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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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-o% the ,%o(Sample t(,est 3or.s4
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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-o% the ,%o(Sample t(,est 3or.s4
)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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,he ,%o(Sample Pooled" t(,est
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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,he ,%o(Sample Pooled" t(,est
' $al"e of t2*etween9121 an% 9121 is!onsistent witheF"ality of means
@t is possi*le for the
means to *e eF"al an%t2to ex!ee% either
9121 or 9121# *"t itwo"l% *e a >rareevent? & lea%s to the
!on!l"sion that themeans are %ifferent
o"l% also "se theP(valueapproa!h
t2 -92
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,he ,%o(Sample Pooled" t(,est
The P-valueis the risk of %rongly re6ectingthe n"llhypothesis of eF"al means it meas"res rareness of the e$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.