttodeterminetheweightofthreeitems,A,BandC.Aweighingresult
canbedesignatedinthesamewayasdescribedabove.Forexample"a"designatesthe
resultoftheweighingwhereitemAisontheweightalone,while(1)designatesweighing
withoutanyitembeingontheweight,i.e.,thezeropointadjustment.
Thesimplestexperimentconsistsindoingthefollowing4singleexperiments:
(1)
a
b
c
thatis,thatonemeasurementobtainedwithoutanyitemontheweightisobtained�rst,
andthethreeitemsareweighedseparately.
Theestimatesfortheweightofthethreeitemsare:
b A=[�(1)+a]
;
b B=[�(1)+b]
;
b C=[�(1)+c]
Thiskindofdesignisprobablyfrequently(butunfortunately)usedinpractice.Itcanbe
brie ycharacterisedas\one-factor-at-a-time".
Onecandirectly�ndthevarianceintheestimates:
Varfb Ag=Varfb Bg=Varfb Cg=2��
2
Abasiccharacteristicofgoodexperimentaldesignsisthatalldataareusedinestimates
foralle�ects.Thisisseennottobethecasehere,andonecanaskifonecouldpossibly
�ndanexperimentaldesignthatismore\eÆcient"thantheoneshown.
Theexperimentsthatcanbecarriedoutare:
(1)
a
b
ab
c
ac
bc
abc
c hs.
DesignofExperiments,Course02411,IMM,DTU
34
Thecompletefactorialexperimentofcourseconsistsofdoingall8singleexperiments,
andtheestimatesforthee�ectsarefoundaspreviouslyshown.InthecaseoffactorA,
wegetthee�ectestimate: b A=
[�(1)+a�b+ab�c+ac�bc+abc]=4
whichisthustheestimatefortheweightofitemA.Thevarianceforthisestimateis�
2=2
(namely8�
2=42).
OnecaneasilyconvinceoneselfthattheweightofitemBanditemCarebalancedout
oftheestimateforA.Thesameappliestoapossiblezeropointerrorinthescaleofthe
weight(�).
Asanalternativetothesetwoobviousexperiments,wecanconsiderthefollowingexper-
iment:
(1)
ab
bc
ac
Theexperimentthusconsistsofweighingtheitemstogethertwobytwo.Forexamplethe
estimatefortheweightofAis:
b A=[A]=2=[�(1)+ab+ac�bc]=2
Varfb Ag=�
2
Notethatthezeropoint�aswellastheweightsofitemsBandCareeliminatedinthis
estimate.
Onealsonotesthatinrelationtotheprimitive\one-factor-at-a-time"experiment,inthis
designwecanuseall4observationstoestimatetheAe�ect,thatis,theweightofitem
A.ThesameobviouslyappliestotheestimatesfortheBandCe�ects.Inaddition,
thevarianceoftheestimatehereisonlyhalfthevarianceoftheestimatesinthe\one-
factor-at-a-time"experiment.Theexperimentisthereforeappreciablybetterthanthe
\one-factor-at-a-time"experiment.
Theexperimentiscalleda
1 2�23factorialexperimentora23
�
1
factorialex-
periment,asitconsistspreciselyofhalfthecomplete23factorialexperiment.
Finallyasmallnumericalexample:
(1)=6:78g
ab=28:84g
ac=20:66g
bc=18:12g
c hs.
DesignofExperiments,Course02411,IMM,DTU
35
b A=
(�6:78+28:84+20:66�18:12)=2
=
12.30g
b B=
(�6:78+28:84�20:66+18:12)=2
=
9.76g
b C=
(�6:78�28:84+20:66+18:12)=2
=
1.58g
Letussupposethatthemanufacturerhasstatedthattheweighthasanaccuracycorre-
spondingtothestandarddeviation�=0:02g.WiththisisfoundVarfb Ag=4�0:02
2=22=
0:02
2g2.ThestandarddeviationoftheestimatedAweightisthus0.02g.Thesamestan-
darddeviationisfoundfortheweightsofBandC.
A95%con�denceintervalfortheweightofAis12.30�2�0.02g=[12.26,12.34]g.
Endofexample2.1
Wewillnowdiscusswhatcangenerallybeestimatedinanexperimentasdescribedinthe
aboveexample.Ifonecanassumethatitisonlythemaine�ectsthatareimportantin
theexperiments,therearenoproblemsestimatingthese.Intheexample,onecantakeit
thattheweightofthetwoitemsisexactlythesumoftheweightsofthetwoitems,which
correspondstosayingthatthereisnointeraction.
Alternatively,wenowimaginethatthefollowinggeneralmodelappliesforthedescribed
experimentwiththethreefactors,A,BandC:
Yijk�=�+Ai+Bj+ABij+Ck+ACik+BCjk+ABCijk+Eijk�
wherei;j;k=(0;1)and�=(1;:::;r)withtheusualrestrictions.Completerandomisa-
tionisassumed.
ThequantityEijk�
designatestheexperimentalerrorinthe�'threpetitionofthesingle
experimentindexedby(i;j;k).
Forthesingleexperiment"(1)"inthedescribedexperiment,allindicesareonlevel\0",
anditsexpectedvalueis:
Ef(1)g=�+A0+B0+AB00+C0+AC00+BC00+ABC000
ByusingthefactthatA0=�A1
andcorrespondinglyfortheothertermsofthemodel,
we�nd
Ef(1)g=��A1�B1+AB11�C1+AC11+BC11�ABC111
Efabg=�+A1+B1+AB11�C1�AC11�BC11�ABC111
c hs.
DesignofExperiments,Course02411,IMM,DTU
36
Efacg=�+A1�B1�AB11+C1+AC11�BC11�ABC111
Efbcg=��A1+B1�AB11+C1�AC11+BC11�ABC111
Inthisway,fortheAcontrastwecannow�nd
Ef[A]g=Ef�(1)+ab+ac�bcg=4(A1�BC11)
ThismeansthatifthefactorsBandCinteract,soBC116=0,theestimateforthemain
e�ectoffactorAwillbea�ectedinthishalfexperiment.Thee�ectsAandBCare
thereforeconfoundedintheexperiment.Itholdstruegenerallyinthisexperimentthat
thee�ectsareconfoundedingroupsoftwo.
Thisisformallyexpressedthroughthealiasrelation"A=BC".Therelationexpresses
thatthee�ectsAandBCactsynchronouslyintheexperimentandthattheytherefore
areconfounded.TheAandBCe�ectscannotbedestinguishedfromeachotherinthe
experiment.
Thealiasrelationsforthewholeexperimentare
I
=
ABC
A
=
BC
B
=
AC
C
=
AB
wherethe�rstrelation,I=ABC,iscalledthede�ningrelationoftheexperimentand
ABCcalledthede�ningcontrast-inthesamewayasintheconstructionofaconfounded
blockexperiment(cf.page25).Thisexpressesthatthethree-factor-interactionABC
doesnotvaryintheexperiment,buthasthesamelevelinallthesingleexperiments
(namely�ABC111).
Theotheraliasrelationsaresimplyderivedbymultiplyingbothsidesofthede�ning
relationwiththee�ectsofinterest,andthenreducingtheexponentsmodulo2.For
example,thealiasrelationfortheAe�ectisfoundasA�I=A�ABC
i.e.A
=
A2BC
�!BC,where"I"isheretreatedasa\one"andthe2-exponentinA
2BC
is
reducedto0(modulo2reduction).
Ifwerecalltheconfoundedblockexperiment,whereacomplete23factorialexperiment
couldbelaidoutintwoblocksaccordingtothede�ningrelationI=ABC,weseethat
ourexperimentispreciselytheprincipalblockinthatexperiment.Ifitisacaseofa
1 2�2kfactorialexperiment,thefractionthatcontains"(1)"canbecalledtheprincipal
fraction.
c hs.
DesignofExperiments,Course02411,IMM,DTU
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Wecancheckwhetherfromtheotherhalfofthecompleteexperimentonecould�nd
estimatesthatarejustasgoodasinthehalfwetreatedinourexample.Theexperiment
is
a
b
c
abc
[A]=[a�b�c+abc]
Ef[A]g=Efa�b�c+abcg=4(A1+BC11)
Notethattheconfoundinghastheoppositesigncomparedwithearlier.Ifoneaddsthe
twocontrasts,thatis
[�(1)+ab+ac�bc]+[a�b�c+abc]
one�ndspreciselytheAcontrastforthecompleteexperiment,whilesubtractingthem,
thatis
�[�(1)+ab+ac�bc]+[a�b�c+abc]
�ndspreciselytheBCcontrast.
Thetwoalternativehalfexperimentsarecalledcomplementaryfractionalfactorials,as
togethertheyformthecompletefactorialexperiment.
Wewillnowshowhowonechoosesforexamplea
1 2�23factorialexperimentinpractice.
Wenotethata
1 2�23factorialexperimentconsistsof22measurements.Theexperiment
thatistobederivedcanthereforebeunderstoodasa22experimentwithanextrafactor
putin.Letusthereforeconsiderthecomplete22experimentwiththefactorsAandB.
Themathematicalmodelforthisexperimentis:
Yij�=�+Ai+Bj+ABij+Eij�
;
i=(0;1);j=(0;1);�=(1;2;:::;r)
Ifwesuspectthatall4parametersinthismodelcanbeimportant,furtherfactorscannot
beputintotheexperiment,butifweassumethattheinteractionABisnegligible,asin
theweighingexperiment,wecanintroducefactorC,sothatitisconfoundedwithjust
AB.
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DesignofExperiments,Course02411,IMM,DTU
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WethereforechoosetoconfoundCwithAB,thatisusingthealiasrelationC=AB.
Thisaliasrelationcan(only)bederivedfromthede�ningrelationI=ABC,which
canbeseenbymultiplyingthealiasrelationC=ABonbothsideswithC(orABfor
thatmatter)andreducingallexponentsmodulo2.
C=AB
=)
I
=
ABC(thede�ningrelation)
A
=
BC
B
=
AC
AB
=
C
(thegeneratorequation)
WeshallcallC=ABthegeneratorequationsinceitisthealiasrelationfromwhich
thedesignisgenerated.
Theprincipalfractionismadeupofallsingleexperimentsthathaveanevennumber
oflettersincommonwithABC,i.e.,theexperiments(1),ab,ac,bc.Alternatively,the
complementaryfractioncouldbechosen,whichcontainsallsingleexperimentsthathave
anunevennumberoflettersincommonwithABC,i.e.,a,b,candabc.
Withthislastmethod,wherethestartingpointisthecompletefactorialexperimentforthe
two(�rst)factorsAandB,itissaidthattheseformanunderlyingcompletefactorial
forthefractionalfactorialdesign.Wewillreturntothisimportantconceptlater.
Letusnowsupposethatwechoosetheexperimentcorrespondingto\uneven":
a
b
c
abc
To�ndthesignfortheconfoundings,itisenoughtoconsideroneofthealiasrelations,
forexampleC=ABandcomparethiswithoneoftheexperimentsthatistobedone,
forexampletheexperiment\a".
Fortheexperiment"a",thee�ectChasthevalueC0
(sincefactorCison0level),and
thee�ectABhasthevalueAB10.TheconfoundingisthereforeC0
=AB10.Sincewe
calculateonthebasisofthe\high"levelsC1andAB11,theseareputin.
SinceC0=�C1
andAB10=�AB11,we�nallygetthatthealiasrelationisC1=AB11.
Therestofthealiasrelationsgetthesamesignwhentheyareexpressedinthehighlevels.
ForexampleonegetsA1=BC11.
Onewritesforexample
+C=+AB
=)
+I
=
+ABC(thede�ningrelation)
+A
=
+BC
+B
=
+AC
+AB
=
+C
(thegeneratorequation)
c hs.
DesignofExperiments,Course02411,IMM,DTU
39
Whethertheconstructedexperimentisasuitableexperimentdependsonwhetherthe
aliasrelationstogethergivesatisfactorypossibilitiesforestimatingthee�ects,which,a
priori,areconsideredinteresting.
Example2.2:
A
1/4�2
5
factorialexperiment
.
We�nishthissectionbyshowinghow,withthehelpoftheintroducedideas,onecan
constructa1=4�25factorialexperiment,i.e.,anexperimentthatconsistsonlyof23=8
measurements,butincludes5factors.Thesearecalled(always)A,B,C,DandE(for
1st,2nd,3rd,4thand5thfactor).
Thecompletefactorialexperimentwith3factorscontainsinadditiontothelevel�the
e�ectsA,B,AB,C,AC,BC,andABC.
SupposenowthatitcanbeassumedthatfactorsBandCdonotinteract,i.e.that
BC=0.AreasonableinferencefromthiscouldbethatalsoABC=0.Therebyitwould
benaturaltochoosetwogeneratorequations,namelyD
=
BC
andE
=
ABC.
ThesegiveI 1=BCDandI 2=ABCE,respectively.Theprincipalfractionconsistsofthe
singleexperimentsthathaveanevennumberoflettersincommonwithboththede�ning
contrastsBCDandABCE.Thesesingleexperimentsare:
(1)
ae
bde
abd
cde
acd
bc
abce
Adirectandeasymethodtoconstructthisexperimentistowriteoutatableasfollows:
A
B
C
D=�BC
E=ABC
Code
�1
�1
�1
�1
�1
(1)
+1
�1
�1
�1
+1
ae
�1
+1
�1
+1
+1
bde
+1
+1
�1
+1
�1
abd
�1
�1
+1
+1
+1
cde
+1
�1
+1
+1
�1
acd
�1
+1
+1
�1
�1
bc
+1
+1
+1
�1
+1
abce
NotethatforthefactorsA,BandCtheorderingofthelevelscorrespondtothestandard
order:(1),a,b,ab,c,ac,bc,abc,asusedinYatesalgorithm,forexample.
TheminussigninD=�BCensuresthattheexperiment(1)isobtainedasthe�rstone,
iftheprincipalfractioniswanted.
Thistabularmethodofwritingouttheexperimentcanbeusedquitegenerallyaswillbe
demonstratedinthefollowing.Afurtheradvantageisthatthesignsoftheconfoundings
areobtaineddirectly.
Analternativeexperimentisfoundbyconstructingoneoftheother\fractions".Iffor
c hs.
DesignofExperiments,Course02411,IMM,DTU
40
exampleonewantsanexperimentthatcontainsthesingleexperiment"a",thecorre-
spondingfractioncanbefoundbymultiplyingtheprincipalfractionthroughwith\a"
andreducingtheexponentsmodulo2.Inthiswayonegets:
a
e
abde
bd
acde
cd
abc
bce
ThesameexperimentwouldhavebeenobtainedbychangingthesigninE=ABCso
thatE=�ABCisusedintheabovetabularmethod.
Now,to�ndthealiasrelationsintheexperiment,wewillagainusethetwode�ning
relations.
Theinteractionofthetwode�ningcontrastsisfoundbymultiplyingthemtogetherand
againreducingallexponentsmodulo2:
D=BC,E=ABC=)I 1=BCD,I 2=ABCE
andI 3=I 1�I 2=AB
2
C2DE!ADE
sothatthede�ningrelationandthealiasrelations(withoutsigns)oftheexperimentare:
I
=
BCD
=
ABCE
=
ADE
A
=
ABCD
=
BCE
=
DE
B
=
CD
=
ACE
=
ABDE
AB
=
ACD
=
CE
=
BDE
C
=
BD
=
ABE
=
ACDE
AC
=
ABD
=
BE
=
CDE
BC
=
D
=
AE
=
ABCDE
ABC
=
AD
=
E
=
BCDE
Roughlyspeaking,theexperimentisonlyagoodexperimentifonecanassumethatthe
interactionsarenegligible(inrelationtothemaine�ects).
Thesignsfortheconfoundingscanagainbefoundbyconsideringanaliasrelation,e.g.
A=ABCD=BCE=DE,togetherwithoneofthesingleexperimentsthatarepartof
thechosenexperimentaldesign.
Forexample"a"isintheexperimentanditcorrespondstoasingleexperimentwith
indices(1,0,0,0,0)forthefactorsA,B,C,DandE,respectively.Thus
A1=ABCD1000=BCE000=DE00()+A1=�ABCD1111=�BCE111=+DE11
Thissignpatternisrepeatedinallthealiasrelations:
c hs.
DesignofExperiments,Course02411,IMM,DTU
41
+I
=
�BCD
=
�ABCE
=
+ADE
+A
=
�ABCD
=
�BCE
=
+DE
+B
=
�CD
=
�ACE
=
+ABDE
+AB
=
�ACD
=
�CE
=
+BDE
+C
=
�BD
=
�ABE
=
+ACDE
+AC
=
�ABD
=
�BE
=
+CDE
+BC
=
�D
=
�AE
=
+ABCDE
+ABC
=
�AD
=
�E
=
+BCDE
Weneednowto�ndestimatesandsumsofsquares.Thiscanbedonebyagainusing
thefactthattheexperimentisformedonthebasisofthecompleteunderlyingfactorial
structurecomposedoffactorsA,BandC.Inthisstructurewenowestimateallthee�ects
correspondingtothethreefactors.
Inordertosubsequently�ndtheDe�ectweonlyneedtolookuptheBCrow,wherethe
De�ectappearswiththeoppositesign.Dataaregroupedinstandardorderaccordingto
thefactorsA,BandC.Thisisdonebyignoring\d"and\e".Thenthecontrastscanbe
calculated,withtheuseofYates'algorithm,forexample.Onegets:
2 6 6 6 6 6 6 6 6 6 6 6 6 6 4I
=
�BCD
=
�ABCE
=
ADE
A
=
�ABCD
=
�BCE
=
DE
B
=
�CD
=
�ACE
=
ABDE
AB
=
�ACD
=
�CE
=
BDE
C
=
�BD
=
�ABE
=
ACDE
AC
=
�ABD
=
�BE
=
CDE
BC
=
�D
=
�AE
=
ABCDE
ABC
=
�AD
=
�E
=
BCDE
3 7 7 7 7 7 7 7 7 7 7 7 7 7 5
=2 6 6 6 6 6 6 6 6 6 6 6 6 6 41
1
1
1
1
1
1
1
�1
1
�1
1
�1
1
�1
1
�1
�1
1
1
�1
�1
1
1
1
�1
�1
1
1
�1
�1
1
�1
�1
�1
�1
1
1
1
1
1
�1
1
�1
�1
1
�1
1
1
1
�1
�1
�1
�1
1
1
�1
1
1
�1
1
�1
�1
13 7 7 7 7 7 7 7 7 7 7 7 7 7 52 6 6 6 6 6 6 6 6 6 6 6 6 6 4e a bd
abde
cdacde
bce
abc
3 7 7 7 7 7 7 7 7 7 7 7 7 7 5
Notethattherowe;a;bd;abde;cd;acde;bce;abc,becomestherow(1);a;b;ab,c;ac;bc;abc,
ifoneleavesoutdande,i.e.thestandardorderforthecomplete23factorialexperiment
forA,BandC.
Theexperimentwhichwe�ndbyignoringthefactorsDandE,i.e.thecomplete23
factorialexperimentincludingA,BandC,isagainanunderlyingcompletefactorial
experimentandA,BandCconstituteanunderlyingcompletefactorialstructure.
c hs.
DesignofExperiments,Course02411,IMM,DTU
42
WecaneasilycheckthatforexampletheABCe�ectisconfoundedwith-E.Onewayto
dothisistoconsidertheABCcontrast:
[ABC]=�e+a+bd�abde+cd�acde�bce+abc
wherewenotethatalldatawithEatthehighlevel,i.e.,e,abde,acdeandbce,appear
with-1ascoeÆcient,whiletheremainder,i.e.a,bd,cdandabcappearwith+1.The
contrastthereforecontainsacontributionof�4(E1)+4(�E1)=�8E1fromthefactorE.
Thesuggestedexperimentcouldbedoneintwoblocksof4byforexampleconfounding
theABinteractionwithblocks.Thatwouldgivethegrouping:
abc
e
cd
abde
a
bce
bd
acde
block0
block1
Theconfoundingsinthisexperimentwouldbe:
I
=
�BCD
=
�ABCE
=
ADE
A
=
�ABCD
=
�BCE
=
DE
B
=
�CD
=
�ACE
=
ABDE
AB
=
�ACD
=
�CE
=
BDE
=
Blocks
C
=
�BD
=
�ABE
=
ACDE
AC
=
�ABD
=
�BE
=
CDE
BC
=
�D
=
�AE
=
ABCDE
ABC
=
�AD
=
�E
=
BCDE
wherethecontrastsarecalculatedaspreviously,butwherethecontrastthatappearsin
theABrownowcontainspossiblefactore�ectsaswellastheblocke�ects.
Endofexample2.2
2.5
Factorson2and4levels
Inmanycaseswhereseveralfactorsareanalysed,itcanbedesirableandperhapseven
necessaryforsinglefactorsthattheycanappearon3orperhaps4levelstogetherwith
the2levelsoftheotherfactors.Incasethereisaneedforamixtureof2and3levels,itis
diÆculttoconstructgoodexperimentaldesigns,butinthetextbookbyOscarKempthorne
(1952):TheDesignandAnalysisofExperiments,Wiley,NewYork,therearehowever
somesuggestionsforthis.
