LECTURE 6_Treatment Structure 1 Factorial and Nested New
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Transcript of LECTURE 6_Treatment Structure 1 Factorial and Nested New
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Treatment structures
How the treatments arearranged or combined in the
design structure
Presentation 5
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Recapitulation
So far our discussion on CRD, RCBD and LT!"S#$R% ha&e been limited to '"% treatmentfactor (but with a few le&els) onl*+
e+g+ ariet*- ., /, 0
1rom now on we will consider more than onetreatment factor (use T2'), each with a few
le&els+
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2hat to do if there are more than onefactor3
How are we going to arrange the factors (and 4tinto the CRD and RCBD designs)3
Treatmentstructures
Designstructures
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!+ 1actorial arrangement
factorial eperiment6 treatments consist ofall possiblecombinations of the
le&els of se&eralfactors
$seful in eplorator*wor7 when little is7nown about theoptimum le&els of thefactors
1actor
. /
1actorB
B.A1B1
A2B1
B/A1B2
A2B2
%ample of a/8/
factorial
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%amples
The e9ect of temperature (/5 and 55 C) and altitude (.:; ft,0: ;) on the current
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d&antages of multifactor studieso&er one factor at a time approach
%>cienc*
ar* the temp (/5 ? 55C), 7eep altitude constant (.:;)+@et results, repeat eperiment to estimate error&ariabilit*+
Do the same at 0:; n alternati&e, use factorial arrangement combining
temp+ and altitude+
The amount of eperimentation is less when use factorial+
mount of !nformation Readil* in&estigate the Aoint e9ects or interaction+
2hen the factors interact, factorial eperiments can
estimate the interaction+ 'neatattime eperiments cannot estimate interaction+
$se of oneatatime eperiments in the presence ofinteraction can lead to serious misunderstanding of howthe response &aries as a function of the factors+
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alidit* of 1inding
part from being more e>cient and readil* pro&idinginformation on interaction e9ects, multifactor studiesalso strengthen the &alidit* of results+
e+g+The e9ects of selling price (R= .:, /:, 0:) and t*peof ad&ertising campaign ("ewspaper, Radio) on sales ofa product+
primar* interest is e9ect of price on sales+!f onl* used newspaper ad&ertising, doubts might eistwhether or not the price e9ect di9ers for otherad&ertising campaigns+ B* using other t*pe ofad&ertising, management can get info on thepersistence of price e9ect with di9erent promotionalcampaign+
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Caution
The ad&antages of multifactor eperiment Austdescribed should not be mista7en that the morefactors are included in a stud*, the better+
%periments in&ol&ing man* factors with each atnumerous le&els become more comple tointerpret, costl* and timeconsuming+
The better strateg* is to begin with onl* a fewfactors (the important ones), then etend thestud* in accordance with the results obtainedthereof+
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1actorial rrangement in CRD
To stud* the in
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To stud* the in
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Data
.B. .B/ /B. /B/ Total
E+50 .F+50 0G+. 0/+:
/:+50 /.+:F /I+/ /0+E
./+50 /:+E: 0.+00 /E+EF
.+:: .F+00 5+E /5+:I
.:+E: /:+:F :+/ /G+00
J II+0G GI+E .E/+IF .0G+:I E+G/
J/ GI0+EE .EEF+:/ IG.0+I0 0G./+.F .0IFI+F
.0+/E .G+0I 0I+50 /F+E. /+/5Y
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Statistical linear model
Recall basic CRD with one treatment factor
Yij= + ti + eij
1or factorial with / factors in CRD, treatment nowconsists of two factors
Yij= + ti + eij
Yijk= + i+ j+ ij+ eijk
i = the e9ect of ithfactor
j = the r9ect ofjthfactor B
ij = the interaction of ithfactor andjthfactor B
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Statistical =odel
Yijk= + i+ j +ij+ ijk
where i= 1,,a; j=1,..,b; k=1,...,r
Computation (Basic CRD)
C1 Y2/ rab E/K/:SST' Y2ijk CF .0IFI+F ..F5F+0 .G.G+0
SSTR Y2ij. /r CF (II+0/ ++.0G+:/)K5 6 ..F5F+0 .50G+SS% b* di9erence .G.G+0 6 .50G+ 0FG+G
Decomposition of SSTR in factorial into its components
SSTR is made up of two components, SSTR SS SSB SSB
SS Y2i../rb CF (.I0+./ 0/.+F0/) K.: 6 C1 ./5I+FSSB Y2.j./ra CF (/G+:/ /05+E/) K.: 6 C1 E+FSSB SSTR 6 SS 6 SSB .50G+ ./5I+F E+F /F0+G
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"'
basic no&a table for CRD
Source df SSTreatment ab. 0 .50G+%rror (r.)ab.I 0FG+G
Total rab..G .G.G+0
no&a Table =odi4ed for 1actorial rrangement in CRDBrea7down of treatment e9ectsSource df SS =S 1
Time a.. ./5I+F ./5I+F 50MHormone b.. E+F E+F N.
