DOE Course - Parts 1-4o

7/24/2019 DOE Course - Parts 1-4o

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L. M. Lye 1

Design and Analysis of

Multi-Factored ExperimentsEngineering 9516

Dr. Leonard M. Lye !.Eng F"#"E FE"!rofessor and Associate Dean $%raduate #tudies&

Faculty of Engineering and Applied #cience Memorial 'ni(ersity of)e*foundland

#t. +o,ns )L A1 /05


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L. M. Lye

D2E - 3

3ntroduction


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L. M. Lye /

Design of Engineering Experiments

Introduction

4 %oals of t,e course and assumptions

4 An are(iated historyof D2E

4 ,e strategyof experimentation

4 #ome asic principlesand terminology

4 Guidelinesfor planning conducting and

analy7ing experiments


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L. M. Lye 8

Assumptions

4 ou ,a(e: a first course in statistics

: ,eard of t,e normal distriution

: ;no* aout t,e mean and (ariance

: ,a(e done some regression analysis or ,eard of it

: ;no* somet,ing aout A)2indo*s or Mac ased computers

4 =a(e done or *ill e conducting experiments

4 =a(e not ,eard of factorial designs fractional

factorial designs ?#M and DA"E.


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L. M. Lye 5

#ome ma@or players in D2E

4 #ir ?onald A. Fis,er - pioneer: in(ented A)2. %. =unter ates

Montgomery Finney etc..


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L. M. Lye 6

Four eras of D2E4 ,e agriculturalorigins 191 : 198Bs

: ?. A. Fis,er C ,is co-*or;ers: !rofound impact on agricultural science

: Factorial designs A)2ilson response surfaces

: Applications in t,e c,emical C process industries4 ,e second industrialera late 19Bs : 199B

: uality impro(ement initiati(es in many companies

: aguc,i and roust parameter design process roustness

4 ,e modernera eginning circa 199B

: >ide use of computer tec,nology in D2E

: Expanded use of D2E in #ix-#igma and in usiness

: 'se of D2E in computer experiments


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L. M. Lye

?eferences

4 D. %. Montgomery $BB& Design and Analysis of

Experiments t, Edition +o,n >iley and #ons: one of t,e est oo; in t,e mar;et. 'ses Design-Expert

soft*are for illustrations. 'ses letters for Factors.

4 %. E. !. ox >. %. =unter and +. #. =unter $BB5&

#tatistics for Experimenters An 3ntroduction to

Design Data Analysis and Model uilding +o,n

>iley and #ons. ndEdition

: "lassic text *it, lots of examples. )o computer aided

solutions. 'ses numers for Factors.

4 +ournal of uality ec,nology ec,nometrics

American #tatistician discipline specific @ournals


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L. M. Lye

3ntroduction >,at is meant y D2EG

4 Experiment -

: a test or a series of tests in *,ic, purposeful c,anges aremade to t,e input variables or factorsof a system so t,at

*e may oser(e and identify t,e reasons for c,anges in

t,e outputresponse$s&.

4 uestion 5 factors and response (ariales: >ant to ;no* t,e effect of eac, factor on t,e response

and ,o* t,e factors may interact *it, eac, ot,er

: >ant to predict t,e responses for gi(en le(els of t,e

factors: >ant to find t,e le(els of t,e factors t,at optimi7es t,e

responses - e.g. maximi7e 1ut minimi7e

: ime and udget allocated for /B test runs only.


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L. M. Lye 9

#trategy of Experimentation

4#trategy of experimentation: est guess approac, $trial and error&

4 can continue indefinitely

4 cannot guarantee est solution ,as een found

: 2ne-factor-at-a-time $2FA& approac,4 inefficient $reHuires many test runs&

4 fails to consider any possile interaction et*een factors

: Factorial approac, $in(ented in t,e 19Bs&

4 Factors (aried toget,er

4 "orrect modern and most efficient approac,

4 "an determine ,o* factors interact

4 'sed extensi(ely in industrial ? and D and for process

impro(ement.


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L. M. Lye 1B

4 ,is course *ill focus on t,ree (ery useful and

important classes of factorial designs

: -le(el full factorial $;&: fractional factorial $;-p& and

: response surface met,odology $?#M&

4 3 *ill also co(er split plot designs and design and analysis of computer

experiments if time permits.4 Dimensional analysis and ,o* it can e comined *it, D2E *ill also e

riefly co(ered.

4 All D2E are ased on t,e same statistical principles

and met,od of analysis - A)2


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L. M. Lye 11

#tatistical Design of Experiments

4 All experiments s,ould e designed experiments

4 'nfortunately some experiments are poorly

designed - (aluale resources are used

ineffecti(ely and results inconclusi(e4 #tatistically designed experiments permit

efficiency and economy and t,e use of statistical

met,ods in examining t,e data result in scientific

o@ecti(ity *,en dra*ing conclusions.


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4 D2E is a met,odology for systematically applying

statistics to experimentation.

4 D2E lets experimenters de(elop a mat,ematical

model t,at predicts ,o* input (ariales interactto

create output (ariales or responses in a process or

system.4 D2E can e used for a *ide range of experiments

for (arious purposes including nearly all fields of

engineering and e(en in usiness mar;eting.

4 'se of statistics is (ery important in D2E and t,easics are co(ered in a first course in an

engineering program.


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4 3n general y using D2E *e can

:Learn aout t,e process *e are in(estigating:#creen important (ariales

:uild a mat,ematical model

:2tain prediction eHuations:2ptimi7e t,e response $if reHuired&

4 #tatistical significance is tested using ANOVAand t,e prediction model is otained using

regression analysis.