If4levelsareused,onecanusetheprocedurebelow,whichisdemonstratedwiththehelp
oftwoexamples,sothepresentationisnottoocomplicated.
c hs.
DesignofExperiments,Course02411,IMM,DTU
43
Example2.3:
A
2�4experimentin2blocks
Supposethatinafactorialexperimenttwofactorsaretobeanalysed,namelyafactor
Athatappearson2levelsandafactorGthatappearson4levels.Themathematical
modelfortheexperimentis
Yil�=�+Ai+G`+AGi`+Ei`�
wherei=(0;1),`=(0;1;2;3)and�=(1;2;:::;r).
Usualparameterrestrictionsareused
1 X i=0
Ai=
3 X `=0
G`=
1 X i=0
AGi`=
3 X `=0
AGi`=0
Toreformulatethemodeltoa2kfactorialstructure,twonewfactorsareintroduced,B
andC,asreplacementsforG.
G=0
G=1
G=2
G=3
B=0,C=0
B=1,C=0
B=0,C=1
B=1,C=1
Yij�=�+Ai+Bj+ABij+Ck+ACik+BCjk+ABCijk+Eijk�
wheretheindexj=
remainderof(`=2)andk=
integerpartof(`=2).Inversely,
`=j+2k.
ThecorrespondencebetweenfactorGandthetwoarti�cialfactorsBandCisthat
G`=Bj+Ck+BCjk
;
`=j+2k
Supposenowthatonewantstodoacompletefactorialexperimentwiththetwofactors
AandG,i.e.a2�4experiment,oratotalof8singleexperiments.
Supposefurtherthatonewantstodotheexperimentin2blockswith4singleexperiments.
Inthereformulatedmodel,wherefactorGisreplacedbyBandC,weseethatthemain
e�ectoffactorGisgivenasB+C+BC.Thee�ectsB,CandBCmustthereforebe
estimatedandcannotbeusedasde�ningcontrastwhendividingintoblocks.
FortheinteractionbetweenfactorAandfactorG,itholdstruethat
AGi`=ABij+ACik+ABCijk
;
`=j+2k
c hs.
DesignofExperiments,Course02411,IMM,DTU
44
andoneofthesethreee�ects(itdoesnotmatterwhich)canbereasonablyusedasde�ning
contrast.WechooseforexampletheABe�ect.
Theexperimenttherebyis
Block1
Block2
(1)
ab
c
abc
a
b
ac
bc
orconvertedtofactorsAandG:
Block1
Block2
(1)
ab
c
abc
a
b
ac
bc
A=0
A=1
A=0
A=1
A=1
A=0
A=1
A=0
G=0
G=1
G=2
G=3
G=0
G=1
G=2
G=3
TheexperimentcanbeanalysedwithYates'algorithm,andonegetsatableofanalysis
ofvariancewhich(inoutline)isbuiltupasthefollowing:
Sourceof
Sumof
Degreesof
S2
F-value
variation
Squares
dom
A
SSQA
1
S2 A
G
SSQB
+SSQC
+SSQBC
3
S2 G
AG-unconfounded
SSQAC
+SSQABC
2
S2 AG
Blocks+AG
SSQAB
1
S2 AG+blocks
Possibleresidual
frompreviousexp.
Total
OneseesthatsomeofthevariationarisingfromtheAGinteractioncanbetakenout
andtested,whiletheremainingpartisconfoundedwithblocks.Ontheotherhand,one
cannotestimatespeci�cAGinteractione�ects,sincethepartdescribedbytheABpart
cannotbeestimated(AG=AB+AC+ABC).
Endofexample2.3
Example2.4:
A
fractional2�2�4factorialdesign
IfforexampleonewantstoevaluatethreefactorsA,BandGwith2,2,and4levels
respectively,twonewarti�cialfactorsareintroduced,CandD,sothatG=C+D+CD,
andinthiscaseonemustkeepthethreee�ectsC,DandCDclearofconfoundings.It
couldbewishedtodosucha2�2�4designwithatotalof16possiblesingleexperiments
asa
1 2�24experiment,usingonly8singleexperiments.
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DesignofExperiments,Course02411,IMM,DTU
45
IfitisassumedthatthethreefactorsA,BandGdonotinteract,theexperimentcanbe
constructedsothatthee�ectsA,B,C,DandCDcanbefound(asG=C+D+CD).
Onecanusethefollowingde�ningcontrastandaliasrelations:
I
=
ABCD
A
=
BCD
B
=
ACD
AB
=
CD
C
=
ABD
AC
=
BD
BC
=
AD
ABC
=
D
wheree�ectsthatareinterestingareunderlined,whilee�ectsconsideredtobewithout
interestarewrittennormally.
Therelationbetweenthefactorsisthat
G=C+D+CD
,
AG=AC+AD+ACD
,
BG=BC+BD+BCD
and
ABG=ABC+ABD+ABCD
Theexperimentwantedcouldbethefollowing(trytoconstructityourself!):
a
b
c
abc
d
abd
acd
bcd
orconvertedtothelevelsofthefactors,astheindexforthefactorGisk+2`,wherek
istheindexforCwhile`istheindexforD:
a
b
c
abc
d
abd
acd
bcd
A=1
A=0
A=0
A=1
A=0
A=1
A=1
A=0
B=0
B=1
B=0
B=1
B=0
B=1
B=0
B=1
G=0
G=0
G=1
G=1
G=2
G=2
G=3
G=3
Ofcourse,onecouldalsohaveusedthecomplementaryexperimentasthestartingpoint:
(1)
ab
ac
bc
ad
bd
cd
abcd
TrytowritethecorrespondingexperimentoutinfactorsA,BandG.
Iftheconstructedexperimentshouldbelaidoutintwoblocksof4singleexperiments,
onecoulduseeitherACorBCasde�ningcontrast.IfACisused,onegetsthedesign:
c hs.
DesignofExperiments,Course02411,IMM,DTU
46
Block1
Block2
b
abc
d
acd
a
c
abd
bcd
A=0
A=1
A=0
A=1
A=1
A=0
A=1
A=0
B=1
B=1
B=0
B=0
B=0
B=0
B=1
B=1
G=0
G=1
G=2
G=3
G=0
G=1
G=2
G=3
Note,forexample,thatallthreefactorsA,BandGarebalancedwithinbothblocks.
Endofexample2.4
c hs.
DesignofExperiments,Course02411,IMM,DTU
47
3
Generalmethodsforpk-factorialdesigns
Inthischapterwewillintroducegeneralmethodsforfactorialexperimentsinwhichthere
arekfactors,eachonplevels.Thepurposeofthisistogeneralizetheconceptsand
methodsthatwerediscussedinthepreviouschapter,whereweconsideredkfactorsof
whicheachwasononlyp=2levels.
Inparticular,wewilllookatexperimentswithmanyfactorsthathavetobeevaluatedon
2or3levels,whicharemostrelevantinpractice.
Ingeneral,noproofsaregiven,butthesubjectispresentedthroughexamplesanddirect
demonstrationinspeci�ccases.
ThemethodwewilldealwithisoftencalledKempthorne'smethod,andtheinterested
readerisreferredtothetextbookbyOscarKempthorne(1952):TheDesignandAnalysis
ofExperiments,Wiley,NewYork.Thisbookhasasomewhatmoremathematicalreview
oftheexperimentalstructuresandmodelsthatwewilldealwithhere.Infairness,it
shouldbesaidthatitwasactuallyR.A.Fischerandotherswho,around1935,formulated
importantpartsofthebasisforKempthorne'spresentation.
3.1
Completepk
factorialexperiments
Wenowconsiderexperimentswithkfactorseachonplevels,wherepiseverywhereas-
sumedtobeaprimenumber.Inaddition,completerandomisationisgenerallyassumed.
Incaseswhereexperimentsarediscussedinwhichthereisusedblocking,completeran-
domisationisassumedwithinblocks.
ThefactorsarealwayscalledA,B,C,etc.FactorAisthe�rstfactor,Bthesecondfactor
etc.Inaddition(tothegreatestpossibleextent)weusetheindicesi,j,k,etc.forthe
factorsA,B,C,etc.,respectively.
Theexperimentisgenerallycalledapkfactorialexperiment,andthenumberofpossible
di�erentfactorcombinationsispreciselyp�p�:::�p=pk.
Foranexperimentwith3factors,A,B,andC,thestandardmathematicalmodelis:
Yijk�=�+Ai+Bj+ABi;j+Ck+ACi;k+BCj;k+ABCi;j;k+Eijk�
wherei;j;k=(0;1;::;p�1)and�=(1;2;::;r).
Theindex�=(1;2;::;r)givesthenumberofrepetitionsofeachsingleexperimentinthe
experiment.Theotherindicesassumethevalues(0,1,2,..,p�1).Itshouldbenoted
thattheindexalwaysrunsfrom0uptoandincludingp�1.
c hs.
DesignofExperiments,Course02411,IMM,DTU
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Forsuchexperimentsweintroduceastandardnotationforthesingleexperimentsinthe
samewayaswiththe2kexperiment.Inthecasewherep=
3andk=
3wehave
thefollowingtable,whichshowsallthesingleexperimentsinthecomplete33factorial
experiment:
A
A
A
0
1
2
0
1
2
0
1
2
B=0
(1)
a
a2
c
ac
a2c
c2
ac2
a2c2
B=1
b
ab
a2b
bc
abc
a2bc
bc2
abc2
a2bc2
B=2
b2
ab2
a2b2
b2c
ab2c
a2b2c
b2c2
ab2c2
a2b2c2
C=0
C=1
C=2
Aspreviously,weuseoneoftheseexpressionsasthetermforacertain"treatment"or
factorcombinationinasingleexperiment,aswellasforthetotalresponsefromthesingle
experimentsdonewiththisfactorcombination.Thus,forexample
ab2c=
r X �=1
Y121�=T121�
orjustT121
Forthe2kfactorialexperiment,wearrangedthesetermsinwhatwascalledastandard
order.Wecanalsodothisforthepkexperimentingeneral.Thesestandardordersare:
2k:(1);a;b;ab;c;ac;bc;abc;d;ad;bd;abd;cd;:::
3k:(1);a;a
2;b;ab;a
2b;b2;ab2;a
2b2;c;ac;a
2c;::;a
2b2c2;d;:::
5k:(1);a;a
2;a
3;a
4;b;ab;a
2b;a
3b;a
4b;b2;ab2;a
2b2;a
3b2;a
4b2;:::;a
4b4;
c;ac;a
2c;:::;a
4b4c4;d;ad;:::
7k:(1);a;a
2;:::;a
6;b;ab;a
2b;:::;a
6b6;c;ac;:::;a
6b6c6;d;ad;:::
Forexampleinthe3kfactorialexperiment,anewfactorisaddedbymultiplyingallthe
termsuntilnowwiththefactorinthe�rstpowerandinthesecondpowerandadding
boththesenewrowstotheoriginalorder.
Theseterms,ofcourse,canperfectlywellbeusedasnamesforfactorcombinationsin
completelygeneralfactorialexperiments,buttheresultswewillshowareonlygenerally
applicabletoexperimentsthatcanbeformulatedaspk
factorialexperimentswherepis
aprimenumber.
Beforewecontinue,itwouldbeusefultolookmorecloselyata32factorialexperiment
andshowhowthetotalvariationinthisexperimentcanbedescribedandfoundwiththe
helpofaGraeco-Latinsquare.Inadditionwewillintroducesomemnemonictermsfor
newarti�ciale�ects,whichwilllaterprovetobepracticalintheconstructionofmore
sophisticatedexperimentaldesigns.
c hs.
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Example3.1:
MakingaGraeco-Latinsquareina32factorialexperiment
Theexperimenthas3�3=9di�erentsingleexperiments:
A=0
A=1
A=2
B=0
(1)
a
a2
B=1
b
ab
a2b
B=2
b2
ab2
a2b2
Themathematicalmodelfortheexperimentis
Yij�=�+Ai+Bj+ABi;j+Eij�
;i=(0;1;2);j=(0;1;2);�=(1;:::;r)
2 X i=0
Ai=
2 X j=0
Bj=
2 X i=0
ABi;j=
2 X j=0
ABi;j=0
Inthisexperimentwecanintroducetwoarti�cialfactors,whichwecancallXandZ.We
letthesefactorshaveindicessandt,respectively,whichwedeterminewith
s=(i+j)3
andt=(i+2j) 3
wherethedesignation(:) 3nowstandsfor"modulo3",i.e."remainderof(.)afterdivision
by3".
Wewillnowseehowtheindicessandtforthede�nednewe�ectsXandZvarythroughout
theexperimentwiththeindicesiandjofthetwooriginalfactorsAandB.
Thisisshowninthetablebelow,asi+jandi+2jarestillcalculated"modulo3".
i
j
s=(i+j) 3
t=(i+2j) 3
0
1
2
0
0
0
0
1
2
0
1
2
0
1
2
1
1
1
1
2
0
2
0
1
0
1
2
2
2
2
2
0
1
1
2
0
Ai
Bj
Xi+j
Zi+2j
Wenotethatifwe�xoneofthelevelsforoneofthe4indices,eachoftheother3indices
appearspreciselywiththevalues0,1and2withinthislevel.Asanexampleofthis,we
considerthesingleexperimentswhereZ'sindext=(i+2j) 3=1:
i
j
s=(i+j) 3
t=(i+2j) 3
1
0
1
1
2
1
0
1
0
2
2
1
Ai
Bj
Xi+j
Zi+2j
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DesignofExperiments,Course02411,IMM,DTU
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Inthedesignshown,the4factorsA,B,XandZareobviouslyinbalanceinrelationto
eachother.ThedesignisaGraeco-LatinsquarewiththeintroducedfactorsXandZ
insidethesquareandwiththefactorsAandBatthesides.Sincethevariationbetween
the9singleexperimentsor"treatments"intheexperimenthasatotalof9-1degrees
offreedom,andthe4factorsareinbalance,asdescribed,itcanbeshownthatthese4
factorscandescribethewholevariationbetweenthesingleexperiments.
AandBareidenticalwiththeoriginalmaine�ects,anditcanbeshownthatXand
ZtogetherpreciselymakeuptheinteractiontermABi;jinthe"natural"mathematical
modeloftheexperiment.
Wewillnotprovethisresult,butonlyillustrateitwithanexample,whereweimagine
thata32experimentwithoneobservationpercellhasresultedinthefollowingdata:
A=0
A=1
A=2
sum-B
sum-X
sum-Z
B=0
(1)=10
a=15
a2=18
43
X=0
35
Z=0
33
B=1
b=8
ab=12
a2b=16
36
X=1
34
Z=1
36
B=2
b2=5
ab2=9
a2b2=11
25
X=2
35
Z=2
35
sum-A
23
36
45
104
104
104
Theusualtwo-sidedanalysisofvarianceforthesedatawiththefactorsAandBgives:
ANOVA
Sourceof
Sumof
Degreesof
F-value
variation
squares
freedom
A
81.556
(3-1)=2
B
54.889
(3-1)=2
AB
1.778
(3-1)(3-1)=4
Residual
0.000
0
Total
138.223
(9-1)=8
Totherightofthedatatablearesumsforthetwoarti�cialfactorsXandZ.Forexample
the(X=0)sumisfoundas10+16+9=35,i.e.thesumofthedatawheretheindex
(i+j)3=0asXhastheindex=(i+j)3.
Fromthiswe�ndthefollowingsumsofsquaresanddegreesoffreedom,wherethe4
factorsA,B,XandZconstituteaGraeco-Latinsquare:
SSQ(treatments)
=
102+15
2+18
2+82+::+11
2�1042=9
=
138.222
,
f=9-1
SSQ(A)
=
(232+36
2+45
2)=3�1042=9
=
81.556
,
f=2
SSQ(B)
=
(252+36
2+43
2)=3�1042=9
=
54.889
,
f=2
SSQ(X)
=
(352+34
2+35
2)=3�1042=9
=
0.222
,
f=2
SSQ(Z)
=
(332+36
2+35
2)=3�1042=9
=
1.556
,
f=2
SSQ(A)+SSQ(B)+SSQ(X)+SSQ(Z)
=
138.223
,
f=8
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Itisseen(exceptfortherounding)thatfortheinteractionABandcorrespondingdegrees
offreedomwehave:
SSQ(AB-interaction)=SSQ(X)+SSQ(Z),andf(AB-interaction)=f(X)+f(Z)
Further,itcanbegenerallyshownthatfortheinteractiontermitappliesthat
ABi;j=Xi+j+Zi+2j
;
i=(0;1;2);j=(0;1;2)
Thiscanbeillustratedby�ndingtheestimatesfortheinteractiontermsaswellasfor
thearti�ciale�ectsXandZ.Asanexamplewecan�ndtheinteractionestimatefor
(A=1,B=2),i.e.AB1;2.
^�=104=9=11:556
;
b A 1=36=3�104=9=0:444
;
b B 2=25=3�104=9=�3:222
=)d AB1;2=Y1;2�^��b A 1�b B 2=9:000�11:556�0:444�(�3:222)=0:222
c X 1+2=c X 3!c X 0=35=3�104=9=0:111
b Z 1+2�2=
b Z 5!b Z 2=35=3�104=9=0:111
sothatc X 1+2+b Z 1+2�2=d AB1;2,aspostulated(rememberthatindicesarestillcalculated
"modulo3").TrytoworkoutwhetheritiscorrectthatAB2;2
=X2+2+Z2+2�2,when
theseareestimated.
Inordertousetheresultsoftheexamplegenerally,itispracticaltointroducesomemore
mnemonicnamesforthetwointroducede�ectsXandZ.Wethusset
Xi+j=ABi+j
andZi+2j=AB
2 i+2j
Correspondingly,wewritetheoriginalmodelontheform
Yij�=�+Ai+Bj+ABi+j+AB
2 i+2j+Eij�
where
ABi+j+AB
2 i+2j=ABi;j
;
i=(0;1;2);j=(0;1;2)
Itappliesthatwiththisnewformalnotation:
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DesignofExperiments,Course02411,IMM,DTU
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2 X i=0
Ai=
2 X j=0
Bj=
2 X r=0
ABr=
2 X s=0
AB
2 s=0
;
r=(i+j)3
;s=(i+2j) 3
whereallindicesarestillcalculated\modulo3".
Thetwoe�ectsABi+jandAB
2 i+2jinthiswaydesignatethearti�ciallyintroducede�ects,
whichenableadecompositionoftheusualinteractiontermABi;j
fromthetraditional
modelformulation.Theexponent"2"onAB
2
shouldonlybeconsideredasamnemonic
helpandnotasanexpressionofraisingtoapowerof2.
Finallynotehowtheindicesandnamesfortheintroducedarti�ciale�ectsmatch.
Endofexample3.1
Sucharti�ciale�ectscanbede�nedforgeneralpk
factorialexperiments.Tobeableto
keepthee�ectsinorder,weintroducea"standardorder"fore�ects.Foranexperiment
whereallfactorshaveplevels,thearti�ciale�ectswilllikewiseallhaveplevels:
2k:I;A;B;AB;C;AC;BC;ABC;D;AD;BD;ABD;CD;ACD;BCD;ABCD;E;AE;::
3k:I;A;B;AB;AB2;C;AC;AC2;BC;BC2;ABC;ABC2;AB2C;AB2C2;D;AD;AD2,
BD;BD2;:::;AB2C2D2;E;:::
5k:I;A;B;AB;AB2;AB3;AB4;C;AC;AC2;:::;BC;:::;AB4C;:::;AB4C4;D;:::
Thesee�ectshaveindicesaccordingtothesamerulesthatwereusedintheprevious
example.Thatis,forexample,thatinthe5kexperimentwithfactorsA,BandC,each
with5levels,thee�ectAB
3Chasindex=(i+3j+k) 5,i.e.(i+3j+k)modulo5.
FactorAisthe�rstfactor,BthesecondfactorandCthethirdfactor.
Notethatthisstandardordercanbederivedfromthestandardorderforsingleexperi-
mentsbychangingtoupper-caselettersandleavingoutthetermswheretheexponent
onthe�rstfactorinthee�ectisgreaterthan1.Forexample,AB
3Cshouldbeincluded,
whileforexampleB
2CDshouldbeleftout.
Example3.2:
Latincubesin33experiments
Lettherebeacompletelyrandomised33experimentwithrrepetitionsofeachsingle
experiment.Wehaveintheusualmodelformulation:
Yijk�=�+Ai+Bj+ABi;j+Ck+ACi;k+BCj;k+ABCi;j;k+Eijk�
wherei=(0;1;2);j=(0;1;2);k=(0;1;2)and�=(1;2;:::;r)
Thecellsoftheexperimentorsingleexperimentsmakeupacube,thelengthofitsedge
being3.
