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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D2E - 3
3ntroduction
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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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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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#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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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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?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 $BB5&
#tatistics for Experimenters An 3ntroduction to
Design Data Analysis and Model uilding +o,n
>iley and #ons. ndEdition
: "lassic text *it, lots of examples. )o computer aided
solutions. 'ses numers for Factors.
4 +ournal of uality ec,nology ec,nometrics
American #tatistician discipline specific @ournals
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3ntroduction >,at is meant y D2EG
4 Experiment -
: a test or a series of tests in *,ic, purposeful c,anges aremade to t,e input variables or factorsof a system so t,at
*e may oser(e and identify t,e reasons for c,anges in
t,e outputresponse$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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#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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#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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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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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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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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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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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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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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,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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L. M. Lye 9
D2E $33&
Factorial (s 2FA
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L. M. Lye /B
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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L. M. Lye //
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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L. M. Lye 8
5E%'$6*EA%EPlot
7n(f)89A+ "eynold !s:;9.+ k
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* / = ? 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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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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D2E $333&
asic "oncepts
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L. M. Lye 5B
Design of Engineering Experiments
#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
$sample&1! !
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Graphical Vie of the DataDot Diagram
-orm 3 -orm B
30=?
31=?
3@=?
5otplots of -orm 3 and -orm B(means are indicated &y lines)
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#ox %lots
-orm 3 -orm B
30=2
31=2
3@=2
.oxplots of -orm 3 and -orm B(means are indicated &y solid circles)
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L. M. Lye 58
&he 'ypothesis &esting Frameor(
4 !tatistical hypothesis testingis a useful
frame*or; for many experimental
situations4 2rigins of t,e met,odology date from t,e
early 19BBs
4 >e *ill use a procedure ;no*n as t,e to)sample t)test
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&he 'ypothesis &esting Frameor(
4 #ampling from a normaldistriution4 #tatistical ,ypot,eses
B 1
1 1
"
"
=
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Estimation of %arameters
1
1
1estimates t,e population mean
1$ & estimates t,e (ariance
1
n
i
i
n
i
i
! !n
# ! !n
=
=
=
=
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!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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'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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'o the &o)!ample t)&est or(s/
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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'o the &o)!ample t)&est or(s/
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
&he &o)!ample 0%ooled1 t)&est
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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&he &o)!ample 0%ooled1 t)&est
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,
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L. M. Lye 68
&he &o)!ample 0%ooled1 t)&est
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$O B.BBBBBBB/
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3initab &o)!ample t)&est 4esults
wo-Sa)ple -est and CI: "or) /0 "or) 1Two-sample T for Form 1 vs Form 2
N Mean StDev SE Mean
Form 1 10 16.764 0.316 0.10
Form 2 10 17.922 0.24 0.07
D!fferen"e # m$ 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
$h (i A ti
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$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
$oodness of -it
Tension .ond trength 5ataM7 Estimates
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L. M. Lye 6
Importance of the t)&est
4 !ro(ides an ob2ecti"eframe*or; for simple
comparati(e experiments
4 "ould e used to test all rele(ant ,ypot,esesin a t*o-le(el factorial design ecause all
of t,ese ,ypot,eses in(ol(e t,e mean
response at one IsideJ of t,e cue (ersus t,emean response at t,e opposite IsideJ of t,e
cue
h t If &h A 3 &h
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L. M. Lye 6
hat If &here Are 3ore &han
&o Factor 6e"els74 ,e t-test does not directly apply
4 ,ere are lots of practical situations *,ere t,ere are eit,er
more t,an t*o le(els of interest or t,ere are se(eral factors of
simultaneous interest
4 ,e analysis of "ariance$A)2
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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
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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
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&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
&he Analysis of Variance
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&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
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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
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L. M. Lye 8
&he Analysis of Variance
4 &otal "ariabilityis measured y t,e total sum of
sHuares
4 ,e asic A)2
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&he Analysis of Variance
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
&he Analysis of Variance
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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
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&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, , >
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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/
".
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&he 4eference Distribution/
Graphical Vie of the 4esults
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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
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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
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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
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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
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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
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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
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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
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&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
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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 ?
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!ample !i;e Determination
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!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
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D2E $3
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Design of Engineering Experiments
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
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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/
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L. M. Lye 98
5B 1 B 8B1
8B 1 B 5B 9
1 B 8B 5B9
A A
7 7
A ! !
! !
A
+
+
+ += = =
+ += = =
+ += =
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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
= + +
+ +
= + +
+
+ +
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&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
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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
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Factorial Experiment
ale(els of factorAP ble(els of factorP nreplicates
,is is a completely randomi;ed design
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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
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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
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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
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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
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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
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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
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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?
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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
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L. M. Lye 1B
Buantitati"e and Bualitati"e Factors
4 ,e asic A)2
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L. M. Lye 1B
Buantitati"e and Bualitati"e Factors
Response:2ife
999 ARNING: he Cu&ic *odel is Aliased; 999
Se6uential *odel Su) of S6uares
Su) of *ean "
Source S6uares (" S6uare 3alue Pro& 8 "
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=
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L. M. Lye 1B9
Buantitati"e and Bualitati"e Factors
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&
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L. M. Lye 11B
Buantitati"e and Bualitati"e Factors
2ac. of "it ests
Su) of *ean "
Source S6uares (" S6uare 3alue Pro& 8 "
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=
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L. M. Lye 111
Buantitati"e and Bualitati"e Factors
*odel Su))ar4 Statistics
Std< Ad,usted Predicted
Source (e%
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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 "
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 0.0020
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
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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
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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
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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
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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
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