ANOVA and Simple Comparative Experiment

7/25/2019 ANOVA and Simple Comparative Experiment

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DOX 6E Montgomery 1

Design of Engineering Experiments

Part 2 Basic Statistical Concepts Simple comparativeexperiments

The hypothesis testing framework

The two-sample t-test

he!king ass"mptions# $ali%ity

omparing more that two fa!tor le$els&theanalysis of variance '(O)' %e!omposition of total $aria*ility

Statisti!al testing + analysis he!king ass"mptions# mo%el $ali%ity

,ost-'(O)' testing of means

Sample size%etermination


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Portland Cement Formulation page 2!"


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#raphical $ie% of the Data

Dot Diagram& Fig' 2()& pp' 2*


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Box Plots& Fig' 2(!& pp' 2+


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,he -ypothesis ,esting Frame%or.

Statistical hypothesis testingis a "sef"l

framework for many experimental

sit"ations

Origins of the metho%ology %ate from the

early 122s

3e will "se a pro!e%"re known as the t%o(

sample t(test


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,he -ypothesis ,esting Frame%or.

Sampling from a normal%istri*"tion Statisti!al hypotheses4

2 1

1 1

4

4

H

H

=


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Estimation of Parameters

1

1

1estimates the pop"lation mean

16 7 estimates the $arian!e

1

n

i

i

n

i

i

y yn

S y yn

=

=

=

=


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Summary Statistics pg' !+"

1

1

1

1

16956

29122

29.16

12

y

S

S

n

=

=

=

=

Formulation )

/0e% recipe1

Formulation 2

/riginal recipe1

1

1

1

1

1592/29261

29/8

12

yS

S

n

==

=

=


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-o% the ,%o(Sample t(,est 3or.s4

1

y

:se the sample means to %raw inferen!es a*o"t the pop"lation means

16956 1592/ 298

Differen!e in sample means

Stan%ar% %e$iation of the %ifferen!e in sample means

This s"ggests a statisti!4

y y

n

= =

=

1 2

1

1

; y y

n n

=

+


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1 1

1

1

1

1

1 1

1

:se an% to estimate an%

The pre$io"s ratio *e!omes


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)al"es of t2 that are near =ero are !onsistent with the n"ll

hypothesis

)al"es of t2 that are $ery %ifferent from =ero are !onsistent

with the alternati$e hypothesis t2 is a >%istan!e? meas"re-how far apart the a$erages are

expresse% in stan%ar% %e$iation "nits

(oti!e the interpretation of t2 as a signal(to(noiseratio

1 -2

1 -

The test statisti! is

1 1

p

y yt

Sn n

=

+


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,he ,%o(Sample Pooled" t(,est

1 1

1

1 2

1

17 17 291227 29261729281

12 12

298/

16956 1592/ 92

1 1 1 1298/

12 12

The two sample means are a little o$er two stan%ar% %e$iations apart

@s t

p

p

p

n S n S S

n n

S

y yt

Sn n

+ += = =

+ +

=

= = =

+ +

his a AlargeA %ifferen!eB


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So far# we ha$enCt really%one any >statisti!s?

3e nee% an o56ective*asis for %e!i%ing howlarge the test statisti! t2really is

@n 128# 39 S9 osset%eri$e% the referencedistri5utionfor t2 &

!alle% the t%istri*"tion Ta*les of the t

%istri*"tion - text# page626

t2 -92


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' $al"e of t2*etween9121 an% 9121 is!onsistent witheF"ality of means

@t is possi*le for the

means to *e eF"al an%t2to ex!ee% either

9121 or 9121# *"t itwo"l% *e a >rareevent? & lea%s to the

!on!l"sion that themeans are %ifferent

o"l% also "se theP(valueapproa!h

t2 -92


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The P-valueis the risk of %rongly re6ectingthe n"llhypothesis of eF"al means it meas"res rareness of the e$ent7

TheP-$al"e in o"r pro*lem isP 292/

t2 -92


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7inita5 ,%o(Sample t(,est 8esults


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Chec.ing 9ssumptions

,he 0ormal Pro5a5ility Plot


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:mportance of the t(,est

,ro$i%es an o56ectiveframework for simple

!omparati$e experiments

o"l% *e "se% to test all rele$ant hypotheses

in a two-le$el fa!torial %esign# *e!a"se all

of these hypotheses in$ol$e the mean

response at one >si%e? of the !"*e $ers"s themean response at the opposite >si%e? of the

!"*e


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Confidence :ntervals See pg' *!"


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3hat :f ,here 9re 7ore ,han

,%o Factor ;evels< The t-test %oes not %ire!tly apply

There are lots of pra!ti!al sit"ations where there are either

more than two le$els of interest# or there are se$eral

fa!tors of sim"ltaneo"s interest

The analysis of variance'(O)'7 is the appropriate

analysis >engine? for these types of experiments hapter

.# text*ook

The '(O)' was %e$elope% *y Kisher in the early 12s#

an% initially applie% to agri!"lt"ral experiments

:se% extensi$ely to%ay for in%"strial experiments


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9n Example See pg' +="

'n engineer is intereste% in in$estigating the relationship *etween theLK power setting an% the et!h rate for this tool9 The o*Ge!ti$e of an

experiment like this is to mo%el the relationship *etween et!h rate an%LK power# an% to spe!ify the power setting that will gi$e a %esire%

target et!h rate9 The response $aria*le is et!h rate9

She is intereste% in a parti!"lar gas K67 an% gap 2982 !m7# an%

wants to test fo"r le$els of LK power4 1623# 1823# 223# an% 239She %e!i%e% to test fi$e wafers at ea!h le$el of LK power9

The experimenter !hooses / levelsof LK power 1623# 1823# 223#an% 23

The experiment is replicated0 times r"ns ma%e in ran%om or%er


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9n Example See pg' +2"

Does changingthe

power !hange the

mean et!h rateB

@s there an optimumle$el for powerB


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,he 9nalysis of $ariance Sec' !(2& pg' +!"