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Itthuslookslikethis:
A
A
A
0
1
2
0
1
2
0
1
2
B=0
(1)
a
a2
c
ac
a2c
c2
ac2
a2c2
B=1
b
ab
a2b
bc
abc
a2bc
bc2
abc2
a2bc2
B=2
b2
ab2
a2b2
b2c
ab2c
a2b2c
b2c2
ab2c2
a2b2c2
C=0
C=1
C=2
Withthestatedarithmeticrulesforindicesusedonthestandardorderfortheintroduced
arti�ciale�ectsfora33factorialexperiment,wecannow�ndtheindexvaluesforallthe
e�ects.Theindexvaluesarefoundasshowninthefollowingtables:
i=0
i=1
i=2
i=0
i=1
i=2
i=0
i=1
i=2
j=0
0
1
2
0
1
2
0
1
2
indexfor
j=1
0
1
2
0
1
2
0
1
2
Ai
j=2
0
1
2
0
1
2
0
1
2
k=0
k=1
k=2
i=0
i=1
i=2
i=0
i=1
i=2
i=0
i=1
i=2
j=0
0
0
0
0
0
0
0
0
0
indexfor
j=1
1
1
1
1
1
1
1
1
1
Bj
j=2
2
2
2
2
2
2
2
2
2
k=0
k=1
k=2
i=0
i=1
i=2
i=0
i=1
i=2
i=0
i=1
i=2
j=0
0
1
2
0
1
2
0
1
2
indexfor
j=1
1
2
0
1
2
0
1
2
0
ABi+j
j=2
2
0
1
2
0
1
2
0
1
k=0
k=1
k=2
i=0
i=1
i=2
i=0
i=1
i=2
i=0
i=1
i=2
j=0
0
1
2
0
1
2
0
1
2
indexfor
j=1
2
0
1
2
0
1
2
0
1
AB2 i+2j
j=2
1
2
0
1
2
0
1
2
0
k=0
k=1
k=2
i=0
i=1
i=2
i=0
i=1
i=2
i=0
i=1
i=2
j=0
0
0
0
1
1
1
2
2
2
indexfor
j=1
0
0
0
1
1
1
2
2
2
Ck
j=2
0
0
0
1
1
1
2
2
2
k=0
k=1
k=2
i=0
i=1
i=2
i=0
i=1
i=2
i=0
i=1
i=2
j=0
0
1
2
1
2
0
2
0
1
indexfor
j=1
0
1
2
1
2
0
2
0
1
ACi+k
j=2
0
1
2
1
2
0
2
0
1
k=0
k=1
k=2
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DesignofExperiments,Course02411,IMM,DTU
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i=0
i=1
i=2
i=0
i=1
i=2
i=0
i=1
i=2
j=0
0
1
2
2
0
1
1
2
0
indexfor
j=1
0
1
2
2
0
1
1
2
0
AC2 i+2k
j=2
0
1
2
2
0
1
1
2
0
k=0
k=1
k=2
i=0
i=1
i=2
i=0
i=1
i=2
i=0
i=1
i=2
j=0
0
0
0
1
1
1
2
2
2
indexfor
j=1
1
1
1
2
2
2
0
0
0
BCj+k
j=2
2
2
2
0
0
0
1
1
1
k=0
k=1
k=2
i=0
i=1
i=2
i=0
i=1
i=2
i=0
i=1
i=2
j=0
0
0
0
2
2
2
1
1
1
indexfor
j=1
1
1
1
0
0
0
2
2
2
BC2 j+2k
j=2
2
2
2
1
1
1
0
0
0
k=0
k=1
k=2
i=0
i=1
i=2
i=0
i=1
i=2
i=0
i=1
i=2
j=0
0
1
2
1
2
0
2
0
1
indexfor
j=1
1
2
0
2
0
1
0
1
2
ABCi+j+k
j=2
2
0
1
0
1
2
1
2
0
k=0
k=1
k=2
i=0
i=1
i=2
i=0
i=1
i=2
i=0
i=1
i=2
j=0
0
1
2
2
0
1
1
2
0
indexfor
j=1
1
2
0
0
1
2
2
0
1
ABC2 i+j+2k
j=2
2
0
1
1
2
0
0
1
2
k=0
k=1
k=2
i=0
i=1
i=2
i=0
i=1
i=2
i=0
i=1
i=2
j=0
0
1
2
1
2
0
2
0
1
indexfor
j=1
2
0
1
0
1
2
1
2
0
AB2Ci+2j+k
j=2
1
2
0
2
0
1
0
1
2
k=0
k=1
k=2
i=0
i=1
i=2
i=0
i=1
i=2
i=0
i=1
i=2
j=0
0
1
2
2
0
1
1
2
0
indexfor
j=1
2
0
1
1
2
0
0
1
2
AB2C2 i+2j+2k
j=2
1
2
0
0
1
2
2
0
1
k=0
k=1
k=2
Forexample,wecanlookatthetermBCj+k
andnotewhereithastheindexvalue1.
Thisisstatedbelow:
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DesignofExperiments,Course02411,IMM,DTU
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i=0
i=1
i=2
i=0
i=1
i=2
i=0
i=1
i=2
j=0
1
1
1
indexfor
j=1
1
1
1
BCj+k
j=2
1
1
1
k=0
k=1
k=2
Foranyotherarbitrarilychosenterm,therewillbeanequalnumberof0's,1'sand2'sin
these9places.Asanexample,wetakethetermABC
2,wherethecorrespondingplaces
areshownbelow:
i=0
i=1
i=2
i=0
i=1
i=2
i=0
i=1
i=2
j=0
2
0
1
indexfor
j=1
1
2
0
ABC2 i+j+2k
j=2
0
1
2
k=0
k=1
k=2
Wehavethusconstructed13e�ects,eachwith3levels(0,1,and2),whichareinbalance
witheachotherinthesamewayaswiththeGraeco-Latinsquareintheexamplementioned
earlier.
Itcanbederivedfromthisthatwecanre-writeouroriginalmodelwiththehelpofthe
newarti�ciale�ects:
Yijk�=�+Ai+Bj+ABi+j+AB
2 i+2j+Ck+ACi+k+AC
2 i+2k+BCj+k
+BC
2 j+2k+ABCi+j+k+ABC
2 i+j+2k+AB
2Ci+2j+k+AB
2C
2 i+2j+2k+Eijk�
wherethetermsinthemodelaredecompositionsoftheconventionalmodelterms:
Ai
)
Ai
Bj
)
Bj
ABi;j
)
ABi+j+AB
2 i+2j
Ck
)
Ck
ACi;k
)
ACi+k+AC
2 i+2k
BCj;k
)
BCj+k+BC
2 j+2k
ABCi;j;k
)
ABCi+j+k+ABC
2 i+j+2k+AB
2Ci+2j+k+AB
2C
2 i+2j+2k
withtheusualmeaningtotheleftandthearti�ciale�ectstotheright.
Inthe33experiment,thereare27cellsorsingleexperiments.Todescribethemeanvalues
inthesecells,27parametersshouldbeused,ofwhichoneis�,sothatthereshouldbe
27�1=26degreesoffreedomforfactore�ect.
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The13termsinthestandardorderallhave3levels,whichsumupto0.Thus3�1=2
freeparameters(degreesoffreedom)areconnectedtoeachofthe13terms,oratotalof
13�(3�1)=26freeparameters(degreesoffreedom).
Itisfurtherseenthat,becauseofthebalance,allparametersareestimatedbyforming
theaverageinthesamewayasforthemaine�ectsandcorrectingwiththetotalaverage.
Forexample:
d AC2 0
=
133�
1X i
X j
X k
� Yijk:�Æ i+2k;0�� Y::::;
where
Æ r;s=
( 1
forr=s
0
forr6=s
astheindicatorÆ i+2k;0
pointsoutthedatawheretheindexforAC
2
is0(zero),i.e.
(i+2k) 3=0,while� Yijk:givestheaverageresponseincells(i;j;k),and� Y::::givesthe
averageresponseforthewholeexperiment.
Thus,inordertoestimateAC
2 0
thecellswherethecorrespondingindex,namely,i+2k
modulo3is0(zero)areincluded.Correspondingly,i+2khastobe1,respectively2to
gointotheestimatesforAC
2 1
andAC
2 2,respectively:
d AC2 0
=1 9(� Y000:+� Y010:+� Y020:+� Y101:+� Y111:+� Y121:+� Y202:+� Y212:+� Y222:)�� Y::::
d AC2 1
=1 9(� Y100:+� Y110:+� Y120:+� Y201:+� Y211:+� Y221:+� Y002:+� Y012:+� Y022:)�� Y::::
d AC2 2
=1 9(� Y200:+� Y210:+� Y220:+� Y001:+� Y011:+� Y021:+� Y102:+� Y112:+� Y122:)�� Y::::
Endofexample3.2
3.2
CalculationsbasedonKempthorne'smethod
Wehaveseenwithexamplesthattheintroducednewe�ects/parameters,whichobviously
donotreferdirectlyforexampletocertaintreatments(apartfromthemaine�ects),give
risetoamathematicaldecompositionoftheinteractions.Thisprocedureandthe
methodsderivedfromitaregenerallycalled"Kempthorne's"method,afterthenameof
thestatisticiantowhomitsoriginisoftenascribed,andwhohasdescribedit(cf.thelist
ofliteraturesuggestionsatthebeginningofthesenotes).
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Wehaveillustratedthatthecorrespondingestimatesareindependentbecauseofthe
describedbalance,and�nallywesawinexample3.1,page50,thatwecanalsoform
sumsofsquare,whichareindependentandsumtothetotalsumofsquares.
Wenowconsideranarbitrarilychosene�ectinthestandardorder.Wegenerallycallthis
e�ectF:
Ft=A
�B
�:::C
t
wheretheindexis
t=i��+j��+:::+k� ;
modulop
Withthenotationintroduced,whereYij:::k�
givestheresponseinthesingleexperiment
no.�withthefactorcombination(ij:::k),wehavethat
Tij:::k=a
i bj:::c
k=
r X �=1
Yij:::k�
andestimatesare:
b F l=P ij
:::kTij:::k�Æ l;t
N=p
�P ij
:::kTij:::k
N
forl=(0;1;:::;p�1);wheret=(i��+j��+:::+k� ) pandN=r�p
k
SSQ(F)=
P p�1
l=0
� Pij:::kTij:::k�Æ l;t
� 2
N=p
�� P
ij:::kTij:::k
� 2
N
=(N=p)�p
�
1 X l=0
b F2 l
wherewestillusetheindicator Æ r
;s=
( 1
forr=s
0
forr6=s
;
Thee�ectestimatecanbeexpressedinwords
b F l=sumofdata,wheret=l
numberofdata,wheret=l�averageafalldata;l=(0;1;:::;p�1)
Example3.3:
EstimationandSSQ
inthe3
2-factorialexperiment
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Ifweagainconsiderthe32experimentinexample3.1page50,whereweletTijgivethe
sumofdataincells(i;j)forexample,we�nd:
b A i=
Ti0+Ti1+Ti2
r�32�
1
�T::
r�32
SSQ(A)=r�3
2�
1
2 X i=0
b A2 i
b B j=
T0j+T1j+T2j
r�32�
1
�T::
r�32
d AB0=
T00+T21+T12
r�32�
1
�T::
r�32
d AB1=
T10+T01+T22
r�32�
1
�T::
r�32
d AB2=
T20+T11+T02
r�32�
1
�T::
r�32
SSQ(AB)=r�3((d AB0)2+(d AB1)2+(d AB2)2)
d AB20=
T00+T11+T22
r�32�
1
�T::
r�32
d AB21=
T10+T21+T02
r�32�
1
�T::
r�32
d AB22=
T20+T01+T12
r�32�
1
�T::
r�32
SSQ(AB
2)=r�3((d AB20)2+(d AB21)2+(d AB22)2)
wheretheinnermost2exponentissymbolic-mnemonic,whiletheoutermosthereisthe
usuallysquaring.
Endofexample3.3
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3.3
Generalformulationofinteractionsandarti�ciale�ects
ConsiderapkfactorialexperimentwithfactorsA,B,...,C,andpisaprimenumber.
Theinteractionbetweenanyfactorsisdecomposedaswehaveseeninthepreviousex-
amples:
ABi;j=ABi+j+AB
2 i+2j+:::+AB
p�
1
i+(p�
1)j
Ageneralnotationcanbeintroducedfortheinteractione�ectsinafactorstructureby
introducinganoperator"�"inthefollowingway
A�B=AB+AB
2+:::+AB
p�
1
andA�I=A.Laterwewillneedthefurtherarithmeticrulethat
(A+B)�=A
�+B
�
sothatforexample
(A�B)�=(AB+AB
2+:::+AB
p�
1)�=(AB)�+(AB
2)�+:::+(AB
p�
1)�
Inadditionanevenmoregeneraloperator"�"canbeintroduced,workinginthefollowing
way:
A�B=A+B+A�B
Inthiswaytheoperator"�"generatesallthetermsinthestandardorderforthefactors
onwhichitworks.
Foranycompletepkfactorialexperiment,thefactormodelcanthenbewritten
Y=�+A�B�:::�C+E
=�+A+B+A�B+:::+C+A�C+B�C+:::+A�B�:::�C+E
Ifp=3,thedecompositionisaspreviously.Forexample
A�B
=
AB+AB
2
A�B�C
=
(AB+AB
2)�C=ABC+ABC
2+AB
2C+AB
2C
2
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SupposeonebeginswithBasthe�rstfactor.Inthe32casethiswouldgivethedecom-
position
BAj;i=B�A=BAj+i+BA
2 j+2i
butthenitemergesthattheindicesforthissetofarti�ciale�ectsvarysynchronously
withtheindicesforthee�ectsABi+jandAB
2 i+2j.
Wecanillustratethiswiththefollowing
Example3.4:
Indexvariationwithinversionofthefactororder
Takea32experimentandconsiderthefollowingtable:
A
B
AB
AB
2
BA
BA
2
i
j
i+j
i+2j
j+i
j+2i
0
0
0
0
0
0
1
0
1
1
1
2
2
0
2
2
2
1
0
1
1
2
1
1
1
1
2
0
2
0
2
1
0
1
0
2
0
2
2
1
2
2
1
2
0
2
0
1
2
2
1
0
1
0
NotethatfortheindicesofthetwotermsAB
2
andBA
2,itappliesthat
(i+2j) 3=0()(j+2i) 3=(2i+j)3=0
(i+2j) 3=1()(j+2i) 3=(2i+j)3=2
(i+2j) 3=2()(j+2i) 3=(2i+j)3=1
aswestillcalculate"modulo3".
ThismeansthattheindicesforAB
2
andBA
2
varysynchronouslysothatAB
2 0
�BA
2 0,
AB
2 1
�BA
2 2,andAB
2 2
�BA
2 1orsaidinanotherway:Inordertoextractthepropersum
ofsquaresweonlyneedoneofthem,ofwhichwehavechosenAB
2.
Endofexample3.4
Theexampleillustratestherulethatweshouldonlyincludee�ectswheretheexponent
onthe�rstfactorinthee�ectequals1.Incertainsituations,however,onecancometo
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e�ectsthatdonotful�lthiscondition.Butfortunatelyitiseasyto�ndthee�ectfrom
thestandardorderwithwhichitcanbereplaced.
3.4
Standardisationofgenerale�ects
Weconsiderageneralnon-standardisede�ect.Asexamplewecantakeane�ectsuchas
CA
2B
4D
3froma3kfactorialexperiment.To�ndthee�ectfromthestandardorderwith
anindexvariationthatvariessynchronouslywiththis,oneproceedsasfollows:
1.ArrangethefactorsinthefactororderA,B,C,D,...etc
CA
2B
4D
3�!A
2B
4CD
3
2.Reduceallexponentsmodulop
A2B
4CD
3�!A
2BCD
0�!A
2BC(p=3her)
3.Iftheexponentonthe�rstfactorinthee�ect(hereA)is1,weare�nished.Oth-
erwise,thewholee�ectisliftedtothesecondpowerandtheexponentsareagain
reducedmodulop:
A2BC�!A
4B
2C
2�!AB
2C
2
4.Step3isrepeateduntilthe�rstfactorinthee�ecthastheexponent1.
Withthehelpofthisalgorithm,onecanalwaysmaketheexponentonthe�rstfactor
inaKempthornee�ectbe1,andbyusinga�xedorderoffactors,onegetsanunequiv-
ocalstandardorder.Forexampleweseethattheindexforthee�ectCA
2B
4D
3
varies
synchronouslywiththeindexforthee�ectAB
2C
2
-calculatedmodulo3.
Example3.5:
Generalisedinteractionsandstandardisation
Ifwehavetwoe�ectsina35experiment,forexampleABC
2
andABDE,we�ndtheir
generalisedinteractionas(wherep=3)
ABC
2�ABDE=ABC
2(ABDE)+ABC
2(ABDE)2=ABCD
2E
2+CDE
wherewehavealsousedtheabove-mentionedsquaringmethodtogettheexponent1on
the�rstfactorinthee�ect.
Inthisconnectionitcanbeusefultorememberthatinasquareexperiment,onefactor
canbemoved"outofthesquare"andouttotheedge,wherebytheedgeinquestion
"movesintothesquare".Fromthe32experimentdealtwithinexample3.1page50:
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i!
0
1
2
0
1
2
0
1
2
0
1
2
j
0
0
1
2
0
0
0
0
1
2
0
1
2
#
1
0
1
2
1
1
1
1
2
0
2
0
1
2
0
1
2
2
2
2
2
0
1
1
2
0
Ai
Bj
ABi+j
AB
2 i+2j
Themaine�ectsAiandBjconstitutetheedgesinthesquareandABi+jandAB
2 i+2jare
"insidethesquare".IfwenowmoveABi+jandAB
2 i+2jouttothesides,weseethatAi
andBjmove"intothesquare":
i+j!
0
1
2
0
1
2
0
1
2
0
1
2
i+2j
0
0
1
2
0
0
0
0
2
1
0
2
1
#
1
0
1
2
1
1
1
2
1
0
1
0
2
2
0
1
2
2
2
2
1
0
2
2
1
0
ABi+j
AB
2 i+2j
Ai
Bj
Examples:(i+j=1andi+2j=2))(i=0andj=1),(i+j=2andi+2j=0))
(i=1andj=1).
AB�AB
2=AB(AB
2)+AB(AB
2)2=A
2B
3+A
3B
5=A
2+B
2=A+B
Therefore,byusingtherulesabove,wecouldhaveforeseenthatAandB
wouldcome
intothesquareasgeneralisedinteractionsforthearti�ciale�ectsABandAB
2.
Thefoure�ectsA,B,ABandAB
2togetherformtheelementsinwhatiscalledagroup.
Inbrief,itdistinguishesitselfbythefactthatwiththeintroducedarithmeticruleswe
cancreatenewelementsfromotherelements,andallelementscreatedwillbelongtothe
group.
Endofexample3.5
Example3.6:Latinsquaresin23factorialexperimentsandYates'algorithm
Inchapter2wewentthroughthe2kexperiment,whilehere-exempli�edwiththe3k
experiment-wehaveintroducedmoregeneralpkexperiments.Wewillnowshowbrie y
howtheintroducedmoregeneralmethodslookina2kexperiment.
Withthreefactors,A,B,andC,themathematicalmodelintheintroducedformulation,
withp=2,is:
Yijk�=�+Ai+Bj+ABi+j+Ck+ACi+k+BCj+k+ABCi+j+k+Eijk�
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wherei;j;k=(0;1)and�=(1;::;r).
Theusualrestrictionsare:
1 X i=0
Ai=
1 X j=0
Bj=
1 Xi+j=0
ABi+j=
1 X k=0
Ck=
1 Xi+k=0
ACi+k=
1 Xj+k=0
B
Cj+k=
1 Xi+j+k=0
AB
Ci+j+k=
0
Theconnectionbetweenthetraditionalmodelformulationandtheformulationintroduced
hereis(asalsoshownonpage56forthe3kexperiment)that(withtheusualformulation
totheleftandthenewformulationtotheright):
Ai
)
Ai
Bj
)
Bj
ABi;j
)
ABi+j
Ck
)
Ck
ACi;k
)
ACi+k
BCj;k
)
BCj+k
ABCi;j;k
)
ABCi+j+k
IfdataareanalysedwiththehelpofYates'algorithm,onemustensurethatthee�ect
estimatesgetthecorrectsign.Yates'algorithmalwaysgivesestimatescorrespondingto
thelevelwhereallfactorsintheparameterareonlevel"1".FortheABCinteraction,
Yates'algorithmgivesthat
dABC1;1;1=[ABCkontrast]=(2
k�r)
Ifthedataareanalysedaccordingtotheintroducedmodel,itisfoundthat
dABC1;1;1)dABC1+1+1!dABC3!dABC1
thatis,thatthealgorithm�ndstheABCi+j+k-parameterlevel"1".