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L. M. Lye 18

Applications of D2E in Engineering Design

4 Experiments are conducted in t,e field of

engineering to

: e(aluate and compare asic design configurations

: e(aluate different materials: select design parameters so t,at t,e design *ill *or;

*ell under a *ide (ariety of field conditions $roust

design&

: determine ;ey design parameters t,at impactperformance


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L. M. Lye 15

PROCESS:

A Blending of

Inputs which

Generates

Corresponding

Outputs

INPUS

!"actors#

$ %aria&les

OUPUS

!Responses#

' %aria&les

People

Materials

Equipment

Policies

Procedures

Methods

Environment

responses relatedto performing a

service

responses relatedto producing a

produce

responses relatedto completing a task

Illustration of a Process


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L. M. Lye 16

PROCESS:

(isco%ering

Opti)al

Concrete

*i+ture

INPUS

!"actors#

$ %aria&les

OUPUS

!Responses#

' %aria&les

Type of

cement

Percent water

Type of

Additives

Percent

Additives

Mixing Time

Curing

Conditions

% Plas ticier

compressivestrength

modulus of elasticity

modulus of rupture

Opti)u) Concrete *i+ture

Poisson!s ratio


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L. M. Lye 1

PROCESS:

*anufacturing

In,ection

*olded Parts

INPUS

!"actors#

$ %aria&les

OUPUS

!Responses#

' %aria&les

Type of "aw

Material

Mold

Temperature

#olding

Pressure

#olding Time

$ate ie

crew peed

Moisture

Content

thickness of moldedpart

% shrinkage frommold sie

num&er of defectiveparts

*anufacturing In,ection *olded

Parts


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L. M. Lye 1

PROCESS:

Rainfall-Runoff

*odel

Cali&ration

INPUS

!"actors#

$ %aria&les

OUPUS

!Responses#

' %aria&les

'nitial storage

(mm)

Coefficient of

'nfiltration

Coefficient of

"ecession

oil Moisture

Capacity(mm)

"*square+

Predicted vs,&served -its

*odel Cali&ration

'mpermea&le layer

(mm)

'nitial oil Moisture

(mm)


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L. M. Lye 19

PROCESS:

*a.ing the

Best

*icrowa%e

popcorn

INPUS

!"actors#

$ %aria&les

OUPUS

!Responses#

' %aria&les

.rand+

Cheap vs Costly

Time+

/ min vs 0 min

Power+

12% or 344%

#eight+

,n &ottom or raised

Taste+cale of 3 to 34

.ullets+$rams of unpopped

corns

*a.in )icrowa%e o corn


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L. M. Lye B

Examples of experiments from daily life

4 !,otograp,y

: Factors speed of film lig,ting s,utter speed

: ?esponse Huality of slides made close up *it, flas, attac,ment

4 oiling *ater

: Factors !an type urner si7e co(er

: ?esponse ime to oil *ater

4 D-day

: Factors ype of drin; numer of drin;s rate of drin;ing time

after last meal

:?esponse ime to get a steel all t,roug, a ma7e

4 Mailing

: Factors stamp area code time of day *,en letter mailed

: ?esponse )umer of days reHuired for letter to e deli(ered


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1

More examples

4 "oo;ing

: Factors amount of coo;ing *ine oyster sauce sesame oil

: ?esponse aste of ste*ed c,ic;en

4 #exual !leasure

: Factors mari@uana screec, sauna

: ?esponse !leasure experienced in suseHuent you ;no* *,at

4 as;etall

: Factors Distance from as;et type of s,ot location on floor

: ?esponse )umer of s,ots made $out of 1B& *it, as;etall

4 #;iing: Factors #;i type temperature type of *ax

: ?esponse ime to go do*n s;i slope


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asic !rinciples

4 #tatistical design of experiments $D2E&

:t,e process of planning experiments so t,at

appropriate data can e analy7ed y statisticalmet,ods t,at results in (alid o@ecti(e and

meaningful conclusions from t,e data

:in(ol(es t*o aspects design and statistical

analysis


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4 E(ery experiment in(ol(es a seHuence of

acti(ities:"on@ecture - ,ypot,esis t,at moti(ates t,e

experiment

:Experiment - t,e test performed to in(estigatet,e con@ecture

:Analysis - t,e statistical analysis of t,e data

from t,e experiment

:"onclusion - *,at ,as een learned aout t,e

original con@ecture from t,e experiment.


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L. M. Lye 8

,ree asic principles of #tatistical D2E

4 ?eplication

: allo*s an estimate of experimental error

: allo*s for a more precise estimate of t,e sample mean

(alue

4 ?andomi7ation: cornerstone of all statistical met,ods

: Ia(erage outJ effects of extraneous factors

: reduce ias and systematic errors

4 loc;ing

: increases precision of experiment

: Ifactor outJ (ariale not studied


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%uidelines for Designing Experiments

4 ?ecognition of and statement of t,e prolem: need to de(elop all ideas aout t,e o@ecti(es of t,e

experiment - get input from e(eryody - use team

approac,.

4 ",oice of factors le(els ranges and response

(ariales.

:)eed to use engineering @udgment or prior test results.

4 ",oice of experimental design: sample si7e replicates run order randomi7ation

soft*are to use design of data collection forms.


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4 !erforming t,e experiment

: (ital to monitor t,e process carefully. Easy to

underestimate logistical and planning aspects in acomplex ? and D en(ironment.

4 #tatistical analysis of data

:pro(ides o@ecti(e conclusions - use simple grap,ics

*,ene(er possile.