@n general# there will *e alevelsof the fa!tor# or atreatments& andnreplicatesof the experiment# r"n in randomorder>a !ompletelyran%omi=e% %esignC8D"

N = antotal r"ns

3e !onsi%er the fixed effects!ase&the random effects!ase will *e%is!"sse% later

O*Ge!ti$e is to test hypotheses a*o"t the eF"ality of the a treatment means


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,he 9nalysis of $ariance The name >analysis of $arian!e? stems from a

partitioningof the total $aria*ility in the response$aria*le into !omponents that are !onsistent with a

modelfor the experiment

The *asi! single-fa!tor '(O)' mo%el is

-

1#-#999##

1#-#999#

an o$erall mean# treatment effe!t#

experimental error# 62# 7

ij i ij

i

ij

i ay

j n

ith

NID

== + +

=

= =

=


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7odels for the Data

There are se$eral ways to write a mo%el for

the %ata4

is !alle% the effe!ts mo%el

Met # then

is !alle% the means mo%elLegression mo%els !an also *e employe%

ij i ij

i i

ij i ij

y

y

= + +

= +

= +


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,he 9nalysis of $ariance

,otal varia5ilityis meas"re% *y the total s"m of

sF"ares4

The *asi! '(O)' partitioning is4

99

1 1

6 7a n

T ij

i j

SS y y= =

=

99 9 99 9

1 1 1 1

9 99 9

1 1 1

7 N 7 7

7 7

a n a n

ij i ij i

i j i j

a a n

i ij i

i i j

T Treatments E

y y y y y y

n y y y y

SS SS SS

= = = =

= = =

= +

= +

= +


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,he 9nalysis of $ariance

' large $al"e of SSTreatments refle!ts large %ifferen!es in

treatment means

' small $al"e of SSTreatments likely in%i!ates no %ifferen!es intreatment means

Kormal statisti!al hypotheses are4

T Treatments E SS SS SS = +

2 1

1

4

4 't least one mean is %ifferent

aH

H

= = =L


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,he 9nalysis of $ariance 3hile s"ms of sF"ares !annot *e %ire!tly !ompare% to test the hypothesis of eF"al means# mean s?uares!an *e !ompare%9

' mean sF"are is a s"m of sF"ares %i$i%e% *y its %egrees of free%om4

@f the treatment means are eF"al# the treatment an% error mean sF"ares will *e theoreti!ally7 eF"al9

@f treatment means %iffer# the treatment mean sF"are will *e larger than the error mean sF"are9

1 1 6 17

#1 6 17

Total Treatments Error

Treatments E Treatments E

df df df

an a a n

SS SS MS MS

a a n

= +

= +

= =


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,he 9nalysis of $ariance is

Summarized in a ,a5le

omp"ting&see text# pp 66-52

The reference distri5utionforF2 is theFa-1#an-17%istri*"tion

8e6ectthe n"ll hypothesis eF"al treatment means7 if

2 # 1# 17a a nF F >


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90$9 ,a5le

Example !()


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,he 8eference Distri5ution4


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'(O)' !al!"lations are "s"ally %one $ia !omp"ter

Text exhi*its sample !al!"lations from two

$ery pop"lar software pa!kages# Design-

Expert an% Minita* See page for Design-Expert# page 122

for Minita*

Text %is!"sses some of the s"mmarystatisti!s pro$i%e% *y these pa!kages


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7odel 9de?uacy Chec.ing in the 90$9

,ext reference& Section !(*& pg' @A Chec.ing assumptionsis important

(ormality

onstant $arian!e @n%epen%en!e


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7odel 9de?uacy Chec.ing in the 90$9

Examination of residuals

see text# Se!9 .-/# pg9 507

Design-Expert generates

the resi%"als

8esidual plotsare $ery

"sef"l 0ormal pro5a5ility plot

of resi%"als

9

Pij ij ij

ij i

e y y

y y

=

=


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ther :mportant 8esidual Plots


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Post(90$9 Comparison of 7eans

The analysis of $arian!e tests the hypothesis of eF"al treatmentmeans

'ss"me that resi%"al analysis is satisfa!tory

@f that hypothesis is reGe!te%# we %onCt know %hichspecificmeansare %ifferent

Determining whi!h spe!ifi! means %iffer following an '(O)' is!alle% the multiple comparisons pro5lem

There are lotsof ways to %o this&see text# Se!tion .-0# pg9 85

3e will "se pairwise t-tests on means&sometimes !alle%KisherCs east Signifi!ant Differen!e or KisherCs ;SD7 Metho%


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Design(Expert utput


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#raphical Comparison of 7eans

,ext& pg'


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,he 8egression 7odel


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3hy Does the 90$9 3or.

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ANOVA and Simple Comparative Experiment