lecture 6_treatment structure 1 factorial and nested new

7/26/2019 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,..+

lecture 6_treatment structure 1 factorial and nested new

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