4 "onclusion and recommendations

: follo*-up test runs and confirmation testing to (alidate

t,e conclusions from t,e experiment.4 Do *e need to add or drop factors c,ange ranges

le(els ne* responses etc.. GGG


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'sing #tatistical ec,niHues in

Experimentation - t,ings to ;eep in mind

4 'se non-statistical ;no*ledge of t,e prolem:p,ysical la*s ac;ground ;no*ledge

4 Keep t,e design and analysis as simple as possile

: Dont use complex sop,isticated statistical tec,niHues: 3f design is good analysis is relati(ely straig,tfor*ard

: 3f design is ad - e(en t,e most complex and elegant

statistics cannot sa(e t,e situation

4 ?ecogni7e t,e difference et*een practical andstatistical significance

: statistical significance practically significance


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4 Experiments are usually iterati(e

: un*ise to design a compre,ensi(e experiment at t,estart of t,e study

: may need modification of factor le(els factors

responses etc.. - too early to ;no* *,et,er experiment

*ould *or;: use a seHuential or iterati(e approac,

: s,ould not in(est more t,an 5 of resources in t,e

initial design.

: 'se initial design as learning experiences to accomplis,t,e final o@ecti(es of t,e experiment.


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D2E $33&

Factorial (s 2FA


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Factorial (.s. 2FA

4 Factorial design - experimental trials or runs areperformed at all possile cominations of factor

le(els in contrast to 2FA experiments.

4 Factorial and fractional factorial experiments are

among t,e most useful multi-factor experiments

for engineering and scientific in(estigations.


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4 ,e aility to gain competiti(e ad(antage reHuiresextreme care in t,e design and conduct of

experiments. #pecial attention must e paid to @oint

effects and estimates of (ariaility t,at are pro(ided

y factorial experiments.

4 Full and fractional experiments can e conducted

using a (ariety of statistical designs. ,e design

selected can e c,osen according to specific

reHuirements and restrictions of t,e in(estigation.


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Factorial Designs

4 3n a factorial experiment allpossible combinationsoffactor le(els are tested

4 ,e golf experiment: ype of dri(er $o(er or regular&: ype of all $alata or /-piece&

: >al;ing (s. riding a cart

: ype of e(erage $eer (s *ater&

: ime of round $am or pm&: >eat,er

: ype of golf spi;e

: Etc etc etc


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Factorial Design


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Factorial Designs ith !e"eral Factors


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Erroneous 3mpressions Aout Factorial

Experiments

4 >asteful and do not compensate t,e extra effort *it,additional useful information - t,is fol;lore presumes t,at

one ;no*s $not assumes& t,at factors independently

influence t,e responses $i.e. t,ere are no factor

interactions& and t,at eac, factor ,as a linear effect on t,e

response - almost any reasonale type of experimentation

*ill identify optimum le(els of t,e factors

4 3nformation on t,e factor effects ecomes a(ailale only

after t,e entire experiment is completed. a;es too long.

Actually factorial experiments can e loc;ed and

conducted seHuentially so t,at data from eac, loc; can e

analy7ed as t,ey are otained.


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2ne-factor-at-a-time experiments $2FA&

4 2FA is a pre(alent ut potentially disastrous type ofexperimentation commonly used y many engineers and

scientists in ot, industry and academia.

4 ests are conducted y systematically c,anging t,e le(els

of one factor *,ile ,olding t,e le(els of all ot,er factorsfixed. ,e IoptimalJ le(el of t,e first factor is t,en

selected.

4 #useHuently eac, factor in turn is (aried and its

IoptimalJ le(el selected *,ile t,e ot,er factors are ,eldfixed.


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2ne-factor-at-a-time experiments $2FA&

4 2FA experiments are regarded as easier to implementmore easily understood and more economical t,an

factorial experiments. etter t,an trial and error.

4 2FA experiments are elie(ed to pro(ide t,e optimum

cominations of t,e factor le(els.4 'nfortunately eac, of t,ese presumptions can generally e

s,o*n to e false except under (ery special circumstances.

4 ,e ;ey reasons *,y 2FA s,ould not e conducted

except under (ery special circumstances are:Do not provide adequate information on interactions

:Do not provide efficient estimates of the effects


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Factorial (s 2FA $ -le(els only&

4 factors 8 runs

: / effects

4 / factors runs: effects

4 5 factors / or 16 runs

: /1 or 15 effects

4 factors 1 or 68 runs

: 1 or 6/ effects

4 factors 6 runs

: effects

4 / factors 16 runs: / effects

4 5 factors 96 runs

: 5 effects

4 factors 51 runs

: effects

Factorial 2FA


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Example Factorial (s 2FA

Factor A

lo* ,ig,

lo*

Factor

,ig,

E.g. Factor A ?eynolds numer Factor ;ND

,ig,

lo*

lo* ,ig,

A

2FAFactorial


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L. M. Lye 8B

Example Effect of ?e and ;ND on friction factor f4 "onsider a -le(el factorial design $&

4 ?eynolds numer O Factor AP ;ND O Factor

4 Le(els for A 1B8$lo*& 1B6$,ig,&

4 Le(els for B.BBB1 $lo*& B.BB1 $,ig,&

4 ?esponses $1& O B.B/11 a O B.B1/5 O B.B/

a O B.BBB

4 Effect $A& O -B.66 Effect $& O B. Effect $A& O B.1

4 contriution A O 8.5 O 9.8 A O 5.6

4 ,e presence of interactions implies t,at one cannot

satisfactorily descrie t,e effects of eac, factor using main

effects.


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L. M. Lye81

5E%'$6* EA%E Plot

7n(f)

8 9 A+"eynold!s:

; 9 . + k < 5

5esign Points

. * 4 = 4 4 4

.> 4 =443

k


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5E%'$6*EA%EPlot

7n(f)89A+ "eynold !s:;9.+ k


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L. M. Lye 8/

* / = ? 4 2 4 1

* / = 4 @ ? @

* ? = @ 0 B 1 B

* ? = 0 / 3 2 2

* ? = / B 4 ? @

7

n

(f)

/ = 4 4 4

/ = 2 4 4

2 = 4 4 4

2 = 2 4 4

0 = 4 4 4

4 = 4 4 4 3

4 = 4 4 4 ?