Ifontheotherhand,theinteractionABisconsidered,Yates'algorithm�nds
d AB1;1)d AB1+1!d AB2!d AB0=�d AB1
thatisplusABi+jparameterlevel"0"orminusitslevel"1".
ItthusgenerallyappliesthatYates'algorithm
usedfora2kfactorialexperimentfor
theintroducedgenerale�ectswithanunevennumberoffactorsgivestheparameters'
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level"1",whilethealgorithmforparameterswithanevennumberoffactorsgivesthe
parameters'level"0".
Endofexample3.6
3.5
Block-confoundedpk
factorialexperiment
Inthissectionwewillgeneralisethemethodsthatwereintroducedinsection2.2page20,
andwestartwiththefollowing
Example3.7:
23factorialexperimentin2blocksof4singleexperiments
Weconsidera23factorialexperimentwithfactorsA,BandCandwithrrepetitionsper
factorcombination.
Thetraditionalmathematicalmodelforthisexperimentis
Yijk�=�+Ai+Bj+ABi;j+Ck+ACi;k+BCj;k+ABCi;j;k+Eijk�
wherei;j;k=(0;1)and�=(1;::;r)
Wehavepreviouslyseenthatsuchanexperimentcanbelaidoutintwoblocksbychoosing
toconfoundoneofthefactore�ectswithblocks,andwehaveseenthatthisisformalised
bychoosingade�ningcontrast.Thee�ectcorrespondingtothiswillbeconfoundedwith
blocks.Inordertousetheintroducedmethodforanalysisofp
kfactorialexperiments,we
willwritethemodelonthegeneralform,whichforp=2is:
Yijk�=�+Ai+Bj+ABi+j+Ck+ACi+k+BCj+k+ABCi+j+k+Eijk�
wherei;j;k;=(0;1)and�=(1;:::;r)
Todividetheexperimentintotwoparts,wenowchoosea
De�ningrelation:I=ABC
where,asanexample,wechoosetoconfoundthe3-factorinteractionABCwithblocks.
Thise�ecthasindex=(i+j+k) 2,whichthustakesthevalues0or1.Welettheblock
numberfollowthisindex,i.e.thatinBlock0areplacedtheexperimentswhereitapplies
that(i+j+k) 2=0.Correspondingly,experimentswhere(i+j+k) 2=1areputin
block1.To�ndtheprincipalblock,wemustinotherwords�ndallthesolutionstothe
equation:
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(i+j+k)(modulo2)
=0
Wetry:
i=0;j=0=)k=0:experiment=(1)
i=1;j=0=)k=1:experiment=ac
i=0;j=1=)k=1:experiment=bc
i=1;j=1=)k=0:experiment=ab
Thelastsolutioncouldbefoundbyaddingthetwoprevioussolutionstoeachother:
i
j
k
1
+
0
+
1
=
2
!
0
ac
0
+
1
+
1
=
2
!
0
bc
1
+
1
+
2!0
=
2
!
0
ac�bc=ab
Onenotesthatthisindexadditioncorrespondsto"multiplying"thetwosolutionsacand
bcbyeachother.
Theotherblockisconstructedby�ndingthesolutionsto(i+j+k) 2=1.Thesesolutions
are(a;b;c;abc).
Inthiswaytheblockingisfound:
Block0
Block1
(1)
ab
ac
bc
a
b
c
abc
Wenotethatthissolutionisexactlythesameastheonewefoundinsection2.2using
forexamplethetabularmethod.
Endofexample3.7
WehavenowseenasimpleexampleoftheuseofKempthorne'smethodtomakeblock
experiments.Theprincipleisstillthatwelettheblockvariablevarysynchronouslywith
thelevelsforthefactore�ectthatwewillconfoundwithblocks.
Example3.8:
32factorialexperimentin3blocks
Supposewehavetheexperiment
A=0
A=1
A=2
B=0
(1)
a
a2
B=1
b
ab
a2b
B=2
b2
ab2
a2b2
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whereweagainwritethemodelonthegeneralform,whichforp=3is:
Yij�=�+Ai+Bj+ABi+j+AB
2 i+2j+Eij�
Wenowwanttocarryouttheexperimentbein3blocks,eachwith3singleexperiments.
Forthatpurposewecanlettheblockindexfollowtheindexforthearti�ciale�ectABi+j,
wherebyitisstillpossibletoestimatethetwomaine�ectsAandB:
De�ningrelation:I=AB
Index=i+j:
i=0
i=1
i=2
j=0
0
1
2
j=1
1
2
0
j=2
2
0
1
Block0
Block1
Block2
(1)
ab2
a2b
a
b
a2b2
a2
b2
ab
Block0isgivenwithallsolutionstotheequation(i+j)3=0.Theothertwoblocksare
givenwith(i+j)3
=1and(i+j)3
=2respectively.
Thedesigncouldhavebeencomputeddirectlyusingthefollowingtabularmethod:
i
j
code
Block=(i+j)3
0
0
(1)
0
1
0
a
1
2
0
a2
2
0
1
b
1
1
1
ab
2
2
1
a2b
0
0
2
b2
2
1
2
ab2
0
2
2
a2b2
1
Ifweonlywantedtheprincipalblockwecanusethemethodshownintheprevious
example,whichconsistsofsolvingtheequation:
(i+j)modulo3=0
i=0
)
j=0
Experiment:(1)
i=1
)
j=�1!�1+3
=2
Experiment:ab2
i=2
)
j=�2!�2+3
=1
Experiment:a
2b
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Itisseenthattheprincipalblockhastheappearance:
Block0
(1)
x
x2
wherex=ab2)x
2=(ab2)2=a
2b4=a
2b
wherexcanrepresentanychosensolutionto(i+j)3
=0except(i;j)=(0;0).
Theothertwoblocksarecanbefoundbytheabovetabularmethodor,equivalently,by
solvingtheequations(i+j)3=1and(i+j)3
=2respectively.Forexample,theterma
isthesolutionto(i+j)3
=1inthati=1,andj=0correspondtoa.
Theblockthatcontainstheexperimentacanbeconstructedby"multiplying"aonthe
principalblockfound:
Block0
Block1
a�(1)
ab2
a2b
=)
a
a2b2
b
principalblock
Thelastblockisconstructedby�ndingasolutiontotheequation(i+j)3=2,forexample
a2
andmultiplyingthisontheprincipalblock:
Block0
Block
a2�(1)
ab2
a2b
=)
a2
b2
ab
principalblock
Itiseasytoshowthatwiththisblocking,theonlye�ectinourmodelthatisconfounded
withtheblocke�ectispreciselytheABi+je�ect.
Ifwewantedanalternativeblockgrouping,wheretheAB
2
e�ectwasconfoundedwith
blocks,wewouldusethede�ningrelationI=AB
2anddeterminetheprincipalblockby
solvingtheindexequation(i+2j)=0.Onesolutionisab,andtheprincipalblockis
therefore(1),ab,(ab)
2
=
(1),ab,a
2b2.Afterthisone�ndstheblocking
Block0
Block1
Block2
(1)
ab
a2b2
a
a2b
b2
a2
b
ab2
principalblock
(i+2j) 3=0
Tryityourself!
Endofexample3.8
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Wehaveseenhowapkfactorialexperimentcanbedividedintopblockssothatane�ect
choseninadvanceisconfoundedwithblocks.
Wecangeneralisethismethodtoadivisionintopqblocks,whereq<k.Todothis,we
startbydividinga23experimentinto2�2=22=4blocks.
Example3.9:
Divisionofa23factorialexperimentinto22blocks
Lettherebea23factorialexperimentwithfactorsA,BandC.Withtheintroduced
formulationthemodelis,inthatp=2:
Yijk�=�+Ai+Bj+ABi+j+Ck+ACi+k+BCj+k+ABCi+j+k+Eijk�
whereallindicesi;j;k=(0;1)and�=(1;::;r)
Tode�ne4(=2�2)blocks,weuse2de�ningrelations,forexample
I 1=AB
and
I 2=AC
aspreviouslyshownonpage27.
Thestructureofthe4blockscanbeillustrated
I 1=AB
i+j=0
i+j=1
I 2=AC
i+k=0
Block(0,0)
Block(1,0)
i+k=1
Block(0,1)
Block(1,1)
IftheindexforbothABi+jandACi+kis0forexample,thesingleexperimentsareplaced
inblock(0,0).
Inthisway,orbyusingthetabularmethod,one�ndstheblocking:
I 1=AB
i+j=0
i+j=1
I 2=AC
i+k=0
(1)
abc
b
ac
i+k=1
ab
c
a
bc
Iftheblocke�ectsaremodelledasa2�2design,wecanwritethattheblockscontribute
with
Blocks=�+Ff+Gg+FGf+g
;
wheref=(i+j)2
andg=(i+k) 2
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Itisclearthatthee�ectABvariessynchronouslywithFandthatthetwoe�ectsare
confounded.Correspondingly,ACisconfoundedwithG.Thatpartoftheblockvariation,
whichisherecalledFG,hastheindex(f+g) 2=((i+j)+(i+k))2=(j+k) 2,whichis
preciselytheindexforthetermBCinthemodelfortheresponseoftheexperiment.
Thereforeitcanbeconcludedthatthee�ectBCwillalsobeconfoundedwithblocks,
whichcanalsobeseenfromthefollowingtable,wheretheindexoftheBCe�ectis0on
onediagonaland1ontheotherone:
I 1=AB
i+j=0
i+j=1
I 2=AC
i+k=0
j+k=0
j+k=1
i+k=1
j+k=1
j+k=0
Moreformallywecanwrite:
Blocks=AB+AC+AB�AC=AB+AC+BC
Endofexample3.9
3.6
Generalisationofthedivisionintoblockswithseveralde�n-
ingrelations
LetI 1=A�B�:::C
denoteade�ningrelation,thatdividesapk
factorialexperiment
intopblocks.Further,letI 2=A
aB
b:::C
cdenoteade�ningrelationthatlikewisedivides
thepkexperimentintopblocks.
Inthedivisionoftheexperimentintop�pblocksonthebasisofthesede�ningrelations,
bothe�ects
I 1=A�
B�...C
andI 2=AaBb...Cc
willbeconfoundedwithblocks.Inaddition,theirgeneralisedinteractionwillbecon-
foundedwithblockssothatbesidesI 1andI 2thee�ectsgivenintheexpression:
I 1�I 2=(A
�
B�...C
)�(A
aB
b...C
c)
willbeconfoundedwithblocks.Alltermsintheexpression:
I 1�I 2=I 1+I 2+I 1�I 2=I 1+I 2+I 1I 2+I 1(I2)2+...++I 1(I2)(p�
1)
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areconfoundedwithblocks.
Wecangenerallywriteuptheconfoundingsforanydivisionofap
kfactorialexperiment
inpqblocks.
Ifwehavethecorrespondingde�ningrelationsgivenbyI 1,I 2,I 3,..,I q,allthee�ectsin
theequation
I 1�I 2�I 3�...�I q=I 1+I 2+I 1�I 2+I 3+...+I 1�I 2�I 3�...�I q
willbeconfoundedwithblocks.Theoperators"�"and"�"workasstatedinsection3.3
page60
Example3.10:
Dividinga33factorialexperimentinto9blocks
Lettherebea33factorialexperimentwithfactorsA,BandC.
Asanexamplewedividetheexperimentinto3�3blocksusing
I 1=ABC
2
andI 2=AC
Thereby,ABC
2
andACtogetherwiththeirgeneralisedinteractionareconfoundedwith
blocks,thatis,allthee�ectsintheexpression(wherep=3):
ABC
2
�AC=ABC
2
+AC+ABC
2
�AC
=ABC
2
+AC+ABC
2(AC)+ABC
2(AC)2=ABC
2
+AC+AB
2
+BC
Intheanalysisofvariancetable,thesumsofsquaresforABC
2,AC,AB
2andBCthere-
forealsocontainpossibleblocke�ectsandthustheycannotbeinterpretedasexpressing
factore�ectsalone.
Theprincipalblockinthisexperimentisfoundbysolvingtheequations(p=3):
(i+j+2k) 3=0and(i+k) 3=0
One�ndsforexamplei=1)k=2)j=1,thatisabc
2.Theprincipalblockcontains
33=32=3singleexperiments.Thismeansthatitsatis�esto�ndonesolutioninorder
todeterminetheblock.Ifthissolutioniscalled"x",theprincipalblockexperimentsare
(1),xandx
2.
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Inourcase,wethenforx=abc
2getthethreeexperiments(1),abc
2and(abc
2)2=a
2b2c.
Onecancheckthat(i=2;j=2;c=1)isalsoasolutiontothetwoindexequations.
Theotherblocksarefoundby�ndingthesolutionstotheindexequationsfortheright-
handsidesequalto(0,1,2)inthecaseofbothequations,i.e.atotalof9di�erentcases,
correspondingtothe3�3blocks.
Foraanyoneoftheseblocks,itappliesthattheycanbefoundwhenjustoneexperiment
isfoundintheblock.Bymultiplyingthisexperimentontheprincipalblock,thewhole
blockisdetermined.
Endofexample3.10
Example3.11:
Divisionofa25experimentinto23blocks
LetthefactorsbeA,B,C,DandE,whichallappearon2levels.Todividetheexperiment
into2�2�2blocks,3de�ningrelationsareused,f.ex.
I 1=ABC,I 2=BDEandI 3=ABE
Therebyalle�ectsinthefollowingexpressionareconfoundedwithblocks:
I 1�I 2�I 3=I 1+I 2+I 1�I 2+I 3+I 1�I 3+I 2�I 3+I 1�I 2�I 3
Thatis,inadditiontoABC,BDEandABE,thefollowingterms(sincep=2):
(I1
�I 2)
=
ABCBDE
=
ACDE
(I1
�I 3)
=
ABCABE
=
CE
(I2
�I 3)
=
BDEABE
=
AD
(I1
�I 2�I 3)
=
ABCBDEABE
=
BCD
Thedesigncanbefoundbythetabularmethod(sef.ex.page25).
Ifonlytheprincipalblockiswantedwecansolvetheequations
(i+j+k) 2=0
;
(j+l+m) 2=0
;
(i+j+m) 2=0
Oneblockcontains25=23=22=4singleexperiments.Therefore2solutionshavetobe
found.Ifthesesolutionsarecalledxandy,theprincipalblockis:(1),x,yandxy
Aningeniouswayto�ndthesesolutionsistotrywith(i=1,j=0)and(i=0,j=1),
whichcorrespondtotheelementsaandbinthefactorstructureforfactorsAandB.The
methodworksifthemaine�ectsAandBarenotconfoundedwitheachotherorwith
blocks.
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DesignofExperiments,Course02411,IMM,DTU
72
We�nd (i
=1;j=0)
)
k=1,
m=1,
l=1,
theexperimentisx=acde
(i=0;j=1)
)
k=1,
m=1,
l=0,
theexperimentisy=bce
Theprincipalblockistherefore
Principalblock
Block(0,0,0)
(1)
x
y
xy
=
(1)
acde
bce
abd
Notethatallexperimentsintheprincipalblockhaveanevennumberoflettersincommon
withthe3de�ningcontrasts,ABC,BDEandABE.Theremainingblockscannowbe
foundbymultiplyingwithelementsthatarenotintheprincipalblock.
Fortheblockcorrespondingtotheequations
(i+j+k) 2=1
;
(j+l+m) 2=0
;
(i+j+m) 2=0
thatisblock(1,0,0),thereisasolution:(i;j;k;l;m)=(1;0;0;1;1)=ade(startwith
(i;j)=(1;0),whichistheeasiestmethod).Therestoftheblockisfoundbymultiplying
thissolutionontotheprincipalblock:
Principalblock
Block(1,0,0)
ade�(1)
acde
bce
abd
=)
ade
c
abcd
be
Theremaining6blockscanbefoundbysettingtheright-handsidesoftheequationsto
(0,1,0),(1,1,0),(0,0,1),(1,0,1),(0,1,1)and(1,1,1),respectively.
Endofexample3.11
Example3.12:
Divisionof3kexperimentsinto33blocks
Lettherebea3kfactorialexperimentandsupposethatI 1,I 2andI 3de�neadivisionof
theexperimentinto3�3�3=27blocks.
Theconfoundingistherebygivenwith
I 1�I 2�I 3=I 1+I 2+I 1�
I 2+I 3+I 1�
I 3+I 2�
I 3+I 1�
I 2�
I 3
=I 1+I 2+I 1I 2+I 1I2 2
+I 3+I 1I 3+I 1I2 3
+I 2I 3+I 2I2 3
+I 1(I2
�
I 3)+I 1(I2
�
I 3)2
whereexponentsarereducedmodulo3.Forthetwolasttermswehave:
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DesignofExperiments,Course02411,IMM,DTU
73
I 1(I2
�
I 3)=I 1(I2
I 3+I 2I2 3)=I 1I 2I 3+I 1I 2I2 3
and
I 1(I2
�
I 3)2=I 1(I2
I 3+I 2I2 3)2=I 1I2 2
I2 3
+I 1I2 2
I4 3
=I 1I2 2
I2 3
+I 1I2 2
I 3
Thetermsfoundwillallbeconfoundedwithblocks.Eachofthecorrespondinge�ects
hasprecisely3levels,i.e.thevariationbetweenthese3levelshas2degreesoffreedom.A
totalof13termswith2degreesoffreedomarefound,i.e.atotalof26degreesoffreedom,
whichcorrespondexactlytothevariationbetween27blocks.
Thedesigncanbefoundbythetabularmethod(sef.ex.page67).
Endofexample3.12
3.6.1
Constructionofblocksingeneral
Wehaveseenabovethatinapkfactorialexperiment,onede�ningrelation,I 1,dividesthe
experimentintopblocks,whileqrelations,forexampleI 1,...,I q,dividetheexperiment
intopqblockseachcontainingpreciselypk�
qsingleexperiments.
Thiscorrespondstothefactthatinoneblockarejustasmanysingleexperimentsas
thereareinacompletep
k�
qexperiment,thatis,anexperimentwithk�qfactorseach
onplevels.
We�rstconstructtheprincipalblockamongthesepqblocks,and,onthebasisofthis,
theremainingblockscanbedetermined,asisshownintheexamples.
Themethodwewilluseisinbrief:
1)::Lettherebeqde�ningcontrastsI 1,I 2,...,I q,andagainletallthesingleexperi-
mentsbedesignatedwith(1);a;a
2;:::;ap�
1;b;ab;:::;ap�
1b;b2;ab2;:::;ap�
1bp
�
1,...,
etc.
2):Determinek�qofthesesingleexperiments,whichareintheprincipalblock.For
exampletheycanbe:f1,f2,...,fk�
q.Itisrequiredforthesesingleexperimentsthat
thecorrespondingsolutionstotheindexequationsarelinearlyindependent.
3):Theprincipalblockcannowbeconstructedbyusingthesesingleexperimentsas
"basicexperiments"andmakingacompletepk�
q
factorialexperiment,i.e.the
standardorderforthesingleexperimentsonthebasisoff1,,f2,...,fk�
q:
(1);f1;f1
2;:::;f1
p�
1;f2;f1f2;f1
2f2;:::;f1
p�
1f2
p�
1;:::;f1
p�
1f2
p�
1��fk�
qp�
1
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DesignofExperiments,Course02411,IMM,DTU
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Thiscollectionofsingleexperimentsthenmakesuptheprincipalblock.Duringthe
formation,allexponentsarereducedmodulop.
4):Changethelevelforoneoftheindexequationsandthereby�ndanewsingleex-
perimentthatisnotintheprincipalblockandmultiplytheexperimentonallthe
experimentsintheprincipalblock.Inthiswayanewblockisformed.
5):Continuewith4)untilallblocksareformed.
Inordertoassurethatallthesingleexperimentsintheprincipalblockaredi�erent,we
mustrequirefortheoriginal(k�q)solutionsthattheyarelinearlyindependent(where
allarestillcalculatedmodulop).
Otherwisetheprincipalblockwillnotbecompletelydetermined,andthesamesingle
experimentswillbefoundseveraltimeswhentryingto�ndtheexperimentsintheblock.
Withthesamekindofargumentation,itcanbeshownhow,onthebasisofoneexperiment
belongingtoanalternativeblock,therestofthatblockcanbeformedbymultiplyingit
ontotheprincipalblock.
Forexampleif,withthehelpofaspreadsheetoracomputerprogram,onewantsto
�ndablockdistribution,thesimplestmethodistorunthroughallsingleexperiments
instandardorderandforeachsingleexperimentcalculatethevalueoftheindicesofthe
de�ningcontrasts,thatistousethetabularmethod.
Example3.13:
Dividinga34factorialexperimentinto32blocks
Letthenotationbeasusual.Thesingleexperimentsaregivenby:
(1);a;a
2
;b;ab;a
2b;b2;ab2;a
2b2;:::;a
2b2c2d
2
Takeforexamplethede�ningrelations:
I 1=ABi+jandI 2=BCD
2 j+k+2l
Theprincipalblockconsistsoftheexperimentswhereboth(i+j)3=0and(j+k+2l) 3=0.