Time Horm+ (a.)(b.). /F0+G /F0+G ..+5M%rror (r.)ab.I 0FG+G /0+F
Total rab. .G.G+0
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More on Interaction
=odelJiA7 u i BA (MB)iA eiA7
Test h*pothesis
Ho- Test whether or not main e9ects arepresentHo- Test whether or not main e9ects B arepresent
Ho- Test whether or not the two factors interact
To illustrate the meaning of the model elements,consider a simple twofactor stud* on the e9ectsof S%8 and @% on L%R"!"@ of a tas7+
S%8- male (.) and female (/)
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Interaction plot : To chec7 whether the factorsare or are are not independent+
"o interaction
AGE and SEX effects, with no
interactions
Graphical presentation
The curves of mean responses for
the different levels of a factor are
parallel.
AGE effect but no SEX effect, with nointeractions
Graphic presentation
The zero slope of each curve indicates that
SEX has no effect.
The differences in heihts of the curves
show the AGE effects.
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!nteraction present
AGE and SEX effects, with important
interactions
Graphical presentation
The treatment mean curves for the two
SEXES are not parallel
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!actorial with one observation per cell
Yij= + i+ j+ eij
!actorial with more than one observations per
cell "have replicates#
Yijk= + i+ j+ ij+ eijk
. / 0
B. J.. J/. J0.
B/ J./ J// J0/
. / 0
B. J...J../
J/..J/./
J0..J0./
B/ J./.J.//
J//.J///
J0/.J0//
"ote- =odel summar*
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1actorial arrangement in RCBD
Oust as factorial eperiment can be done inCRD when eperimental units or setting arehomogeneous, it can also be carried out
using RCBD when the units of eperimentare not homogeneous (the bloc7ing criteriaha&e been discussed pre&iousl*)+
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simple eample of // factorial inRCBD
Consider an eperiment to stud* the e9ects oftwoconcentration le&els of a substrate (factor )and tworeaction temperatures (factor B) on the*ield of a chemical product+
%ach of these four combinations is to be run inrandom order on each of three da*s+
The primar* interest is to loo7 at the e9ects ofconcentrations and reaction temperatures+
Howe&er since onl* four reactions (eperiments)can be done in a gi&en da*, the etraneous e9ectof da*s will be bloc7ed+
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La*out and Data
DJ.
.
B.
/
B.
/
B/
.
B/
DJ/
/
B.
.
B.
/
B/
.
B/
DJ0
.
B/
/
B/
.
B.
/
B.