4 = 4 4 4 0

4 = 4 4 4 @

4 = 4 4 3 4

" e y n o l d !s :

k


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>it, t,e addition of a fe* more points

4 Augmenting t,e asic design *it, a center point

and 5 axial points *e get a central compositedesign $""D& and a nd order model can e fit.

4 ,e nonlinear nature of t,e relations,ip et*een

?e ;ND and t,e friction factor f can e seen.4 3f )i;uradse $19//& ,ad used a factorial design in

,is pipe friction experiments ,e *ould need far

less experimental runsQQ

4 3f t,e numer of factors can e reduced ydimensional analysis t,e prolem can e made

simpler for experimentation.


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L. M. Lye85

5E%'$6*E8PE"TPlot

7og34(f)

8 9 A+ "E

; 9 .+ k 4=443

. + k < 5

'n te ra c t io n $ ra p h

A + " E

7

o

g

3

4

(f)

/=B? /=0/0 2 =444 2 =?2/ 2=141

*3=1@/

*3=13B

*3=0?

*3=201

*3=/2

5E%'$6* E8PE" T P l o t

7og34(f)8 9 A + "E; 9 . + k


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L. M. Lye 86

* 3 = 1 @ ?

* 3 = 1 B 2

* 3 = 0 0 @

* 3 = 0 3 3

* 3 = 2 2 /

7

o

g

34

(f)

/ = B ? / = 0 / 0

2 = 4 4 4 2 = ? 2 /

2 = 1 4 1

4 = 4 4 4 ? 3 1 B

4 = 4 4 4 / 2 @ 0

4 = 4 4 4 0 4 4 4

4 = 4 4 4 1 / 3 /

4 = 4 4 4 @ @ B @

A + " E

. + k< 5


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5E'$6*E8PE"T Plot

7og34(f)5esignPoints

8 9A+"E; 9.+k


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L. M. Lye 8

5E%'$6*E8PE"T Plot

7og34(f)

Ac tu a l

Pr

e

d

icte

d

P red ic ted vs = A c tua l

*3=1@?

*3=133

*3=0?

*3=200

*3=//

* 3=1@? *3=133 * 3=0? * 3=200 * 3=//


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L. M. Lye 89

D2E $333&

asic "oncepts


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L. M. Lye 5B


#asic !tatistical $oncepts4 #imple comparati"eexperiments

: ,e ,ypot,esis testing frame*or;

: ,e t*o-sample t-test

: ",ec;ing assumptions (alidity

4 "omparing more t,an t*o factor le(elstheanalysis of "ariance: A)2


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L. M. Lye 51

%ortland $ement Formulation

1.1516.51B

1.9616.599

1.9B1.15

1.16.96

1.51.B86

1.616.55

1.BB16./58

1.51.1/

1.6/16.8B

1.5B16.51

'nmodified Mortar

$Formulation &

Modified Mortar

$Formulation 1&

2ser(ation

$sample&1! !


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L. M. Lye 5

Graphical Vie of the DataDot Diagram

-orm 3 -orm B

30=?

31=?

3@=?

5otplots of -orm 3 and -orm B(means are indicated &y lines)


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#ox %lots

-orm 3 -orm B

30=2

31=2

3@=2

.oxplots of -orm 3 and -orm B(means are indicated &y solid circles)


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L. M. Lye 58

&he 'ypothesis &esting Frameor(

4 !tatistical hypothesis testingis a useful

frame*or; for many experimental

situations4 2rigins of t,e met,odology date from t,e

early 19BBs

4 >e *ill use a procedure ;no*n as t,e to)sample t)test


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L. M. Lye 55

&he 'ypothesis &esting Frameor(

4 #ampling from a normaldistriution4 #tatistical ,ypot,eses

B 1

1 1

"

"

=


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L. M. Lye 56

Estimation of %arameters

1

1

1estimates t,e population mean

1$ & estimates t,e (ariance

1

n

i

i

n

i

i

! !n

# ! !n

=

=

=

=


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L. M. Lye 5

!ummary !tatistics

1

1

1

1

16.6

B.1BB

B./16

1B

!

#

#

n

=

=

=

=

1.9

B.B61

B.8

1B

!

#

#

n

=

=

=

=

Formulation *

+Ne recipe,

Formulation -

+Original recipe,


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L. M. Lye 5

'o the &o)!ample t)&est or(s/

1

y

'se t,e sample means to dra* inferences aout t,e population means

16.6 1.9 1.16

Difference in sample means

#tandard de(iation of t,e difference in sample means

,is suggests a statistic

! !

n

= =

=

1 B

1

1

R ! !

n n

=

+


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L. M. Lye 59


1 1

1

1

1

1

1 1

1

'se and to estimate and

,e pre(ious ratio ecomes

=o*e(er *e ,a(e t,e case *,ere

!ool t,e indi(idual sample (ariances

$ 1& $ 1&

p

# #

! !

# #

n n

n # n # #

n n

+

= =

+ =

+


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L. M. Lye 6B


4


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L. M. Lye 61

&he &o)!ample 0%ooled1 t)&est

1 1

1

1 B

1

$ 1& $ 1& 9$B.1BB& 9$B.B61&B.B1

1B 1B

B.8

16.D6 1D.9 9.1/

1 1 1 1B.8

1B 1B

,e t*o sample means are a5out 9 standard de(iations apart

3s t,is a Slar

p

p

p

n # n # #

n n

#

! !t

#n n

+ += = =

+ +

=

= = =

+ +

geS differenceG


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L. M. Lye 6


4 #o far *e ,a(ent really done any IstatisticsJ4 >e need an ob2ecti"easis for deciding ,o* large t,e test

statistic tB really is

4 3n 19B >. #. %osset deri(ed t,e referencedistribution

for tB called t,e tdistriution4 ales of t,e tdistriution - any stats text.