Inoneblockthereare34
�
2
=32singleexperiments.
Thecompletedesigncanbeconstructedandwrittenoutbymeansofthetabularmethod
(seepage67).
Ifwewanttheprincipalprincipalblock,forexample,wemustjustdeterminetwo"linearly
independent"singleexperimentsandfromthatformtherestasa32experiment.
Thus:Findtwolinearlyindependentsolutionsto:
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DesignofExperiments,Course02411,IMM,DTU
75
i+j=0andj+k+2l=0
Trywithi=0)j=0)k+2l=0andchoose(k=1;l=1),forexample,giving
(i;j;k;l)=(0;0;1;1)asausablesolution.Theexperimentiscd.
Thentryforexamplewithi=1)j=2,(j+k+2l)=0)(k+2l) 3=(�2)3
=
(�2+3)3=1whereweforexamplechoosel=0andk=1.Notethatonecanalways
addanarbitrarymultipleof"3"toa(negative)numberwhenonehasto�nd"modulo
3"ofthenumber.Thatistosaythatgenerally(x) p=(x+kp) pwhere(:) pheredenotes
"(.)modulop".
Thus(i;j;k;l)=(1;2;1;0)isausablecombinationandtheexperimentis"ab2c".
Checktheindependencebyverifyingthatcd(ab2c)�6=(1)forall�(therelevant�'sare1
and2):OK.
Nowcallf1=cdandf2=ab2c.Theprincipalblockthenis
(1)
f1
f2 1
(1)
cd
(cd)2
f2
f1f2
f2 1f2
=
ab2c
cdab2c
(cd)2ab2c
f2 2
f1f
2 2
f2 1f
2 2
(ab2c)
2
cd(ab2c)
2
(cd)2(ab2c)
2
byorderingtheelements,multiplyingoutandreducingallexponentsmodulo3,theblock
isfound:
(1)
cd
c2d
2
ab2c
ab2c2d
ab2d
2
a2bc
2
a2bd
a2bcd
2
To�ndanalternativeblock,welookforasingleexperimentthatisnotintheblock
alreadyfound.Wecanforexampletake"a".
Thenewblockisthen: (1
)
cd
c2d
2
a
acd
ac2d
2
a�
ab2c
ab2c2d
ab2d
2
=)
a2b2c
a2b2c2d
a2b2d
2
a2bc
2
a2bd
a2bcd
2
bc2
bd
bcd
2
orbymultiplyingwithb:
(1)
cd
c2d
2
b
bcd
bc2d
2
b�
ab2c
ab2c2d
ab2d
2
=)
ac
ac2d
ad
2
a2bc
2
a2bd
a2bcd
2
a2b2c2
a2b2d
a2b2cd
2
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DesignofExperiments,Course02411,IMM,DTU
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Endofexample3.13
Example3.14:
Dividinga53factorialexperimentinto5blocks
A53experimentconsistsofatotalof125singleexperiments.Withthedivisioninto5
blocks,thereare25singleexperimentsineachblock.
ThefactorsareA,BandC,andasde�ningrelationwechooseforexample
I=ABC
3 i+j+3k
Intheprincipalblock,wherep=5,itappliesthat
i+j+3k=0
(modulo5)
Sincethesizeoftheblockis5�5=52,wehaveto�nd2linearlyindependentsolutions
tothisequation.Forexample,
(i;j;k)=(1;0;3)�ac3and(i;j;k)=(0;1;3)�bc
3
canbeused.Asastart,theprincipalblockisthereby
(1)
ac3
a2c6
a3c9
a4c1
2
bc3
abc
6
a2bc
9
a3bc
12
a4bc
15
b2c6
ab2c9
a2b2c1
2
a3b2c1
5
a4b2c1
8
b3c9
ab3c1
2
a2b3c1
5
a3b3c1
8
a4b3c2
1
b4c1
2
ab4c1
5
a2b4c1
8
a3b4c2
1
a4b4c2
4
andafterreductionoftheexponentsmodulo5,one�nallygets
(1)
ac3
a2c
a3c4
a4c2
bc3
abc
a2bc
4
a3bc
2
a4b
b2c1
ab2c4
a2b2c2
a3b2
a4b2c3
b3c4
ab3c2
a2b3
a3b3c3
a4b3c
b4c2
ab4
a2b4c3
a3b4c1
a4b4c4
Itcanbeinterestingtonotethatthisisa5�5Latinsquare,whichwithCinsidethe
squareis:
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DesignofExperiments,Course02411,IMM,DTU
77
A=0
A=1
A=2
A=3
A=4
B=0
0
3
1
4
2
B=1
3
1
4
2
0
B=2
1
4
2
0
3
B=3
4
2
0
3
1
B=4
2
0
3
1
4
whichforinstanceshowsthatthethreefactorsaremutuallybalancedwithintheblock
found.Thesamewillnaturallyapplywithintheother4blocksintheexperiment.One
oftheseblockscanbeeasilyconstructedforexamplebymultiplyingtheprincipalblock
withanexperimentthatisnotincludedintheprincipalblock.Bymultiplyingwitha,
forexample,we�nd
a
a2c3
a3c
a4c4
c2
abc
3
a2bc
a3bc
4
a4bc
2
b
ab2c1
a2b2c4
a3b2c2
a4b2
b2c3
ab3c4
a2b3c2
a3b3
a4b3c3
b3c
ab4c2
a2b4
a3b4c3
a4b4c1
b4c4
whichisthusalsoaLatinsquare.
Theremainingblockscanbefoundinthesameway,butnaturallyonecanalsoleta
programconstructalltheblocksbycalculatingthevalueoftheindex(i+j+3k)forall
singleexperimentsandplacingtheexperimentsaccordingtowhether(i+j+3k)(modulo
5)is0,1,2,3or4,thatisbythetabularmethod.
Endofexample3.14
3.7
Partialconfounding
Partialconfoundingin2kfactorialexperimentswasintroducedinsection2.3page28.
Wewillgiveanotherexampleofpartialconfoundingina2kexperiment,wherewenow
forthesakeofillustrationuseKempthorne'smethodtoformtherelevantblocks.
Example3.15:
Partiallyconfounded2
3
factorialexperiment
Againweconsideranexperimentwith3factorsA,BandC,eachon2levels.Weassume
thattheexperimentscanonlybedoneinblockswhicheachcontain4singleexperiments.
Tobeabletoestimateallthee�ectsinthemodel
Yijk=�+Ai+Bj+ABi+j+Ci+ACi+k+BCj+k+ABCi+j+k+E
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DesignofExperiments,Course02411,IMM,DTU
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itisnecessarytodoapartiallyconfoundedfactorialexperiment.
Supposethatinthe�rstexperimentalserieswechoosetoconfoundthethree-factorin-
teractionABC.
Todividetheexperimentinto2blocks,wehaveto�nd2solutionstotheindexequation
sincetheblocksizeis23
�
1=2�2.
Thereforewehaveto�nd2solutionstotheequation(i+j+k) 2=0.
Bytrialanderror,we�ndforexamplex=acandy=bc.
Theprincipalblockisthen
block1
(1)
x
y
xy
=
(1)
ac
bc
ab
(i+j+k) 2=0
Bymultiplyingwitha,wegettheotherblock,whichofcourseconsistsoftheremaining
singleexperimentsinthecomplete23factorialexperiment:
block2
a�(1)
x
y
xy
=
a
c
abc
b
(i+j+k) 2=1
Analysisofthis�rstblock-confoundedexperimentcanbedonewithYates'algorithm,
whichgivesaresultthatcanalsobeexpressedinmatrixformintheusualway:
2 6 6 6 6 6 6 6 6 6 6 6 6 6 4I (1)
A(1)
B(1)
AB(1)
C(1)
AC(1)
BC(1)
ABC(1)=blocks
3 7 7 7 7 7 7 7 7 7 7 7 7 7 5=2 6 6 6 6 6 6 6 6 6 6 6 6 6 41
1
1
1
1
1
1
1
�1
1
�1
1
�1
1
�1
1
�1
�1
1
1
�1
�1
1
1
1
�1
�1
1
1
�1
�1
1
�1
�1
�1
�1
1
1
1
1
1
�1
1
�1
�1
1
�1
1
1
1
�1
�1
�1
�1
1
1
�1
1
1
�1
1
�1
�1
13 7 7 7 7 7 7 7 7 7 7 7 7 7 52 6 6 6 6 6 6 6 6 6 6 6 6 6 4(1)
a b ab c a
c bc abc
3 7 7 7 7 7 7 7 7 7 7 7 7 7 5
Theindex(:) (1)onthecontrastsreferstothis�rstexperiment.
Wethendoanotherexperiment,butthistimewechoosetoconfoundthee�ectAB.The
blockingoftheexperimentisthus:
block3
block4
(1)
ab
abc
c
and
a
b
ac
bc
(i+j)2=0
(i+j)2=1
c hs.
DesignofExperiments,Course02411,IMM,DTU
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Forthisexperimentwecan�ndcontrastsinthesamewayasinthe�rstexperiment.And
�nallywewillcombinethetwoexperiments.Wehavethefollowingsourcesofvariation:
1)
Factore�ects
whicharenotconfounded
2)
Factore�ects
whicharepartiallyconfounded
3)
Blocke�ects,
i.e.variationbetweenthetotalsofthe4blocks
4)
Residualvariation
Analysisofthetwoexperimentsgivesrespectively:
2 6 6 6 6 6 6 6 6 6 6 6 6 6 4I (1)
A(1)
B(1)
AB(1)
C(1)
AC(1)
BC(1)
ABC(1)=blocks
3 7 7 7 7 7 7 7 7 7 7 7 7 7 5and
2 6 6 6 6 6 6 6 6 6 6 6 6 6 4I (2)
A(2)
B(2)
AB(2)=blocks
C(2)
AC(2)
BC(2)
ABC(2)
3 7 7 7 7 7 7 7 7 7 7 7 7 7 5
whereindex(:) (2)correspondstothesecondexperiment.
Wecan�ndsumsofsquaresanddegreesoffreedomcorrespondingtothefoursourcesof
variation:
c hs.
DesignofExperiments,Course02411,IMM,DTU
80
1)
Unconfoundedfactore�ects
SSQA
=
([A(1)]+[A(2)])
2=(2�2
3),
f=1
SSQB
=
([B(1)]+[B(2)])
2=(2�2
3),
f=1
SSQC
=
([C(1)]+[C(2)])
2=(2�2
3),
f=1
SSQAC
=
([AC(1)]+[AC(2)])
2=(2�2
3),
f=1
SSQBC
=
([BC(1)]+[BC(2)])
2=(2�2
3),
f=1
2)
Partiallyconfounded
factore�ects
SSQAB
(halfprecision)
=
[AB(1)]2=(2
3),
f=1
SSQABC
(halfprecision)
=
[ABC(2)]2=(2
3),
f=1
3)
Blocke�ectsandconfounded
factore�ects
Betweenexperiments
=
([I (1)]�[I(2)])
2=(2�2
3),
f=1
SSQAB+blocks(3-4)
=
[AB(2)]2=(2
3),
f=1
SSQABC+blocks(1-2)
=
[ABC(1)]2=(2
3),
f=1
4)
Residualvariation:
BetweenA-estimates(SSQA,Uncert.)
=
([A(1)]�[A(2)])
2=(2�2
3),
f=1
BetweenB-estimates(SSQB,Uncert.)
=
([B(1)]�[B(2)])
2=(2�2
3),
f=1
BetweenC-estimates(SSQC,Uncert.)
=
([C(1)]�[C(2)])
2=(2�2
3),
f=1
BetweenAC-estimates(SSQAC,Uncert.)
=
([AC(1)]�[AC(2)])
2=(2�2
3),
f=1
BetweenBC-estimates(SSQBC,Uncert.)
=
([BC(1)]�[BC(2)])
2=(2�2
3),
f=1
5)
Totalvariation
=
SSQtotwithdegreesoffreedom
f=15
Generally,thevariationcanbecalculatedbetweenforexampleRA
Aestimatesthatare
allbasedoncontrastsfrom(RA
di�erent)2kexperimentswithrrepetitions(inwhich
theyareallunconfounded)withtheexpression:
SSQA,Uncert.=
[A(1)]2+:::+[A(RA)]2
2k�r
�([A(1)]+:::+[A(RA)])
2
RA
�2k�r
Intheexample,RA
=2,k=3andr=1.Forothernon-confoundedestimates,naturally,
correspondingexpressionsarefound.See,too,page31.
Wehavetherebyaccountedforallthevariationinthetwoexperimentscollectively.Note
thatwehavecalculatedsumsofsquarescorrespondingtoatotalof15sourcesofvariance,
eachwithonedegreeoffreedom.Thiscorrespondspreciselytothetotalvariationbetween
the16singleexperiments,whichgivesriseto(16�1)=15degreesoffreedom.
Iftherearerrepetitionsforeachofthesingleexperiments,alltheSSQ'shavetobedivided
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DesignofExperiments,Course02411,IMM,DTU
81
byr.Inthiscase,onecan,ofcourse,�ndvariationwithineachfactorcombination(a
totalof8+8singleexperimentswith(r�1)degreesoffreedom)andusethem
to
calculateaseparateestimatefortheremaindervariation.Thisestimatecan,ifnecessary,
becomparedwiththementionedestimate,whichwascalculatedabove.
Endofexample3.15
Theexampleshownillustratestheprinciplesforcombiningseveralexperimentswithdif-
ferentconfoundings.Thewholeanalysiscanbesummarisedtothefollowing.Suppose
that,inall,experimentsaremadeinRblocksmednblocksingleexperimentsineachblock.
Thevariationcanthenbedecomposedinthefollowingcontributions,whereTblockigives
thetotalinthei'thblock:
SSQblocks
=
(T2 block1+T
2 block2+:::+T
2 blockR)=nblock�(T
2 tot)=(R�nblock)
=
variationbetweenblocktotals
SSQe�ects
=
SSQforallfactore�ectsbasedonexperimentsinwhich
thee�ectsarenoterconfoundedwithblocks
SSQresid
=
variationbetweene�ectestimatesfromexperiments,where
thee�ectsarenotconfounded
SSQuncertainty
=
variationbetweenrepeatedsingleexperimentswithinblocks
The�rstcontribution,SSQblocksalsocontains,inadditiontothetotalvariationbetween
blocks,thevariationfromconfoundedfactore�ects.
Example3.16:
Partiallyconfounded32factorialexperiment
We�nishthissectionbyshowingtheprinciplesfortheconstructionandanalysisofa
partiallyconfounded32factorialexperiment.Thisexperimentispossiblylittleusedin
practice,butitillustratesthegeneralprocedurewell.Anditshowshowallthemain
e�ectsandinteractionsina3�3experimentcanbedetermined,eventhoughthesizeof
theblockisonly3.
Experiment1:Dividethe32experimentinto3blocksof3accordingtoI=ABi+j
One�nds:
block1
:
(1) (1)
ab2 (1)
a2b (1)
total=T1
block2
:
a(1)
a2b2 (1)
b (1)
total=T2
block3
:
a2 (1)
b2 (1)
ab (1)
total=T3
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DesignofExperiments,Course02411,IMM,DTU
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Index(1)indicatesthatthisisexperiment1.
Experiment2:NowdivideaccordingtoI=AB
2 i+2j.Thisgivestheblocking:
block4
:
(1) (2)
ab (2)
a2b2 (2)
total=T4
block5
:
a(2)
a2b (2)
b2 (2)
total=T5
block6
:
a2 (2)
b (2)
ab2 (2)
total=T6
Index(2)indicatesthatthisisexperiment2.
Wecan�ndthesumofsquaresbetweenthe6blocks,inthatTtot=T1+T2+:::+T6
:
SSQblocks
=
(T2 1
+T
2 2
+:::+T
2 6)=3�T
2 tot=18
Wethenhave
TA0
=
(1) (1)+b (1)+b2 (1)+(1) (2)+b (2)+b2 (2)
TA1
=
a(1)+ab (1)+ab2 (1)+a(2)+ab (2)+ab2 (2)
TA2
=
a2 (1)+a
2b (1)+a
2b2 (1)+a
2 (2)+a
2b (2)+a
2b2 (2)
Thatis,thatforexampleTA0
=thesumofallthemeasurementswherefactorAhas
beenonlevel"0"intheRA
experimentswheree�ectAisnotconfoundedwithblocks,
andcorrespondinglyforlevels"1"and"2".WithTtot;A
=TA0
+TA1
+TA0
wegetquite
generally:
SSQA
=
(T2 A
0
+T
2 A1
+T
2 A2)=(RA
�3k
�
1)�T
2 tot;A=(RA
�3k)
withf=(3�1)=2degreesoffreedom.InourexampleRA
=2andk=2.
TB0
=
(1) (1)+a(1)+a
2 (1)+(1) (2)+a(2)+a
2 (2)
TB1
=
b (1)+ab (1)+a
2b (1)+b (2)+ab (2)+a
2b (2)
TB2
=
b2 (1)+ab2 (1)+a
2b2 (1)+b2 (2)+a
2b2 (2)+a
2b2 (2)
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DesignofExperiments,Course02411,IMM,DTU
83
SSQB
=
(T2 B
0
+T
2 B1
+T
2 B2)=(RB
�3k
�
1)�T
2 tot;B=(RB
�3k)
withf=(3�1)=2degreesoffreedomandRB
=2andk=2.
TAB0
=
(1) (2)+ab2 (2)+a
2b (2)
TAB1
=
a(2)+a
2b2 (2)+b (2)
TAB2
=
a2 (2)+b2 (2)+ab (2)
thatis,sumsfromtheRAB
experimentsinwhichthearti�ciale�ectABi+jisnotcon-
foundedwithblocks,i.e.theexperimentconsistingofblocks4,5and6.WithTtot;AB
=
T4+T5+T6,one�nds
SSQAB
=
(T2 AB0
+T
2 AB1
+T
2 AB2)=(RAB
�3k
�
1)�T
2 tot;AB=(RAB
�3k)
withf=(3�1)=2degreesoffreedomandRAB
=1andk=2.
Finallyone�nds
TAB2 0
=
(1) (1)+ab (1)+a
2b2 (1)
TAB2 1
=
a(1)+a
2b (1)+b2 (1)
TAB2 2
=
a2 (1)+b (1)+ab2 (1)
thatis,sumsfromtheexperimentsinwhichthearti�ciale�ectAB
2 i+jisnotconfounded
withblocks,i.e.theexperimentconsistingofblocks1,2and3.WithTtot;AB2
=T1+T2+T3
one�nds
SSQAB2
=
(T2 AB20
+T
2 AB21
+T
2 AB22)=(RAB2
�3k
�
1)�T
2 tot;AB2=(RAB2
�3k)
withf=(3�1)=2degreesoffreedomandRAB2
=1andk=2.
Theresidualvariationisfoundasthevariationbetweenestimatesfore�ectsthatarenot
confoundedwithblocks.
FromtheAe�ectone�nds,whereSSQA(bothexperiments)istheabovecalculatedsum
ofsquaresfore�ectA,whileSSQA(experiment1)andSSQA(experiment2)arethesums
ofsquaresfore�ectAcalculatedseparatelyforthetwoexperiments:
c hs.
DesignofExperiments,Course02411,IMM,DTU
84
SSQUA
=
SSQA(experiment1)+SSQA(experiment2)-SSQA(bothexperiments)
withf=4�2=2degreesoffreedom.
FromtheBe�ectone�ndscorrespondingly
SSQUB
=
SSQB(experiment1)+SSQB(experiment2)-SSQB(bothexperiments)
likewisewithf=4�2=2degreesoffreedom.
Sincetheremaininge�ectsABi+jandAB
2 i+2jareonly"purely"estimatedonetimeeach,
wedonotgetanycontributiontotheresidualvariationfromthesee�ects.
Insummary,wegetthefollowingvariancedecomposition:
Blocksand/orconfounded
factore�ects
SSQblocks
5
Maine�ectAi
SSQA
2
Maine�ectBj
SSQB
2
InteractionABi+j
SSQAB
2
(halfpr�cision)
InteractionAB
2 i+j
SSQAB2
2
(halfpr�cision)
Residualvariation
SSQUA
+SSQUB
2+2
Total
SSQtotal
17
Notethatwehavederivedvariationcorrespondingto17=18�1degreesoffreedom.
Inconclusionwecangiveestimatesforallthee�ectsinthisexperiment:
c �2 resid=(SSQUA
+SSQUB)=4
(^ A0;^ A1;^ A2)=(T
A0
6
�Ttot
18;
TA1
6
�Ttot
18;
TA2
6
�Ttot
18)
andthemaine�ectBisfoundcorrespondingly.
c hs.