.B. .B/ /B. /B/
DJ. .. .F .. /. 60
DJ/ I .0 .: /I 55
DJ0 I I .: ./ 34
23 36 31 59 149
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Basi !CB"
Yi#= + i+ #+ i#
where i= ith b$%k e&et
#re'rese(ts treat#e(ts a() i* the treat#e(ts are*%r#e) b tw% *at%r *at%ria$ the(
# =j+ k+ jk
s% the %#'$ete #%)e$ *%r 2*at%r F-C!0- i(!CB"
Yijk= + i+ j+ k+ jk+ ijk
Statistical linear model
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Construction of "'
Proceed as usual Compute
C1
SSBloc7
SSTR
SST'
SS% b* di9erence
SST' SSBloc7 SSTR SS%
Basic Anova for RCBD
Source df SS
Bloc7 r. SSBloc7
Trtt. SSTR
%rror (r.)(t.) SS%Total rt. SST'
Anova for factorial in RCBD
Bloc7 bloc7. SSBloc7
a.B b. SSTR SS SSB
B (a.)(b.) SSB
%rror (r.)(ab.) SS%
Total rab. SSTR
tab
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Computations for our eample
C1 .G/K./ .E5:+:E
SST' ../ .F/ + .//6 C1 .E+G/ SSBloc7 (I:/ 55/ 0/)K 6 C1 G5+.F
SS Q(.)/ (/)/KI 6 C1 (5G/ G:/)KI 6 C1 E:+:E
SSB Q(B.)/ (B/)/KI 6 C1 (5/ G5/)KI 6 C1 .:+:E
SSB Q(.B.)/ (.B/)/ (/B.)/(/B/)/K0 6 C1 6 SS 6SSB
SSTR 6 SSSSB (/0/ 0I/0./5G/)K0 C1 6SS 6 SSB .E+F5
SS% SST' 6 SSBloc7 6 SS 6 SSB 6 SSB E+E0
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"' Table
Source D1 SS =S 1 PM
Bloc7 / G5+.F F+5E 0+0F:+.:5
. E:+:E E:+:E 5+II :+:55
B . .:+:E .:+:E G+G. :+:/:
MB . .E+F5 .E+F5 .+00:+/G0
%rror I E+E0 .+.
Total .. .E+G/M P &alues are generated b*computer
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!!+ "ested Structure
Distinction beteenneste! an! factorial
1actorial or crossed factors
%&er* le&el of one factorappears with each le&el of
e&er* other factor
B. B/ B0 B
. U U
/ U U
B. B/ B0 B
. U U U U
/ U U U U
Certain le&els of a factoroccurs onl* with one le&el
of the other factor i+e+, B.and B/ occurs onl* with .+B0 and B occurs onl* with/+
B is said to be nested within
factor
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%ample ? "ested model
!nstructor (B) withinschool ()
. /0
B. B/ B0 B B5BI
@eneral model for twofactors nested
Let JiA7denote kthobser&ation for ith le&el andjth le&el of B
Yijk= + i+ 3 j2i3+ k2ij3
Pots within Species
Sp. Sp/Sp0
Plants within pots
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2or7ed eample
School i !nstructor A
Class 7 !ns Total SchTotal
Sch. !ns . /5 /G 5 FG
!ns / . .. /5
Sch / !ns 0 .. I .F 5F!ns // .E :
Sch 0 !ns 5 .F /: 0F
!ns I 5 / F
Three regional schools for mechanics+ The school ha&etwo instructors each who teaches classes of .:mechanics in a 0wee7 sessions+ Classes are randoml*assigned to instructors in the school+ This was done for /sessions+ J was a suitable measure of learning+
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computation
Compute SST'
/5/ /G/ +//6 .E:/K./ FII
Compute SS for factor , SS
(FG/ 5F/ /)K .E:/K./ .5I+5
To determine SSB(), consider each school separatel*+Compute
Sch .- (5/ /5/)K/ FG/K /.:+/5
Sch /- (.F/ :/)K/ 5F/K .0/+/5
Sch 0 - (0F/ F/)K/ /K //5+:Total SSB() 5IF+5
SS% b* subtraction
SS% SST' 6 SS 6 SSB() /
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"' Table
Source D1 SS =S 1
School / (school 0.) .5I+5 FE+/5FE+/5K.EG+.F :+.
!ns(sch) 0 (/.)(.0) 5IF+5 .EG+.F.EG+.FKF /F+:/
%rror /0I /+: F+::
Total (/I)... FII+:
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Relation between 1actorial or crossedwith nested SS
Suppose for some reason (computer pac7age una&ailabilit*) orpurel* a mista7e, *ou anal*sed the nested eperiment as afactorial eperiment+
Source D1 SS =S
sch / .5I+5 FE+/5inst . .:E+: .:E+:
schMinst / 5G+5 //G+F5
%rror I /+: F+::
Total .. FII+: =SSSKD1
But nested anal*ses do not ha&e interaction SS, so use therelationship
SSB() SSB SSB to get the correct nested "'
SSB() .:E+: 5G+5 5IF+5+ Similarl* for D1/,0,I,..+