4 ,e t-distriution loo;s almost exactly li;e t,e normaldistriution except t,at it is s,orter and fatter *,en t,edegrees of freedom is less t,an aout 1BB.

4 eyond 1BB t,e t is practically t,e same as t,e normal.


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L. M. Lye 6/


4 A (alue of tBet*een

:.1B1 and .1B1 isconsistent *it,eHuality of means

4 3t is possile for t,e

means to e eHual andtBto exceed eit,er

.1B1 or :.1B1 ut it*ould e a Iraree"entJ leads to t,e

conclusion t,at t,emeans are different

4 "ould also use t,eP)"alueapproac,


64/116

L. M. Lye 68


4 ,eP-"alueis the ris( ofrongly re2ectingt,e null,ypot,esis of eHual means $it measures rareness of t,e e(ent&

4 ,e$-(alue in our prolem is$O B.BBBBBBB/


65/116

L. M. Lye 65

3initab &o)!ample t)&est 4esults

wo-Sa)ple -est and CI: "or) /0 "or) 1Two-sample T for Form 1 vs Form 2

N Mean StDev SE Mean

Form 1 10 16.764 0.316 0.10

Form 2 10 17.922 0.24 0.07

D!fferen"e # m$ Form 1 - m$ Form 2

Est!mate for %!fferen"e& -1.1'

9'( )* for %!fferen"e& +-1.42', -0.91

T-Test of %!fferen"e # 0 +vs not #& T-al$e # -9.11

/-al$e # 0.000 DF # 1

ot $se /oole% StDev # 0.24

$h (i A ti


66/116

L. M. Lye 66

$hec(ing Assumptions 5

&he Normal %robability %lot

-orm 3

-orm B

30=2 31=2 3@=2

3

2

34

B4

?4

/4

24

04

14

@4

4

2

5ata

Percent

A5

3=B4

3=?@1

$oodness of -it

Tension .ond trength 5ataM7 Estimates


67/116

L. M. Lye 6

Importance of the t)&est

4 !ro(ides an ob2ecti"eframe*or; for simple

comparati(e experiments

4 "ould e used to test all rele(ant ,ypot,esesin a t*o-le(el factorial design ecause all

of t,ese ,ypot,eses in(ol(e t,e mean

response at one IsideJ of t,e cue (ersus t,emean response at t,e opposite IsideJ of t,e

cue

h t If &h A 3 &h


68/116

L. M. Lye 6

hat If &here Are 3ore &han

&o Factor 6e"els74 ,e t-test does not directly apply

4 ,ere are lots of practical situations *,ere t,ere are eit,er

more t,an t*o le(els of interest or t,ere are se(eral factors of

simultaneous interest

4 ,e analysis of "ariance$A)2


69/116

L. M. Lye 69

An Example

4 "onsider an in(estigation into t,e formulation of a

ne* Isynt,eticJ fier t,at *ill e used to ma;e ropes4 ,e response (ariale is tensile strengt,

4 ,e experimenter *ants to determine t,e IestJ le(elof cotton $in *t & to comine *it, t,e synt,etics

4 "otton content can (ary et*een 1B : 8B *t P somenon-linearity in t,e response is anticipated

4 ,e experimenter c,ooses 5 le"elsof cotton

IcontentJP 15 B 5 /B and /5 *t 4 ,e experiment is replicated5 times : runs made in

random order

An Example


70/116

L. M. Lye B

An Example

4 Does changingt,e

cotton *eig,t percent

c,ange t,e mean

tensile strengt,G

4 3s t,ere an optimum

le(el for cotton

contentG

&h A l i f V i


71/116

L. M. Lye 1

&he Analysis of Variance

4 3n general t,ere *ill e ale"elsof t,e factor or atreatments8 andnreplicatesof t,e experiment run in randomorder9a completelyrandomi7ed design0$4D1

4 % = antotal runs4 >e consider t,e fixed effectscase only

4 2@ecti(e is to test ,ypot,eses aout t,e eHuality of t,e a treatmentmeans



72/116

L. M. Lye

&he Analysis of Variance4 ,e name Ianalysis of (arianceJ stems from a

partitioningof t,e total (ariaility in t,e response

(ariale into components t,at are consistent *it, a

modelfor t,e experiment

4 ,e asic single-factor A)2


73/116

L. M. Lye /

3odels for the Data

,ere are se(eral *ays to *rite a model for

t,e data

is called t,e effects model

Let t,en

is called t,e means model?egression models can also 5e employed

i i i

i i

i i i

!

!

= + +

= +

= +

&h A l i f V i


74/116

L. M. Lye 8


4 &otal "ariabilityis measured y t,e total sum of

sHuares

4 ,e asic A)2


75/116

L. M. Lye 5


4 A large (alue of ##)reatments reflects large differences in

treatment means4 A small (alue of ##)reatments li;ely indicates no differences in

treatment means

4 Formal statistical ,ypot,eses are

reatments * ## ## ## = +

B 1

1

F

F At least one mean is different

a"

"

= = =L



76/116

L. M. Lye 6

&he Analysis of Variance4 >,ile sums of sHuares cannot e directly compared to test t,e ,ypot,esis of eHual means mean s:uarescan e

compared.

4 A mean sHuare is a sum of sHuares di(ided y its degrees of freedom

4 3f t,e treatment means are eHual t,e treatment and error mean sHuares *ill e $t,eoretically& eHual.

4 3f treatment means differ t,e treatment mean sHuare *ill e larger t,an t,e error mean sHuare.