DesignofExperiments,Course02411,IMM,DTU
85
Fortheinteraction,theestimatesarefoundintheblockswheretheyarenotconfounded:
(d AB0;d AB1;d AB2)=(T
AB0
3
�Ttot;2
9
;TAB1
3
�Ttot;2
9
;TAB2
3
�Ttot;2
9
)(fromexperiment2)
(d AB2 0;d AB2 1;d AB2 2)=(T
AB2 0
3
�Ttot;1
9
;TAB2 1
3
�Ttot;1
9
;TAB2 2
3
�Ttot;1
9
)(fromexperiment1)
andwiththehelpoftherelationABi;j=ABi+j+AB
2 i+2jonecan�nally�ndtheparameter
estimatesinthetraditionalmodelformulaYi;j=�+Ai+Bj+ABi;j+E.
Endofexample3.16
3.8
Constructionofafractionalfactorialdesign
WewillnowconcernourselveswithconstructingdesignswherethefactorsformLatin
squares/cubes.Thepresentationisageneralisationoftheresultsinsection2.4,wherewe
introducedfractional2kfactorialdesigns.Wewilllimitthediscussiontoexperimentswith
factorson2oron3levels,sincethesearetheexperimentsthatareusedmostfrequently
inpractice.
Asbefore,mainlyexamplesareusedtoshowthedi�erenttechniques.
Example3.17:
FactorexperimentdoneasaLatinsquareexperiment
Letusassumethatwehavethreefactors,A,BandCandthatwewanttoevaluatethese
eachon3levels.WechoosetomakeaLatinsquareexperimentwiththefactorCinside
thesquare.Accordingtothesameprincipledescribedintheprevioussection,wecanfor
exampleletChaveindexk=(i+j)3.Thismeansthatthemaine�ectCwillhavethe
sameindexastheKempthornee�ectABwithindex(i+j)3.
TheexperimentaldesignwheretheindexforfactorCisinsidethesquareisthus:
A=0
A=1
A=2
B=0
0
1
2
B=1
1
2
0
B=2
2
0
1
IndexforC
orequivalently:
(1)
ac
a2c2
bc
abc
2
a2b
b2c2
ab2
a2b2c
Insteadofthe3�3�3singleexperimentsinthecompletefactorexperiment,wechoose
todoonlythe3�3singleexperimentsintheLatinsquare.
c hs.
DesignofExperiments,Course02411,IMM,DTU
86
Notethatifx=acandy=bcwhichbothcorrespondtoasolutiontotheindexequation
k=(i+j)3,theexperimentcanbewritten:
(1)
x
x2
y
xy
x2y
y2
xy
2
x2y
2
Finally,wecanalsobymeansofthetabularmethodwriteoutthedesign:
Experiment
Experiment
Alevel
Blevel
Clevel
code
no.
sequence
i
j
(i+j)3
1
3
0
0
0
(1)
2
7
1
0
1
ac
3
1
2
0
2
a2c2
4
9
0
1
1
bc
5
5
1
1
2
abc
2
6
6
2
1
0
a2b
7
2
0
2
2
b2c2
8
4
1
2
0
ab2
9
8
2
2
1
a2b2c
Inthepracticalexecutionoftheexperiment,theorderisrandomised,asexempli�edin
thetable.
ThevariationofthemeanvalueduetothetwofactorsAandBcanbewritten
Ai+Bj+ABi;j=Ai+Bj+ABi+j+AB
2 i+2j
wheretheleft-handsideistheconventionalmeaning,whiletheright-handsideisthe
formulationaccordingtoKempthorne'smethod.
TheintroductionoffactorC,asmentioned,wasdonebygivingCtheindexvaluek=
(i+j)3.IfCispurelyadditive,i.e.ifCdoesnotinteractwiththeotherfactors,the
followingmodelwilldescribetheresponse,wherethein uencefromCisputin:
Yijk�=�+Ai+Bj+ABi+j+AB
2 i+2j+Ck=i+j+Eijk�
whereindex�correspondstopossiblerepetitionsofthe9singleexperiments.
IfthereisinteractionbetweenthetwofactorsAandB,thatpartoftheinteraction
describedbythearti�ciale�ectABi+jcannotberegardedasnegligible(thesameistrue
ofcourseforAB
2 i+2j).
c hs.
DesignofExperiments,Course02411,IMM,DTU
87
IfwenowtrytoestimatetheCe�ect,wecannotavoidhavingtheABi+jpartoftheAB
interactionconfoundedtheCestimate,preciselybecauseweusedk=(i+j)3
Thee�ectsCkandABi+jinotherwordsareconfoundedintheexperiment.
Wecanalsodemonstratethisbydirectcalculation.Using� Ytotfortheaverageofthe9
measurements,wehave,since(1)=Y0;0;0,a
2b=Y2;1;0,andab2=Y1;2;0:
b C 0=(Y0;0;0+Y2;1;0+Y1;2;0)=3�� Ytot
whichhastheexpectedvalue
Efb C 0g=Ef(Y0;0;0+Y2;1;0+Y1;2;0)=3�� Ytotg=C0+AB0
Furtheritisfound
Efb C 1g=Ef(Y0;1;1+Y1;0;1+Y2;2;1)=3�� Ytotg=C1+AB1
and
Efb C 2g=Ef(Y0;2;2+Y1;1;2+Y2;0;2)=3�� Ytotg=C2+AB2
InthisexperimentwehaveacertainpossibilityofevaluatingwhethertheABinteraction
canberegardedasnegligible,becausewecanexaminetheAB
2 i+2je�ect.Ifthisisneg-
ligible,onecanperhapsallowoneselftoconcludethattheABe�ectasawholecanbe
zero.
Insummary,onecanseethatonlyifthefactorsAandBdonotinteract,istheexperiment
suitableforestimatingC.
Asweshallseebelow,weneedtoassumethatalltwo-factorinteractionsarezeroinorder
toestimatethemaine�ectsA,BandCinthedescribed(1=3)�33=33
�
1
experiment.
Endofexample3.17
Wehaveseenaboveandpreviouslyinsection2.4thatitisnotwithoutproblemsto
putfurtherfactorsintoanexperimentintheform
ofasquareexperiment.Butwith
appropriateassumptionsaboutthelackofinteractions,itcanbedone.Inthefollowing,
wewilltrytoshowhowitisdoneinpractice.
Example3.18:Confoundingsina3
�
1�3
3factorialexperiment,aliasrelations
Consideragaintheaboveexample.Ifweshouldhavedoneanordinary33factorial
experiment,itwouldhaveconsistedof3�3�3=27singleexperiments.Theexperiment
wedidisonly1/3ofthis,namelyatotalof33=3=33
�
1=9singleexperiments.
Ifingeneralthereareinteractionsbetweenallthefactors,thee�ectsoftheexperiment
willbeconfoundedwitheachotheringroupsof3e�ects,analogouswithforexamplethe
c hs.
DesignofExperiments,Course02411,IMM,DTU
88
23�
1experiment,wheretheywereconfoundedingroupsof2.Thecompletemathematical
modelforthe33factorialexperimentcanbewritten,onceagain:
Yijk�=�+Ai+Bj+ABi+j+AB
2 i+2j+Ck+ACi+k+AC
2 i+2k+BCj+k
+BC
2 j+2k+ABCi+j+k+ABC
2 i+j+2k+AB
2Ci+2j+k+AB
2C
2 i+2j+2k+Eijk�
Forthe9singleexperimentsweconsideredinthepreviousexample,weusedk=i+j,
correspondingtotheconfoundingCk=ABi+j.
Thisgeneratorequationcanbechangedtoade�ningrelationbymultiplyingonboth
sidesoftheequationsignwithC
2
,andthenreorganisingtheexpressionsandreducing
theexponentsmodulo3.TheresultisC
2C=C
2AB�!I=ABC
2.Thuswehavethe
De�ningrelation:I=ABC
2
Ifonenowwantswhiche�ectsinthegeneralmodelanarbitrarye�ectisconfoundedwith,
thede�ningrelationcanbeused.Itismultipliedwiththee�ectinquestionin�rstand
secondpower(becausethefactorsareon3levelsandaccordingtotheruleslayedoutin
section3.3).One�ndsfortheAe�ect:
A�(I=ABC
2)�!A=(A)(ABC
2)=(A)2(ABC
2)
SincenowA(ABC
2)=A
2BC
2!A
4B
2C
4!AB
2Cand(A)2(ABC
2)=A
3BC
2!BC
2,
itisfoundthat
A=AB
2C=BC
2
Onecanbeconvincedthatindicesforthesethreee�ectsvarysynchronouslythroughout
theexperiment,becauseitisrequiredthatk=(i+j)3."Modulo3"calculationgives
(tryityourself):
E�ects
A
AB
2C
BC
2
Indices
i
(i+2j+k) 3
(j+2k) 3
0)
0
0
1)
2
2
2)
1
1
Ofcoursethesamecalculationscanbemadeforalle�ectsintheexperimentandonecan
beconvincedthatitwillgenerallyholdtruethatalle�ectsareconfoundedingroupsof
3.c hs.
DesignofExperiments,Course02411,IMM,DTU
89
Thecompletesetofaliasrelationsisfoundonthebasisofthede�ningrelationbymul-
tiplyingwiththee�ectsintheunderlyingfactorstructurein�rstandsecondpower:
GeneratorC=AB=)
De�ningrelationI=ABC
2
Aliasrelations
A
=
AB
2C
=
BC
2
B
=
AB
2C
2
=
AC
2
AB
=
ABC
=
C
AB
2
=
AC
=
BC
ForexampleAB
2�(I=ABC
2)�!AB
2=AC=BC.
Rememberagain,thatthefactorsAandBconstituteanunderlyingcompletefactor
structureandthatfactorCisintroducedintothisstructurebymeansofthegenerator
equationC
=AB.Thisexampli�esthegeneralmethodofconstructionoffractional
factorials.
Thealiasrelationsaremostusefullywrittenupwithonerelationpere�ectintheunder-
lyingfactorstructureandinstandardorder,asisshowninthetable.
Endofexample3.18
3.8.1
Resolutionforfractionalfactorialdesigns
Thetermresolutiondescribeswhichordersofe�ectsareconfoundedwitheachother.
Correspondingtotheexamplepage34wherefractional2kfactorialdesignswereintro-
duced,page37showsaliasrelationsfora23
�
1
factorialexperiment.Itcanbeseenhere
thatthemaine�ects(�rstordere�ects)areconfoundedwith2-factorinteractions(second
ordere�ects).SuchanexperimentiscalledaresolutionIIIexperiment.Itshouldbe
notedthatprecisely3factorsarepresentinthede�ningrelation(I=ABC)forthe
experiment.
Intheaboveexamplea33
�
1
factorialdesignisdescribedwiththede�ningrelationI=
ABC
2.ThisexperimenttooiscalledaresolutionIIIexperiment,sincemaine�ectsare
confoundedwithtwo-factorinteractions(orhigher).Thede�ningrelationinvolvesat
least3factors.
Ifallmaine�ectsinafractionalfactorialdesignareconfoundedwithe�ectsofatleastsec-
ondorder(2-factorinteractions),theexperimentiscalledaresolutionIIIexperiment.
Thiscorrespondstothefactthatnoe�ectinthede�ningrelationoftheexperimentisof
alowerorderthan3.
c hs.
DesignofExperiments,Course02411,IMM,DTU
90
Ifitholdstruethatnoe�ectinthede�ningrelationoftheexperimentisofalowerorder
than4,theexperimentiscalledaresolutionIV
experiment.
InaresolutionIVexperiment,themaine�ectsareallconfoundedwithe�ectsofat
leastthethirdorder,i.e.3-factorinteractions.Two-factorinteractionswillgenerallybe
confoundedwithother2-factorinteractionsand/orinteractionsofahigherorderina
resolutionIVexperiment.
Inmanypracticalcircumstances,onecannotassumeinadvancethatthe2-factorinter-
actionsareunimportantcomparedwiththemainactions.Onewillthereforeoftenwant
anexperimentofresolutionIV-atleast.
InaresolutionVexperiment,themaine�ectsareallconfoundedwithe�ectsofatleast
thefourthorder,i.e.4-factorinteractions.Two-factorinteractionswillgenerallybe
confoundedwith3-factorinteractionsand/orinteractionsofahigherorderinaresolution
Vexperiment.
Asarule,experimentswithahigherresolutionthanVwillnotbeneededtobedone,if
thefactorsinvolvedareofaquantitativenature(temperature,pressure,concentration,
time,densityetc.)wheremaine�ectsand2-factorinteractionsaremostfrequentlyof
considerablygreaterimportancethantheinteractionsofhigherorder.
3.8.2
Practicalandgeneralprocedure
Bymeansofthemethodoutlinedabove,wecannowconstructarbitrary1=pq�pkfactorial
experimentsand�ndtheconfoundings(thealiasrelations)intheexperiment.
kfactorsareconsidered(A,B,C,...,K)andthesefactorsareorderedsothatthe�rst
factorsareaprioriattributedthegreatestimportance.Meaningthattheexperimenter
expectsthatfactorAwillprovetohavethegreatestimportance(e�ect)ontheresponse
Y,andthatBhasthenextgreatestimportanceetc.
Thisorderingofthefactorsbeforetheexperimentisagreathelpbothwithregardto
creatingasuitabledesignandwithregardtoevaluatingtheresultsobtained.
Inthereview,inaddition,onehastomakeadecisionastowhichfactorsthatcouldbe
thoughttointeractandwhichonesthatcanbeassumedtoactadditively.Asthegeneral
rule,interactionsbetweenfactorsthathavealargee�ectwillbelargerthaninteractions
betweenfactorswithmoremoderatee�ects.
Inaddition,onewillgenerallyexpectthatinteractionsofahighorderwillbelessimpor-
tantthaninteractionsofalowerorder.
Inmanycasesoneoftenallowsoneselftoassumethatinteractionsofanorderhigherthan
2(i.e.3-factore�ectssuchasABC,ABD,BCDetc.ande�ectsofevenhigherorder)are
c hs.
DesignofExperiments,Course02411,IMM,DTU
91
assumedtohaveconsiderablylessimportancethanthemaine�ects.
Afterthis,theexperimentismostsimplyconstructedbystartingwiththecompletefacto-
rialexperiment,whichismadeupofthe(k�q)�rst(andexpectedtobemostimportant)
factorsandputtingtheremainingqfactorsintothisfactorstructurebyconfoundingwith
e�ectsregardedasnegligible.The�rst(k�q)andoftenmostimportantfactorswill
therebyformtheunderlyingfactorstructureinthe1=pq�pkfactorialexperimentwanted.
Example3.19:
A
2�
2
�2
5
factorialexperiment
Supposethatoneconsiders5factorsA,B,C,DandE,whichonewantstoevaluateeach
on2levelsina1=22�25factorialexperiment,i.e.in25
�
2=23=8singleexperiments.
WeimaginethatacloserevaluationoftheproblemathandindicatesthatfactorsA,B
andCwillhavethegreateste�ect,andwethuslettheunderlyingfactorstructureconsist
ofpreciselythesefactors.
ThedesignisthengeneratedbyconfoundingfactorsDandEwithe�ectsintheunderlying
factorstructure:
Generators
I A B AB C A
CBC
=
E
ABC
=
D
=)
Aliasrelationer
I
=
ABCD
=
BCE
=
ADE
A
=
BCD
=
ABCE
=
DE
B
=
ACD
=
CE
=
ABDE
AB
=
CD
=
ACE
=
BDE
C
=
ABD
=
BE
=
ACDE
AC
=
BD
=
ABE
=
CDE
BC
=
AD
=
E
=
ABCDE
ABC
=
D
=
AE
=
BCDE
TheexperimentisaresolutionIIIexperiment.
Theexperimentcaneasilybewrittenoutusingthetabularmethodasfollows(ifthe
principalfractionwiththeexperiment"(1)"ischosen)
i
j
k
l=(i+j+k) 2
m=(j+k) 2
Code
0
0
0
0
0
(1)
1
0
0
1
0
ad
0
1
0
1
1
bde
1
1
0
0
1
abe
0
0
1
1
1
cde
1
0
1
0
1
ace
0
1
1
0
0
bc
1
1
1
1
0
abcd
c hs.
DesignofExperiments,Course02411,IMM,DTU
92
whichisprobablytheeasiestwayto�ndtheexperimentandatthesametimetimewrite
outthewholeplan,forexampleusingasimplespreadsheetprogram.
Wemaywritetheplanas:
D=ABC,E=�BC
l=(i+j+k) 2,m=(j+k) 2
(1)
ad
bde
abe
cde
ace
bc
abcd
wheretheindicesarei,j,k,landm
forfactorsA,B,C,DandE,respectively,andthe
constructionoftheexperimentisgivenwiththepreviouslyintroducedsignnotationfor
2kexperimentsaswellaswiththeindexmethodthatisusedinthepresentchapter.See,
too,theexampleonpage40.
Thereare3alternativepossibilities,namely
D=�ABC,E=�BC
l=(i+j+k+1)2,m=(j+k) 2
d
a
be
abde
ce
acde
bcd
abc
D=+ABC,E=+BC
l=(i+j+k) 2,m=(j+k+1)2
e
ade
bd
ab
cd
ac
bce
abcde
D=�ABC,E=+BC
l=(i+j+k+1)2,m=(j+k+1)2
de
ae
b
abd
c
acd
bcde
abce
Aprerequisiteforobtaininga"good"experimentbydoingoneoftheseexperimentsis
thatfactorsBandCdonotinteractwitheachother(BCandABCunimportant),and
thatfactorsDandEdonotinteractwithotherfactorsatallorwitheachother.Factor
AcanbeallowedtointeractwiththetwofactorsBandC,i.e.ABandACcandi�er
from0.
Ifthesepreconditionscannotberegardedasful�lledtoareasonabledegree,theexperi-
mentwillnotbeappropriatetostudythe5factorssimultaneouslyinafractionalfactorial
designwithonly8singleexperiments.
Thealternativeswillthenbeeithertoexcludeoneofthefactors(bykeepingitconstantin
theexperiment)andbeingcontentwitha1=2�24experimentortoextendtheexperiment
toa1=2�25experiment,i.e.anexperimentwith16singleexperiments.Thesecond
alternativecouldreasonablybeconstructedbyputtingfactorEintothefactorstructure
consistingoffactorsA,B,CandDbytherelationABCD=E,withthefollowingalias
relations:
c hs.
DesignofExperiments,Course02411,IMM,DTU
93
I
=
ABCDE
A
=
BCDE
B
=
ACDE
AB
=
CDE
C
=
ABDE
AC
=
BDE
BC
=
ADE
ABC
=
DE
D
=
ABCE
AD
=
BCE
ABD
=
CE
CD
=
ABE
ACD
=
BE
BCD
=
AE
ABCD
=
E
Ifitisassumedthatallinteractionsofanorderhigherthan2areunimportant,onegets
reducedaliasrelations(thefullde�ningrelationisretained)
I
=
ABCDE
A
=
B
=
AB
=
C
=
AC
=
BC
= =
DE
D
=
AD
= =
CE
CD
= =
BE
=
AE
=
E
Onecanseethatall2-factorinteractionscanbetestedinthisdesign.Ifsomeofthese
arereasonablysmall,theirsumsofsquarescouldbepooledintoaresidualsumofsquares
andusedtotesthigherordere�ects.
ThisexperimentisaresolutionVexperiment.
Onecanchooseoneofthetwofollowingcomplementaryexperiments:
E=�ABCD
orm=(i+j+k+l)2
(1)
ae
be
ab
ce
ac
bc
abce
de
ad
bd
abde
cd
acde
bcde
abcd
E=+ABCD
orm=(i+j+k+l+1)2
e
a
b
abe
c
ace
bce
abc
d
ade
bde
abd
cde
acd
bcd
abcde
c hs.
DesignofExperiments,Course02411,IMM,DTU
94
Endofexample3.19
3.8.3
Aliasrelationswith1=pq�pk
experiments
Whenqfactorsareputintoafactorstructureconsistingof(k�q)factors,qgenerator
equationsareused.Eachequationgivesrisetoonede�ningrelation.Thatis,one�nds
qde�ningrelationswithqde�ningcontrasts:I 1,I 2,...,I q.Thecompletede�ning
relationcanthensymbolicallybewrittenas
I=I 1�I2�:::�Iq
wheretheoperator"�"isde�nedonpage60.Bycalculatingtheexpressionandreplacing
all"+"with"=",one�ndsthecompletede�ningrelation:
I=I 1=I 2=I 1I 2=I 1I
2 2
=:::=I 1I
p�
1
2
=:::=I 1I
p�
1
2
���I
p�
1
q
correspondingtothe"standardorder"forqfactors,calledI 1,I 2,...,I q.
Thealiasrelationsoftheexperimentdrawnupcanbefoundforanarbitrarye�ect,F,
bycalculatingtheexpression
F=F�I=)F=F�I1�I2�:::�Iq
andFandallthee�ectsemergingontheright-handsideoftheexpressionwillbecon-
foundedwitheachother.
Fromcalculationoftheexpressionandreplacementof"="with"+"subsequently,one
gets:
F=FI 1=:::=FI
p�
1
1
=FI 2=:::=FI
p�
1
2
=:::=F(I1I
p�
1
2
���I
p�
1
q
)p�
1
Duringthecalculationofthesinglee�ectsintheexpression,itcanbehelpfultousethe
factthatfortwoarbitrarye�ectsX
andY,itholdstruethat
XY
�=X
�Y
suchthatinthecalculationofexpressionswithtwoe�ects,itisofteneasiesttoliftupthe
simpleste�ecttothepowerinquestion.Forexampleina32factorialexperiment,both
(AB
2C)2(AB)and(AB
2C)(AB)2becomeBC.Testthis.
c hs.
DesignofExperiments,Course02411,IMM,DTU
95
Example3.20:
Constructionof3
�
2
�3
5
factorialexperiment
Lettherebe5factorsA,B,C,DandEeachon3levels.Onewantstodoonly1/9of
thewholeexperiment,i.e.atotalof35
�
2=33=27singleexperiments.
Againwestartwithacompletefactorstructureforthreeofthefactors.Anditisassumed
thatitisreasonabletochooseA,BandC.Inthisfactorstructure,afurther2factorsare
putin,namelyDandE. D
esigngenerators
I A B AB
AB2
C AC
BC
ABC
AB2C
AC2
BC2
=
E
ABC2
AB2C2
=
D
=)
I 1
=
AB
2C
2D
2
I 2
=
BC
2E
2
Otheralternativescanbechosen,forexampletoputbothDandEintothe3-factor
interactionABC(whichisdecomposedin4partseachwith2degreesoffreedom)byfor
exampleD=AB
2C
2andE=ABC
2.(Tryto�ndthecharacteristicsofthisexperiment
(aliasrelations)).
Withtheconfoundingchoseninthetable,one�ndsthede�ningrelation
I=AB
2C
2D
2=BC
2E
2=(AB
2C
2D
2)(BC
2E
2)=(AB
2C
2D
2)(BC
2E
2)2
whichafterreductiongives
I=AB
2C
2D
2=BC
2E
2=ACD
2E
2=ABD
2E
Thealiasrelationsoftheexperimentare
c hs.
DesignofExperiments,Course02411,IMM,DTU
96
I
=
AB2C2D2
=BC2E2
=ACD2E2
=ABD2E
=De�ningrelation
A
=
ABCD
=ABC2E2
=AC2DE=AB2DE2
=BCD
=AB2CE=CD2E2
=BD2E
B
=
AC2D2
=BCE=ABCD2E2
=AB2D2E=ABC2D2
=CE=AB2CD2E2
=AD2E
AB
=
ACD
=AB2C2E2
=AB2C2DE=ABDE2
=BC2D2
=ACE=BC2DE=DE2
AB2
=
AB2CD
=AC2E2
=ABC2DE=ADE2
=CD
=ABCE=BCD2E2
=BDE2
C
=
AB2D2
=BE2
=AC2D2E2
=ABCD2E=AB2CD2
=BCE2
=AD2E2
=ABC2D2E
AC
=
ABD
=ABE2
=ACDE=AB2C2DE2
=BC2D
=AB2C2E=DE=BC2D2E
BC
=
AD2
=BE=ABC2D2E2
=AB2CD2E=ABCD2
=CE2
=AB2D2E2
=AC2D2E
ABC
=
AD
=AB2E2
=AB2CDE=ABC2DE2
=BCD2
=AC2E=BDE=CDE2
AB2C
=
AB2D
=AE2
=ABCDE=AC2DE2
=CD2
=ABC2E=BD2E2
=BCDE2
AC2
=
ABC2D
=ABCE2
=ADE=AB2CDE2
=BD
=AB2E=CDE=BCD2E
BC2
=
ACD2
=BC2E=ABD2E2
=AB2C2D2E=ABD2
=E=AB2C2D2E2
=ACD2E
ABC2
=
AC2D
=AB2CE2
=AB2DE=ABCDE2
=BD2
=AE=BCDE=CD2E
AB2C2
=
AB2C2D
=ACE2
=ABDE=ACDE2
=D
=ABE=BC2D2E2
=BC2DE2
Thealiasrelationforexampleofthemaine�ectAisfoundwiththehelpof:
A=A�I1�I2=A�(AB
2C
2D
2)�(BC
2E
2)
whichgives
A=A(AB
2C
2D
2)=A(AB
2C
2D
2)2=A(BC
2E
2)=:::=A(AB
2C
2D
2)2(BC
2E
2)2
TheexpressionsareorganisedinA-B-C-D-Eorderandtheexponentsreducedmodulo3.
Ifnecessarytheexponent1onthe�rstfactorintheexpressionsisfoundbyraisingtothe
powerof2andreducingmodulo3.
Inthesameway,thealiasrelationsforeachoftheothere�ectsarefoundintheunderlying
factorstructureasshowninthetable.
Toelucidatethecharacteristicsoftheexperimentaldesign,alle�ectsconsideredunimpor-
tantcanberemoved.ThisisthecasefortheBCe�ectandallothere�ectsinvolvingmore
than2factors.Forthesakeofclarity,thee�ectsfromtheunderlyingfactorstructureare
retained,butinparenthesisforassuminglyunimportante�ects.
Inthisway,thefollowingtableisfound,whichshowsthat2-factorinteractionsareusually
confoundedwithother2-factorinteractionsorwithmaine�ects.
c hs.
DesignofExperiments,Course02411,IMM,DTU
97
Reducedaliasrelations
I
=
AB2C2D2
=BC2E2
=
ACD2E2
=ABD2E
A
=
B
=
CE
AB
=
DE2
AB2
=
CD
C
=
BE2
AC
=
DE
(BC)
=
AD2
=BE=CE2
(ABC)
=
AD
(AB2C)
=
AE2
=CD2
AC2
=
BD
(BC2)
=
E
(ABC2)
=
BD2
=AE
(AB2C2)
=
D
TheexperimentisaresolutionIIIexperiment.Onecanseethatitisnecessarytoassume
thatanumberofthe2-factorinteractionsareunimportantiftheexperimentistobe
suitable.
If,forexample,onecanfurthermoreignoreinteractionsinvolvingfactorsDandE,allelse
canbetestedandestimated.Thisshowstheusefulnessoforderingthefactorsaccording
toimportance(i.e.maine�ectsandthusinteractionsfromDandErelativelysmall).
IfitholdstruethatDandEhaveonlyadditivee�ectsanddonotinteractwiththe
otherfactors,thealiasrelationscanbereducedtothetableshownonpage96,wherethe
experimentwasconstructed.
Thereare3�3=9possibilitiesforimplementingtheexperiment.If,forexample,we
wantthefractionincluding"(1),theexperimentwillbegivenbytheindexrestrictions
(i+2j+2k+2l) 3=0and(j+2k+2m) 3=0.
Ifwewanttowriteoutatableofindicesforthefactors(thatisthedesign),weusethe
tabularmethodandthegeneratorequationsl=i+2j+2kandm=j+2kasfollows:
c hs.
DesignofExperiments,Course02411,IMM,DTU
98
Factorsandlevels
A
B
C
D
E
Experiment
i
j
k
l=(i+2j+2k) 3
m
=(j+2k) 3
code
0
0
0
0
0
(1)
1
0
0
1
0
ad
2
0
0
2
0
a2d2
0
1
0
2
1
bd2e
1
1
0
0
1
abe
2
1
0
1
1
a2bde
0
2
0
1
2
b2de2
1
2
0
2
2
ab2d2e2
2
2
0
0
2
a2bd2e
0
0
1
2
2
cd2e2
1
0
1
0
2
ace2
...
...
...
...
...
...
...
...
...
...
...
...
2
2
2
1
0
a2b2c2d
Theexperimentis3�3�3=27singleexperiments.Onecanalsoderivethesingle
experimentsbysolvingindexequations.If3experimentswhich(independently)ful�l
indexequationsarecalledx,yandz,theexperimentwillbe:
(1)
x
x2
y
xy
x2y
y2
xy
2
x2y
2
z
xz
x2z
yz
xyz
x2yz
y2z
xy
2z
x2y
2z
z2
xz
2
x2z
2
yz
2
xyz
2
x2yz
2
y2z
2
xy
2z
2
x2y
2z
2
IntheunderlyingfactorstructureA,BandC,x
0
=a,y
0
=bandz
0
=cwillbesolutions,
andthecorrespondingindexsetsare(i;j;k)=(1;0;0),(i;j;k)=(0;1;0)and(i;j;k)=
(0;0;1).
To�ndthreesolutionsx,yandz,wethereforetrywith
x0
:(i;j;k)=(1;0;0)
=)
(l;m)=(1;0)
=)
x=ad
y0
:(i;j;k)=(0;1;0)
=)
(l;m)=(2;1)
=)
y=bd
2e
z0
:(i;j;k)=(0;0;1)
=)
(l;m)=(2;2)
=)
z=cd
2e2
Theexperimentthenconsistsofthesingleexperimentsbelow:
c hs.
DesignofExperiments,Course02411,IMM,DTU
99
(1)
ad
(ad)2
bd2e
ad(bd
2e)
(ad)2(bd
2e)
(bd
2e)
2
ad(bd
2e)
2
(ad)2(bd
2e)
2
cd2e2
adcd
2e2
(ad)2cd
2e2
(bd
2e)cd
2e2
ad(bd
2e)cd
2e2
(ad)2(bd
2e)cd
2e2
(bd
2e)
2cd
2e2
ad(bd
2e)
2cd
2e2
(ad)2(bd
2e)
2cd
2e2
(cd
2e2)2
ad(cd
2e2)2
(ad)2(cd
2e2)2
(bd
2e)(cd
2e2)2
ad(bd
2e)(cd
2e2)2
(ad)2(bd
2e)(cd
2e2)2
(bd
2e)
2(cd
2e2)2
ad(bd
2e)
2(cd
2e2)2
(ad)2(bd
2e)
2(cd
2e2)2
whicharereorganisedandtheexponentsreducedmodulo3:
(1)
ad
a2d
2
bd2e
abe
a2bde
b2de2
ab2d
2e2
a2b2e2
cd2e2
ace
2
a2cde2
bcd
abcd
2
a2bc
b2ce
ab2cde
a2b2cd
2e
c2de
ac2d
2e
a2c2e
bc2e2
abc
2de2
a2bc
2d
2e2
b2c2d
2
ab2c2
a2b2c2d
Inall,thereare9di�erentpossibilitiestoconstructtheexperiment,correspondingtothe
followingtable:
(i+2j+2k+2l)3=
0
(i+2j+2k+2l)3=
1
(i+2j+2k+2l)3=
2
(j+2k+2m
) 3=
0
1=
thedesignshown
2
3
(j+2k+2m
) 3=
1
4
5
6
(j+2k+2m
) 3=
2
7
8
9
Thethreeexperiments"1","4"and"7",forexample,arecomplementarywithregardto
thegeneratorequationBC
2
=E,i.e.thede�ningrelationI 2=BC
2E
2.
Thesameholdstruefor"2","5"and"8",aswellasfor"3","6"and"9".
Ifonecarriesoutoneofthesesetsofcomplementaryexperiments,onebreaksthecon-
foundingsoriginatinginthechoiceofBC
2
=E,andthewholeexperimentwillthenbe
c hs.
DesignofExperiments,Course02411,IMM,DTU
100
a1=3�34experimentwiththede�ningrelationI 1=AB
2C
2D
2andthefactorsA,B,C
andEintheunderlyingfactorstructure.
Endofexample3.20
3.8.4
Estimationandtestingin1=p
q�p
k
factorialexperiments
Theimportantthingtorealiseisthat1=pq�pkfactorialexperimentsareconstructedon
thebasisofacompletefactorstructure,i.e.theunderlyingcompletefactorstructure.
Theanalysisoftheexperimentisthendoneinthefollowingsteps:
1)Forthefractionalfactorialdesign,theunderlyingcompletefactorstructureisiden-
ti�ed.
2)Dataarearrangedinaccordancewiththisunderlyingstructure,andthesumsof
squaresaredeterminedintheusualwayforthefactorsandinteractionsinit.
3)Thealiasrelationsindicatehowallthee�ectsareconfoundedintheexperiment.
Therebythesumsofsquaresarefoundfore�ectsthatarenotintheunderlying
factorstructure.
4)Byconsideringspeci�cfactorcombinationsinasingleexperiment,onecandecide
howtheindexrelationsarebetweenthee�ectsthatarepartofthesamealias
relation.Inthiswayestimatesaredeterminedfortheindividuallevelsforthe
e�ectsthatarenotintheunderlyingstructure.
Asanillustrationofthis,weconsiderthefollowing.
Example3.21:
Estimationina3
�
1
�3
3-factorialexperiment
WehavefactorsA,BandC,allon3levelsandweassumethatA,BandCarepurely
additive,sothatitisrelevanttodoafractionalfactorialexperimentinsteadofacomplete
factorialexperiment.
Astheunderlyingfactorstructure,the(A,B)structureischosen.
Thegeneralised(Kempthorne)e�ectsinthisstructureareA,B,ABandAB
2.Wechoose
toconfoundforexamplewithAB,i.e.ABi+j
=Ck.Thischoiceentailsthefollowing
de�ningrelationandaliasrelations:
I
=
ABC
2
A
=
AB
2C
=
BC
2
B
=
AB
2C
2
=
AC
2
AB
=
ABC
=
C
AB
2
=
AC
=
BC
c hs.
DesignofExperiments,Course02411,IMM,DTU
101
Theindexrestrictionontheprincipalfractionoftheexperiment,i.e.thefractionthat
contains"(1)",is(i+j+2k) 3=0,k=(i+j)3.Twolinearlyindependentsolutions
havetobefoundforthisandbystartingin"a"and"b",one�nds:
(i;j)=(1;0)=)k=(1+0)=1:theexperimentisac
(i;j)=(0;1)=)k=(0+1)=1:theexperimentisbc
Onepossibleexperimentistheprincipalfractioninwhich"(1)"isapart:
(1)
ac
(ac)
2
bc
acbc
(ac)
2bc
(bc)
2
ac(bc)2
(ac)
2(bc)
2
=)
(1)
ac
a2c2
bc
abc
2
a2b
b2c2
ab2
a2b2c
Analternativepossibilityistocarryoutoneofthe(two)otherfractions,forexamplethe
oneofwhichthesingleexperimentaispart.Thisfractionisdeterminedby"multiplying"
theprincipalfractionwitha:
a�
(1)
ac
a2c2
bc
abc
2
a2b
b2c2
ab2
a2b2c
=)
a
a2c
c2
abc
a2bc
2
b
ab2c2
a2b2
b2c
Thisexperimenthastheindexrestriction(i+j+2k) 3=1,k=(i+j+2)3.
Thisexperimentischosenhereanddataareorganisedandanalysednowintheusualway
accordingtofactorsAandB(neglectingC):
A=0
A=1
A=2
B=0
c2
a
a2c
or
(1)
a
a2
B=1
b
abc
a2bc
2
withoutc:
b
ab
a2b
B=2
b2c
ab2c2
a2b2
b2
ab2
a2b2
TA0
=c2+b+b2c
,
TA1
=a+abc+ab2c2
,
TA2
=a
2c+a
2bc
2+a
2b2
TB0
=c2+a+a
2c
,
TB1
=b+abc+a
2bc
2
,
TB2
=b2c+ab2c2+a
2b2
TAB0
=c2+a
2bc
2+ab2c2
,
TAB1
=a+b+a
2b2
,
TAB2
=a
2c+abc+b2c
TAB2 0
=c2+abc+a
2b2
,
TAB2 1
=a+a
2bc
2+b2c
,
TAB2 2
=a
2c+b+ab2c2
c hs.
DesignofExperiments,Course02411,IMM,DTU
102
SSQ(A)
=
([TA0]2+[TA1]2+[TA2]2)=3r�[Ttot]2=9r
,
f=3�1
SSQ(B)
=
([TB0]2+[TB1]2+[TB2]2)=3r�[Ttot]2=9r
,
f=3�1
SSQ(AB)
=
([TAB0]2+[TAB1]2+[TAB2]2)=3r�[Ttot]
2=9r
,
f=3�1
SSQ(AB
2)
=
� [TAB2 0]2+[TAB2 1]2+[TAB2 2]2
� =3r�[Ttot]
2=9r
,
f=3�1
whererindicatesthatatotalofrsinglemeasurementscouldbemadeforeachsingle
experiment.Inthatcase,itisassumedthattheserrepetitionsarerandomisedoverthe
wholeexperiment.
Finally,thee�ectscanbeestimated:
b A 0=TA0=3r�Ttot=9r,
b A 1=TA1=3r�Ttot=9r,
b A 2=TA2=3r�Ttot=9r
b B 0=TB0=3r�Ttot=9r,
b B 1=TB1=3r�Ttot=9r,
b B 2=TB2=3r�Ttot=9r
d AB0=TAB0=3r�Ttot=9r,
d AB1=TAB1=3r�Ttot=9r,
d AB2=TAB2=3r�Ttot=9r
d AB20=TAB2 0=3r�Ttot=9r,
d AB21=TAB2 1=3r�Ttot=9r,
d AB22=TAB2 2=3r�Ttot=9r
To�ndtheconnectionbetweenCandtheAB
e�ect,theindexrelationisfoundfrom
thespeci�cexperimentbyconsideringtwosingleexperiments,forexamplec2(i=0;j=
0;k=2)anda(i=1;j=0;k=0).One�ndsthatitholdstruethat
Index
ABi+j
0
1
2
Ck
2
0
1
whereindexk=(i+j+2)3.Therefore,b C 0=d AB1,b C 1=d AB2
andb C 2=d AB0.
Themathematicalmodeloftheexperimentcouldbewrittenas
Yijk�=�+Ai+Bj+Ck=i+j+2+Eijk�,where�=1;2:::;r
and,if�>1,andthereisusedcompleterandomisationcorrectly,theresidualsumof
squaresis
SSQresid=
3 X i=1
3 X j=1
" r X �=1
Y2 ijk��r�� Y
2 ijk�
#
c hs.
DesignofExperiments,Course02411,IMM,DTU
103
Notethatsumsaremadeoverindicesiandjalone,sinceindexkisofcoursegiven
byiandjinthis33
�
1
factorialexperiment(whichconsistsof9singleexperimentseach
repeatedrtimes).
ThepreconditionofadditivitybetweenthethreefactorsA,BandCcouldbetestedby
testingtheAB
2
e�ectagainstthissumofsquares.
Endofexample3.21
Example3.22:
TwoSASexamples
Thecalculationsshownintheaboveexamplearerelativelyeasytoprogram.Aprogram
canalsobewrittenforthestatisticalpackageSASwhichwilldothework.Thefollow-
ingsmallexamplewithdata(r=1)illustrateshowanalysisofvariancecanbedone
correspondingtofactorsAandBalone,i.e.theunderlyingfactorstructure.
A=0
A=1
A=2
B=0
c2=15.1
a=16.9
a2c=23.0
B=1
b=9.8
abc=12.6
a2bc
2=21.7
B=2
b2c=5.0
ab2c2=10.0
a2b2=12.8
dataexempel1;
input
ABCY;
AB
=mod(A+B,3);
AB2=mod(A+B*2,3);
cards;
00215.1
10016.9
20123.0
0109.8
11112.6
21221.7
0215.0
12210.0
22012.8
; procGLM;
classABABAB2;
model
Y
=ABABAB2;
means
ABABAB2;
run;
Intheexamplestartingonpage96withdatalayoutshownonpage98,aSASjobcould
looklikethefollowing:
dataexempel2;
input
ABCDEY;
c hs.
DesignofExperiments,Course02411,IMM,DTU
104
AB
=mod(A+B,3);
AB2=mod(A+B*2,3);
AC=mod(A+C,3);
AC2=mod(A+C*2,3);
BC
=mod(B+C,3);
BC2=mod(B+C*2,3);
ABC=mod(A+B+C,3);
ABC2=mod(A+B+C*2,3);
AB2C=mod(A+B*2+C,3);
AB2C2=mod(A+B*2+C*2,3);
cards;
00000
31.0
10010
16.0
20020
4.0
01001
23.8
11011
23.6
21021
9.7
.....
.....
.....
12200
12.8
22210
15.1
; procGLM;classABABAB2CACAC2BCBC2ABCABC2
AB2CAB2C2
;
model
Y
=ABABAB2CACAC2BCBC2ABCABC2
AB2CAB2C2
;
means
ABABAB2CACAC2BCBC2ABCABC2
AB2CAB2C2
;
run;
Andsumsofsquaresandestimatesofe�ectsoutsidetheunderlyingfactorstructure
(A,B,C)canbedirectlyfoundusingthealiasrelations.
Endofexample3.22
3.8.5
Fractionalfactorialdesignlaidoutinblocks
Afractionalfactorialdesigncanbelaidoutinsmallerblocksbecauseofawishtoincrease
theaccuracyintheexperiment(di�erentbatches,groupsofexperimentalanimals,several
daysetc.).Otherreasonscouldbethatforthesakeofsavingtimeonewantstodothe
singleexperimentsonparallelexperimentalfacilities(severalovens,reactors,set-upsand
such).