1 1 $ 1&

1 $ 1&

)otal )reatments *rror

)reatments * )reatments *

df df df

an a a n

## ## +# +#

a a n

= +

= +

= =

&he Analysis of Variance is


77/116

L. M. Lye

&he Analysis of Variance is

!ummari;ed in a &able

4 ,e reference distributionfor,B is t,e,a-1a$n-1&distriution4 4e2ectt,e null ,ypot,esis $eHual treatment means& if

B 1 $ 1&a a n, , >


78/116

L. M. Lye

ANOVA $omputer Output

0Design)Expert1

4esponse/!trength

ANOVA for !elected Factorial 3odel

Analysis of "ariance table F

Model 85.6 8 11.98 18.6 V B.BBB1

A 75.7/ 110. 1.7/ 3.3331

!ure Error161.B B .B6

"or otal 6/6.96 8

#td. De(. .8 ?-#Huared B.869

Mean 15.B8 Ad@ ?-#Huared B.696/

".


79/116

L. M. Lye 9

&he 4eference Distribution/

Graphical Vie of the 4esults


80/116

L. M. Lye B

Graphical Vie of the 4esults5E'$6*E8PE"T Plot

trength

8 9A+ CottonDeight %

5esignPoints

tre

n

g

th

, ne - a c to r P lo t

32 B4 B2 ?4 ?2

1

33=2

30

B4=2

B2

BB

BB

BB BB

BB BB

BB


81/116

L. M. Lye 1

3odel Ade:uacy $hec(ing in the ANOVA

4 $hec(ing assumptionsis important

4 )ormality

4 "onstant (ariance4 3ndependence

4 =a(e *e fit t,e rig,t modelG

4 Later *e *ill tal; aout *,at to do if someof t,ese assumptions are "iolated


82/116

L. M. Lye

3odel Ade:uacy $hec(ing in the ANOVA

4 Examination of residuals

4 Design-Expert generates

t,e residuals

4 4esidual plotsare (ery

useful

4 Normal probability plotof residuals

.

Wi i i

i i

e ! !

! !

=

=

5E%'$6*E8PE"T Plot

%trength

6

o

rm

a

l%

p

ro

&

a

&

ility

o rm a p o o re s ua s

* ?=@ * 3=22 4=1 B=2 2=B

3

2

34

B4

?4

24

14

@4

4

2

O h I 4 id l %l


83/116

L. M. Lye /

Other Important 4esidual %lots

BB

BB

BB

BB

BB

BB

BB

P re d ic te d

"

es

iduals

=

*?=@

*3 =22

4= 1

B=2

2= B

=@4 3B=12 3 2=14 3@=02 B3=04

5E'$6*E8PE"T Plottrength

" u n 6 u m & e r

"

es

iduals

*?=@

*3=22

4= 1

B=2

2= B

3 / 1 34 3? 30 3 BB B2


84/116

L. M. Lye 8

%ost)ANOVA $omparison of 3eans

4 ,e analysis of (ariance tests t,e ,ypot,esis of eHualtreatment means

4 Assume t,at residual analysis is satisfactory

4 3f t,at ,ypot,esis is re@ected *e dont ;no* hichspecific

meansare different4 Determining *,ic, specific means differ follo*ing an

A)2e *ill use pair*ise t-tests on meanssometimes calledFis,ers Least #ignificant Difference $or Fis,ers 6!D&Met,od

Design)Expert Output


85/116

L. M. Lye 5

Design Expert Output

&reatment 3eans 0Ad2usted8 If Necessary1

Estimated !tandard

3ean Error1-15 9.B 1.

-B 15.8B 1.

/-5 1.6B 1.

8-/B 1.6B 1.

5-/5 1B.B 1.

3ean !tandard t for '?&reatment Difference DF Error $oeff@? %rob > t

1 (s -5.6B 1 1.B -/.1 B.BB58

1 (s / -.B 1 1.B -8./8 B.BBB/

1 (s 8 -11.B 1 1.B -6.5 V B.BBB1

1 (s 5 -1.BB 1 1.B -B.56 B.5/

(s / -.B 1 1.B -1./ B./8

(s 8 -6.B 1 1.B -/.85 B.BB5

(s 5 8.6B 1 1.B .56 B.B16 / (s 8 -8.BB 1 1.B -./ B.B/5

/ (s 5 6.B 1 1.B /.9 B.BB1

8 (s 5 1B.B 1 1.B 6.B1 V B.BBB1

For the $ase of Buantitati"eFactors8 a


86/116

L. M. Lye 6

4egression 3odelis often Cseful4esponse/!trength

ANOVA for 4esponse !urface $ubic 3odel

Analysis of "ariance table F

Model 881.1 / 18. 15.5 V B.BBB1

A 3.0 1 3.0 .70 3.3351

A2 44.21 1 44.21 4/.4 3.3331

A

4

/.0 1 /.0 /. 3.3152?esidual 195.15 1 9.9

ac6 of ,it 44.5 1 44.5 .21 3.3545

$ure *rror 1/1.23 23 0.3/

"or otal 6/6.96 8

$oefficient !tandard $I $I

Factor Estimate DF Error 6o 'igh VIF3ntercept 19.8 1 B.95 1.89 1.88

A-"otton .1B 1 .59 .1 1/.89 9.B/

A -.6 1 1.86 -11.9 -5./ 1.BB

A/ -.6B 1 . -1/.5 -1.6 9.B/

&he 4egression 3odel5E'$6*E8PE"TPlot

trength

8 9 A+Cotton Deight %

5es ign Points


87/116

L. M. Lye

&he 4egression 3odel

Final E:uation in &erms of

Actual Factors/

#trengt, O 6.611 -

9.B11X >t Y

B.81X >t Z -.6BBE-BB/ X >t Z/

,is is an empirical modelof

t,e experimental results

32=44 B4=44 B2=44 ?4=44 ?2=44

1

33=2

30

B4=2

B2

A + C o tto n D e ig h t %

tren

gth

BB

BB

BB BB

BB BB

BB

5E%'$6*E8PE"T Plot

5esira&ility

89 A+ A

5esign Points


88/116

L. M. Lye

32=44 B4=44 B2=44 ?4=44 ?2=44

4=4444

4=B244

4=2444

4=1244

3=444

A + A

5

e

s

ira

&

ility

, ne - a c to r P lo t

0

0

00000

22222

22222

22222

00000

Pr ed ic t 4 =1 1 B28 B @ =B ?


89/116

L. M. Lye 9

!ample !i;e Determination


90/116

L. M. Lye 9B

!ample !i;e Determination

4 FABin designed experiments

4 Ans*er depends on lots of t,ingsP including *,attype of experiment is eing contemplated ,o* it*ill e conducted resources and desired sensiti"ity

4 #ensiti(ity refers to t,e difference in meanst,at t,eexperimenter *is,es to detect

4 %enerally increasingt,e numer of replicationsincreasest,e sensiti"ityor it ma;es it easier todetect small differences in means


91/116

L. M. Lye 91

D2E $3


92/116

L. M. Lye 9


Introduction to General Factorials

4 General principlesof factorial experiments

4 ,e to)factor factorial*it, fixed effects4 ,e ANOVAfor factorials

4 Extensions to more t,an t*o factors

4 Buantitati"eand :ualitati"efactors :response cur(es and surfaces

!ome #asic Definitions


93/116

L. M. Lye 9/

Definition of a factor effect/ &he change in the mean response hen

the factor is changed from lo to high

8B 5 B /B1

/B 5 B 8B

11

5 B /B 8B1

A A

7 7

A ! !

! !

A

+

+

+ += = =

+ += = =

+ += =

&he $ase of Interaction/


94/116

L. M. Lye 98

5B 1 B 8B1

8B 1 B 5B 9

1 B 8B 5B9

A A

7 7

A ! !

! !

A

+

+

+ += = =

+ += = =

+ += =


95/116

L. M. Lye 95

4egression 3odel &he

Associated 4esponse

!urface

B 1 1

1 1

1

1

1

,e least sHuares fit is

W /5.5 1B.5 5.5

B.5

/5.5 1B.5 5.5

! 8 8

8 8

! 8 8

8 8

8 8

= + +

+ +

= + +

+

+ +


96/116

L. M. Lye 96

&he Effect of Interaction

on the 4esponse !urface

#uppose t,at *e add an

interaction term to t,e

model

1

1

W /5.5 1B.5 5.5

! 8 8

8 8

= + +

+

Interactionis actually

a form of cur"ature

Example/ #attery 6ife Experiment


97/116

L. M. Lye 9

AO Material typePO emperature $A :uantitati"e(ariale&

1. >,at effectsdo material type C temperature ,a(e on lifeG

. 3s t,ere a c,oice of material t,at *ould gi(e long life regardless of

temperature$a robustproduct&G

&he General &o)Factor


98/116

L. M. Lye 9

Factorial Experiment

ale(els of factorAP ble(els of factorP nreplicates

,is is a completely randomi;ed design


99/116

L. M. Lye 99

#tatistical $effects& model

1...

$ & 1 ...

1...

i6 i i i6

i a

! b

6 n

=

= + + + + = =

2t,er models $means model regression models& can e useful

?egression model allo*s for prediction of responses *,en *e,a(e Huantitati(e factors. A)2


100/116

L. M. Lye 1BB

Extension of the ANOVA to Factorials

0Fixed Effects $ase1

... .. ... . . ...

1 1 1 1 1

. .. . . ... .1 1 1 1 1

$ & $ & $ &

$ & $ &

a b n a b

i6 i

i 6 i

a b a b n

i i i6 ii i 6

! ! bn ! ! an ! !

n ! ! ! ! ! !

= = = = =

= = = = =

= +

+ + +

rea;do*n1 1 1 $ 1&$ 1& $ 1&

A A * ## ## ## ## ##

dfabn a b a b ab n

= + + +

= + + +

ANOVA &able 5 Fixed Effects $ase


101/116

L. M. Lye 1B1

ANOVA &able Fixed Effects $ase

Design)Expert*ill perform t,e computations

Most text gi(es details of manual computing

$ug,Q&

Design)Expert Output


102/116

L. M. Lye 1B

Response: 2ife ANO3A for Selected "actorial *odel

Anal4sis of %ariance ta&le 5Partial su) of s6uares7

Su) of *ean "

Source S6uares (" S6uare 3alue Pro& 8 "

Model 2/30=BB @ 1/B1=4? 33=44 4=4443

A 10683.72 2 5341.86 7.91 0.0020B 39118.72 2 19559.36 28.97 < 0.0001

AB 9613.78 4 2403.44 3.56 0.0186

Pure E 3@B?4=12 B1 012=B3

C Total 110/0=1 ?2

td= 5ev= B2=@ "*quared 4=102BMean 342=2? AdF "*quared 4=020

C=G= B/=0B Pred "*quared 4=2@B0

P"E ?B/34=BB Adeq Precision @=31@

4esidual Analysis


103/116

L. M. Lye 1B/

4esidual Analysis5E'$6*E8PE"T Plot7ife

" e s id u a l

6orm

al

%

pro&a&ility

6o rm a l p lo t o f res idua ls

* 04 =12 * ?/ =B2 * 1=12 3@=12 /2 =B2

3

2

34

B4

?4

24

14

@4

4

2

5 E' $ 6 *E8PE" T Plot

7if e

P re d ic te d

"