Intheorganisationofsuchanexperiment,thefractionalfactorialdesignis�rstsetup
withoutregardtothesepossibleblocks,sinceitisimportant�rstandforemosttohave
anoverviewofwhetheritispossibletoconstructagoodfractionalfactorialdesignand
howthefactore�ectsoftheexperimentwillbeconfounded.
Whenasuitablefractionalfactorialdesignhasbeenconstructed,achoiceismadeofwhich
e�ectore�ectswouldbesuitabletoconfoundwithblocks,andacontrolismadethatall
blockconfoundingsaresensible,perhapsthewholeconfoundingtableisreviewed.Inthe
exampleonpage114,anexampleofthisisshown.
c hs.
DesignofExperiments,Course02411,IMM,DTU
105
Bothduringtheconstructionofthefractionalfactorialdesignandinthesubsequent
formationofblocksfortheexperiment,theunderlyingfactorstructureisused,which
mostpracticallyiscomposedofthemostimportantfactors,calledA(�rstfactor),B
(secondfactor)etc.
Inpractice,onecannaturallyimaginealargenumberofvariantsofsuchexperiments,
butthefollowingexamplesillustratethetechniquerathergenerally.
Example3.23:
A
3�
2
�3
5
factorialexperimentin3blocksof9singleexper-
iments
Letusagainconsideranexperimentinwhichthereare5factors:A,B,C,DandE.A
fractionalfactorialdesignconsistsof33=27singleexperiments.Weimaginethatfor
practicalreasons,itcanbeexpedienttodividethese27singleexperimentsinto3blocksof
9;forexampleitcanbediÆculttomaintainuniformexperimentalconditionsthroughout
all27singleexperiments.
The1/32�35factorialexperimentwantedisfoundfromtwogeneratorequations.
Aspreviouslydiscussed,2factors,DandE,areintroducedintoacomplete33factor
structureforthefactorsA,BandC.
Asintheexampleonpage96,wechoosetoputinDandEasinthefollowingtable:
Designgenerators
I A B AB
AB2
C AC
BC
ABC
AB2C
AC2
BC2
=
E
ABC2
AB2C2
=
D
=)
I 1
=
AB2C2D2
I 2
=
BC2E2
Withthisconfounding,onegets(aspreviously)thede�ningrelation
I=AB
2C
2D
2=BC
2E
2=ACD
2E
2=ABD
2E
Constructionoftheexperimentstillfollowstheexampleonpage96,andonecouldperhaps
againchoosetheprincipalfraction(seepage100):
c hs.
DesignofExperiments,Course02411,IMM,DTU
106
(1)
ad
a2d
2
bd2e
abe
a2bde
b2de2
ab2d
2e2
a2b2e2
cd2e2
ace
2
a2cde2
bcd
abcd
2
a2bc
b2ce
ab2cde
a2b2cd
2e
c2de
ac2d
2e
a2c2e
bc2e2
abc
2de2
a2bc
2d
2e2
b2c2d
2
ab2c2
a2b2c2d
Tonowdividethisexperimentconsistingofthe27singleexperimentsinto3blocksof9,
onechoosesyetanothergeneratingrelationwhichindicateshowtheblocksareformed.
Whenthisrelationistobechosen,oneagainstartswiththealiasrelationsoftheexper-
imentinsuchareducedformthatonehasanoverviewofhowthemaine�ectsand/or
interactionsofinterestareconfounded.Ifweagainfollowthesameexample,thesere-
ducedaliasrelationscouldbeasshowninthefollowingtable,inwhichwenowalsoput
intheblocks:
I
=
AB2C2D2
=BC2E2
=ACD2E2
=ABD2E
A
=
B
=
CE
AB
=
DE2
AB2
=
CD
C
=
BE2
AC
=
DE
(BC)
=
AD2
=BE=CE2
=blocks
(ABC)
=
AD
(AB2C)
=
AE2
=CD2
AC2
=
BD
(BC2)
=
E
(ABC2)
=
BD2
=AE
(AB2C2)
=
D
Theeasiestwaytowriteoutthethisexperimentisshownonpage110,however
wewilldiscussthedesignalittleindetail.
Thechoiceofconfoundingwithblocksmeansthatalle�ectsintheunderlyingfactor
structurethatarenotconfoundedwithane�ectofinterestcanbeused.Thee�ectBC
couldbesuchane�ect(but,forexample,notBC
2,why?).
Thede�ningcontrastBChastheindexvalue(j+k) 3.Theblockdivisionisthendeter-
minedbywhether(j+k) 3=0,1or2.
c hs.
DesignofExperiments,Course02411,IMM,DTU
107
To�ndthe3blocks,onecanagainstartwiththeunderlyingfactorstructureanditcan
beseenthattheblockdivisionissolelydeterminedbyindicesforthefactorsBandC,
namelyjandk.
Aswesawinthepreviousexample,theexperiment,asdescribedabove,wasalsofound
onthebasisoftheunderlyingfactorstructure,andtheblocknumbercorrespondingto
thesingleexperimentsisinsertedinthefollowingtable:
Experiment
block
Experiment
block
Experiment
block
(1)
0
ad
0
a2d
2
0
bd2e
1
abe
1
a2bde
1
b2de2
2
ab2d
2e2
2
a2b2e2
2
cd2e2
1
ace
2
1
a2cde2
1
bcd
2
abcd
2
2
a2bc
2
b2ce
0
ab2cde
0
a2b2cd
2e
0
c2de
2
ac2d
2e
2
a2c2e
2
bc2e2
0
abc
2de2
0
a2bc
2d
2e2
0
b2c2d
2
1
ab2c2
1
a2b2c2d
1
To�ndthethreeblocks,wecouldalsosolvetheequations(modulo3):
Generatorer:
Blocks=BC
D=AB
2C
2
E=BC
2
Block0:
j+k=0
i+2j+2k+2l=0
j+2k+2m=0
Block1:
j+k=1
i+2j+2k+2l=0
j+2k+2m=0
Block2:
j+k=2
i+2j+2k+2l=0
j+2k+2m=0
Forexample2solutionshavetobefoundfor"Block0"whichconsistsof3�3=9single
experiments,andafterthatonefurthersolutionforeachoftheothertwoblocks.
Thestructureofblock0canbeillustrated
(1)
u
u2
v
uv
u2v
v2
uv
2
u2v
2
whereuandvrepresentsolutionstotheequationsforblock0.
Forexample,withi=1andj=0,itisfoundfromj+k=0,thatk=0.Further,
i+2j+2k+2l=0indicatesthatl=1,andfromj+2k+2m=0isfoundthatm=0.
Asolutionistherebyu=ad.
Withi=0,j=1,itisfoundthatk=2,l=0andm=2,fromwhichv=bc
2e2isfound.
c hs.
DesignofExperiments,Course02411,IMM,DTU
108
(1)
u=ad
u2=a
2d
2
v=bc
2e2
uv=abc
2de2
u2v=a
2bc
2d
2e2
v2=b2ce
uv
2=ab2cde
u2v
2=a
2b2cd
2e
Anditcanbeseenthatthisispreciselytheblock0foundabove.
To�ndblock1,onesolutionisderivedforj+k=1,i+2j+2k+2l=0andj+2k+2m=0.
Suchasolutionisi=0,j=0,k=1,fromwhichl=2andm=2,correspondingtothe
singleexperimentcd
2e2.
By"multiplying"cd
2e2onthealreadyfoundblock0,block1isformed.Tryityourself.
Block2isfoundbysolvingtheequationsj+k=2,i+2j+2k+2l=0andj+2k+2m=0.
Asolutionisi=0,j=1,k=1,fromwhichl=1andm=0,correspondingtothesingle
experimentbcd.Thissolution"ismultiplied"onblock0,bywhichblock2appears.
Whentheexperimentisanalysed,theblocke�ectisre ectedintheBCe�ecttogether
withtheothere�ectswithwhichBCisconfounded.Inotherwords,theexperimentis
againanalysedonthebasisoftheunderlyingfactorstructuredeterminedbythefactors
A,BandC.
Finallytheexperimentcouldalsobeconstructeddirectlyonthebasisofthegenerators
thatarechosen
I A B AB
AB2
C AC
BC
=
Blocks
ABC
AB2C
AC2
BC2
=
E
ABC2
AB2C2
=
D
andcalculatingthefactorlevelsandblocknumbersasshowninthefollowingtableby
meansofthetabularmethod:
c hs.
DesignofExperiments,Course02411,IMM,DTU
109
Experimentaldesign
D=(A+2B+2C) 3,E=(B+2C) 3
andBlock=(B+C) 3
No.
A
B
C
D
E
Block
Experiment
1
0
0
0
0
0
0
(1)
2
1
0
0
1
0
0
ad
3
2
0
0
2
0
0
a2d2
4
0
1
0
2
1
1
bd2e
5
1
1
0
0
1
1
abe
6
2
1
0
1
1
1
a2bde
7
0
2
0
1
2
2
b2de2
8
1
2
0
2
2
2
ab2d2e2
9
2
2
0
0
2
2
a2b2e2
10
0
0
1
2
2
1
cd2e2
11
1
0
1
0
2
1
ace2
12
2
0
1
1
2
1
a2cde2
13
0
1
1
1
0
2
bcd
14
1
1
1
2
0
2
abcd2
15
2
1
1
0
0
2
a2bc
16
0
2
1
0
1
0
b2ce
17
1
2
1
1
1
0
ab2cde
18
2
2
1
2
1
0
a2b2cd2e
19
0
0
2
1
1
2
c2de
20
1
0
2
2
1
2
ac2d2e
21
2
0
2
0
1
2
a2c2e
22
0
1
2
0
2
0
bc2e2
23
1
1
2
1
2
0
abc2de2
24
2
1
2
2
2
0
a2bc2d2e2
25
0
2
2
2
0
1
b2c2d2
26
1
2
2
0
0
1
ab2c2
27
2
2
2
1
0
1
a2b2c2d
Endofexample3.23
Finallytwoexamplesaregiventhatillustratethepracticalprocedureintheconstruction
oftworesolutionIVexperimentsfor8and7factorsrespectively.Theseexperimentsare
ofgreatpracticalrelevance,sincetheyincluderelativelymanyfactorsinrelativelyfew
singleexperiments,namelyonly16.Atthesametime,theexamplesshowdivisioninto2
and4blocks,enablingtheadvantagessuchblockingcanhave.
Example3.24:
A
2�
4
�2
8
factorialin2blocks
Theexperimentcouldbedoneinconnectionwithastudyofthemanufacturingprocess
foradrug,forexample.
Weimaginethatthegivenfactorsandtheirlevelsarecircumstanceswhich,duringman-
ufacture,onenormallyaimstokeepconstant,oratleastwithingivenlimits.Itisthe
e�ectofvariationwithinthesepermittedlimitsthatwewanttostudy.
Eightfactorsarestudiedina2�
4�28factorialintwoblocks.The8factorsare2waiting
c hs.
DesignofExperiments,Course02411,IMM,DTU
110
timesduringtwophasesoftheprocess,3temperatures,2pHvaluesandthecontentof
zincinthe�nishedproduct.ThefactorsareorderedsothatfactorAisconsideredthe
mostimportant,whileBisthenextmostimportantetc.
TheexperimentisaresolutionIVexperiment.Undertheassumptionofnegligiblethird
orderinteractions,allmaine�ectscanbeanalysedinthisdesign.
Theexperimentisrandomisedwithintwoblocks,asitisassumedthatitisdoneintwo
facilities(R0
andR1)inparallelexperimentsincompletelyrandomorder.
Theexperimentisconstructedasgiveninthefollowingtables.
Factorsandlevelschosen
Factor
Lowlevel
Highlevel
A:Time Solution1+�ltering
(.)70+30min
(a)30+70min
B:TempMix1
(.)20�1
Æ
C
(b)27�1
Æ
C
C:Time Solution2
(.)30min
(c)100min
D:TempSolution2
(.)5�1
Æ
C
(d)17�1
Æ
C
E:Tempproces
(.)5�1
Æ
C
(e)17�1
Æ
C
F:pHrawproduct1
(.)2.65�0.02
(f)3.25�0.02
G:Zink�nalmix
(.)20.0�g/ml
(g)26.0�g/ml
H:pH�nalmix
(.)7.20�0.02
(h)7.40�0.02
Confoundings
I A B AB C A
CBC
ABC
=
H
D AD
BD
ABD
=
G
CD
ACD
=
F
BCD
=
E
ABCD
=
Blocks
Weusethetabularmethodforcalculatingthelevelsofthefactorsandtheblocknumber
onthebasisoftheunderlyingcompletefactorstructureconsistingoffactorsA,B,Cand
D,asshowninthefollowingtable:
c hs.
DesignofExperiments,Course02411,IMM,DTU
111
Experimentaldesign
E=(B+C+D) 2,F=(A+C+D) 2,G=(A+B+D) 2,H=(A+B+C) 2
andFacility=(A+B+C+D) 2
No.
A
B
C
D
E
F
G
H
Experiment
Facility
Randomis.
1
0
0
0
0
0
0
0
0
(1)
R0
9
2
1
0
0
0
0
1
1
1
afgh
R1
4
3
0
1
0
0
1
0
1
1
begh
R1
6
4
1
1
0
0
1
1
0
0
abef
R0
11
5
0
0
1
0
1
1
0
1
cefh
R1
16
6
1
0
1
0
1
0
1
0
aceg
R0
7
7
0
1
1
0
0
1
1
0
bcfg
R0
5
8
1
1
1
0
0
0
0
1
abch
R1
10
9
0
0
0
1
1
1
1
0
defg
R1
12
10
1
0
0
1
1
0
0
1
adeh
R0
3
11
0
1
0
1
0
1
0
1
bdfh
R0
1
12
1
1
0
1
0
0
1
0
abdg
R1
14
13
0
0
1
1
0
0
1
1
cdgh
R0
13
14
1
0
1
1
0
1
0
0
acdf
R1
2
15
0
1
1
1
1
0
0
0
bcde
R1
8
16
1
1
1
1
1
1
1
1
abcdefgh
R0
15
Prescriptionsforthesingleexperiments
Belowareshownthefactorsettingsforthetwo�rstsingleexperimentsandthetwolast
ones.
CarriedoutonfacilityR0
Testnr.FF{1X19
Experiment=bd(fh)
Procesparameter
Levelinexperiment
A:Time Solution1+�ltering
(.)70+30min
B:TempMix1
(b)27�1
Æ
C
C:Time Solution2
(.)30min
D:TempSolution2
(d)17�1
Æ
C
E:Tempproces
(.)5�1
Æ
C
F:pHrawproduct1
(f)3.25�0.02
G:Zink�nalmix
(.)20.0�g/ml
H:pH�nalmix
(h)7.40�0.02
c hs.
DesignofExperiments,Course02411,IMM,DTU
112
CarriedoutonfacilityR1
Testnr.FF{2X19
Experiment=acd(f)
Procesparameter
Levelinexperiment
A:Time Solution1+�ltering
(a)30+70min
B:TempMix1
(.)20�1
Æ
C
C:Time Solution2
(c)100min
D:TempSolution2
(d)17�1
Æ
C
E:Tempproces
(.)5�1
Æ
C
F:pHrawproduct1
(f)3.25�0.02
G:Zink�nalmix
(.)20.0�g/ml
H:pH�nalmix
(.)7.20�0.02
CarriedoutonfacilityR0
Testnr.FF{15X19
Experiment=abcd(efgh)
Procesparameter
Levelinexperiment
A:Time Solution1+�ltering
(a)30+70min
B:TempMix1
(b)27�1
Æ
C
C:Time Solution2
(c)100min
D:TempSolution2
(d)17�1
Æ
C
E:Tempproces
(e)17�1
Æ
C
F:pHrawproduct1
(f)3.25�0.02
G:Zink�nalmix
(g)26.0�g/ml
H:pH�nalmix
(h)7.40�0.02
CarriedoutonfacilityR1
Testnr.FF{16X19
Experiment=c(efh)
Procesparameter
Levelinexperiment
A:Time Solution1+�ltering
(.)70+30min
B:TempMix1
(.)20�1
Æ
C
C:Time Solution2
(c)100min
D:TempSolution2
(.)5�1
Æ
C
E:Tempproces
(e)17�1
Æ
C
F:pHrawproduct1
(f)3.25�0.02
G:Zink�nalmix
(.)20.0�g/ml
H:pH�nalmix
(h)7.40�0.02
Endofexample3.24
c hs.
DesignofExperiments,Course02411,IMM,DTU
113
Example3.25:
A
2�
3
�2
7
factorialexperimentin4blocks
Supposethatthereare7factorswhichonewantsstudiedin16singleexperiments.The
�rstfourfactors,A,B,CandDareusedastheunderlyingfactorstructure.Thefactors
E,FandGareputintothisaccordingtothetablebelowinaresolutionIVexperiment.
Atthesametime,onecouldwantthe16singleexperimentsdonein4blocksof4single
experiments.Since4=2�2blockshavetobeused,2de�ningequationsforblockshave
tobechosen.Asuggestionfortheconstructionoftheexperimentaldesigncouldbe:
Generators
I A B AB C A
CBC
ABC
=
blocks
D AD
BD
ABD
=
G
CD
ACD
=
F
BCD
=
E
ABCD
=
blocks
Withthesechoices,thee�ectsABCandABCD,butalsothee�ectABC�ABCDwillbe
confoundedwithblocks.Now,sinceABC�ABCD=D,thisisnotagoodchoice,because
themaine�ectDisobviouslyconfoundedwithblocks.Abetterchoicecouldbe:
Generators
I A B AB C A
CBC
ABC
=
blocks
D AD
BD
ABD
=
G
CD
ACD
=
F
BCD
=
blocks
ABCD
=
E
c hs.
DesignofExperiments,Course02411,IMM,DTU
114
ThischoicewillentailthatABC,BCDandABC�BCD=ABwillbeconfoundedwith
blocks.Theexperimentaldesigncanbewrittenoutusingthetabularmethod:
Design
E=(A+B+C+D) 2,F=(A+C+D) 2,G=(A+B+D) 2
andBlock=(A+B+C) 2+2�(B+C+D) 2
Nr
A
B
C
D
E
F
G
Experiment
Block
1
0
0
0
0
0
0
0
(1)
0
2
1
0
0
0
1
1
1
aefg
1
3
0
1
0
0
1
0
1
beg
3
4
1
1
0
0
0
1
0
abf
2
5
0
0
1
0
1
1
0
cef
3
6
1
0
1
0
0
0
1
acg
2
7
0
1
1
0
0
1
1
bcfg
0
8
1
1
1
0
1
0
0
abce
1
9
0
0
0
1
1
1
1
defg
2
10
1
0
0
1
0
0
0
ad
3
11
0
1
0
1
0
1
0
bdf
1
12
1
1
0
1
1
0
1
abdeg
0
13
0
0
1
1
0
0
1
cdg
1
14
1
0
1
1
1
1
0
acdef
0
15
0
1
1
1
1
0
0
bcde
2
16
1
1
1
1
0
1
1
abcdfg
3
Endofexample3.25
c hs.
DesignofExperiments,Course02411,IMM,DTU
115
Index
23factorialdesign,16
2kfactorialexperiment,19
3kfactorial,48
aliasrelation,37,41,88,101
confounding,37
generally,95
reduced,94,107
sign,39
balance,21,22,35,78
block,16,70
confounding,20,25,65
construction,25,74,105
e�ect,16,33
level,26
minimal,27
randomisation,48,111
system,26
complementaryexperiment,38,94
completefactorial,9
con�denceinterval,20,36
confounding,24,33,37
block,20,65,70
partial,28,78
contrast,12,13
de�ning,25,65
orthogonal,23
de�ningcontrast,25,37,44,65,73,95
de�ningrelation,25,37,41,70,89
design,9,21
e�ects 2k
experiments,11
generally,11
estimates,12
factore�ects,10
factorialexperiment
2k,9,19
3k,48
pk,48
factors,9
fractional
design,34
factorial,34,86
factorialinblocks,105
generatorequation,39,89,95
Kempthorne,43,48,57
Latinsquare,50,63
modulo2,26,37
modulo3,50
orthogonal,31
contrast,23
pkfactorial,48
parameters,10
partialconfounding,28,78
power(test),32
prime,48,60
principalblock,25,74
principalfraction,37,40,102
pseudocontrast,12
randomisation,36
block,48,111
repetition,16
replication,15
resolution,90
response,9{11
SASexamples,104
signof al
iasrelation,39
standardnotation,11
standardorder,17,62
standardisation,62
tabularmethod,25,40,67,75,87,98,
109,115
underlyingfactorial,39,42,90,92,101
116
Weighingexperiment,34
Yates'algorithm,14,18,42,63
c hs.
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117