esiduals

"e s idua ls vs = P red i c ted

*04=12

*?/=B2

*1=12

3@=12

/2=B2

/=24 10=40 34B=0B 3B =3 322=12

4esidual Analysis


104/116

L. M. Lye 1B8

5E'$6*E8PE"TPlot7ife

" u n 6 u m & e r

"

esi

duals

"e s idua ls vs = "un

*04=12

*?/=B2

*1=12

3@=12

/2=B2

3 0 33 30 B3 B0 ?3 ?0

4esidual Analysis


105/116

L. M. Lye 1B5

es dua a ys s5E'$6*E8PE"TPlot7ife

M a te ri a l

"esiduals

"e sid ua ls vs = M a ter ia l

*04=12

*?/=B2

*1 =12

3@=12

/2=B2

3 B ?

5 E' $ 6 *E8PE" T Plot

7if e

T e m p e ra tu re

"

esiduals

"e sid uals vs= Te m pe rature

*04=12

*?/=B2

*1=12

3@=12

/2=B2

3 B ?

Interaction %lot5E'$6*E8PE"T Plot7ife

89 .+Temperature;9 A+Material

A3 A3AB AB

A? A?


106/116

L. M. Lye 1B6

A + M a te r ia l

'n te ra c t ion $ raph

7ife

. + T e m p e ra tu re

32 14 3B2

B4

0B

34 /

3/ 0

3@ @

B

B

BB

B

B


107/116

L. M. Lye 1B

Buantitati"e and Bualitati"e Factors

4 ,e asic A)2


108/116

L. M. Lye 1B


Response:2ife

999 ARNING: he Cu&ic *odel is Aliased; 999

Se6uential *odel Su) of S6uares

Su) of *ean "


Mean /=44E>442 3 /44E>442

7inear /1B0=? ? 30212=/0 3=44 4=4443 uggested

B-' B?32=4@ B 3321=2/ 3=?0 4=B1?4

Huadratic 10=40 3 10=40 4=4@0 4=114

Cu&ic 1B@=0 B ?0/=?2 2=/4 4=4340 Aliased

"esidual 3@B?4=12 B1 012=B3

Total /=1@2E>442 ?0 3?BB=1

"Sequential Model Sum of Square"+ elect the highest order polynomial where the

additional terms are significant=


109/116

L. M. Lye 1B9


Candidate model

terms from 5esign*

Expert+

'ntercept

A.

.B

A.

.?

A.B

AO Material type

O Linear effect of emperature

O uadratic effect of

emperature

AO Material type : empLinear

AO Material type - empuad

/O "uic effect of

emperature $Aliased&


110/116

L. M. Lye 11B


2ac. of "it ests

Su) of *ean "


7inear 0@=@? 2 3?1=1 B=@1 4=4??? uggested

B-' 1?1/=12 ? B/2@=B2 ?=0/ 4=4B2B

Huadratic 1B@=0 B ?0/=?2 2=/4 4=4340

Cu&ic 4=44 4 Aliased

Pure Error 3@B?4=12 B1 012=B3

"!a# of $it %et"+ Dant the selected model to have insignificant lack*of*fit=


111/116

L. M. Lye 111


*odel Su))ar4 Statistics

Std< Ad,usted Predicted

Source (e%


112/116

L. M. Lye 11

Response: 2ife

ANO3A for Response Surface Reduced Cu&ic *odelAnal4sis of %ariance ta&le 5Partial su) of s6uares7

Su) of *ean "


Model 2/30=BB @ 1/B1=4? 33=44 4=4443

A 10683.72 2 5341.86 7.91 0.0020

B 39042.67 1 39042.67 57.82 < 0.0001B2 76.06 1 76.06 0.11 0.7398

AB 2315.08 2 1157.54 1.71 0.1991

AB2 7298.69 2 3649.35 5.40 0.0106

Pure E 3@B?4=12 B1 012=B3

C Total 110/0=1 ?2

td= 5ev= B2=@ "*quared 4=102BMean 342=2? AdF "*quared 4=020

C=G= B/=0B Pred "*quared 4=2@B0

P"E ?B/34=BB Adeq Precision @=31@

4egression 3odel !ummary of 4esults


113/116

L. M. Lye 11/

4egression 3odel !ummary of 4esults

"inal E6uation in er)s of Actual "actors:

Material A3

7ife 9

>30=?@431

*B=243/2 Temperature

>4=43B@23 TemperatureB

Material AB

7ife 9

>32=0B?1

*4=31??2 Temperature

*2=00330E*44? TemperatureB

Material A?

7ife 9

>3?B=10B/4

>4=4B@ Temperature

*4=434B/@ TemperatureB

4egression 3odel !ummary of 4esults5E'$6*E8PE"TPlot

7ife

89.+Temperature;9A+Material


114/116

L. M. Lye 118

g yA3 A3AB AB

A? A?

A + M a te r i a l

'n te ra c t ion $ rap h

7ife

. + T e m p e ra tu re

32=44 /B=24 14=44 1=24 3B2=44

B4

0B

34 /

3/ 0

3@ @

B

B

BB

B

B

Factorials ith 3ore &han


115/116

L. M. Lye 115

Factorials ith 3ore &han

&o Factors

4 asic procedure is similar to t,e t*o-factor caseP

all abc96ntreatment cominations are run in

random order

4 A)2


116/116

More t,an factors

4 >it, more t,an factors t,e most useful

type of experiment is t,e -le(el factorial

experiment.

4 Most efficient design $least runs&

4 "an add additional le(els only if reHuired

4 "an e done seHuentially4 ,at *ill e t,e next topic of discussion

DOE Course - Parts 1-4o

Documents

Transcript of DOE Course - Parts 1-4o