Syllabus Design of Experiments p#one$
0"""&&&& 0"""&&&&
[email protected] email$ 'i)e$ Pas)asar*ana +,-,.R/ 'i)e 'i)e Hours$ Hours$ umat Saptu
Text$
Design and Analysis of Experiments
1 ont4omer Silabus
1. Konsep Konsep Definisi Definisi (Ch: (Ch: 1 dan 2) 2) 2. Exp with a Single Single Factor: A!"A (Ch: (Ch: #) #. $ando%i&ed $ando%i&ed 'los *atin S+,are (Ch: -) -. Factorial Factorial Desig Design n (Ch:) (Ch:) . /he 2k Factorial Design (Ch:0) 0. /wo leel Fractional Factorial Factorial Design (Ch:) 3. /hree 4ixed 4ixed 5 *eel Factorial and Fractional Fractional Factorial designs (Ch:6) . /ag,chi /ag,chi approach approach (S,%7er (S,%7er lain) lain) Final Grade Based on the Following
Homework 20% Presentation and Related Paper 35% Take Home or Final Final Exam (Simul (Simulation ation !5% Konsep Definisi Definisi (Ch: 1 dan 2) Exp with a Single Factor: A!"A (Ch: #) $ando%i&ed 'los *atin S+,are (Ch: -) Factorial Design Design (Ch:) /he 2k Factorial Design (Ch:0) /wo leel Fractional Factorial Design (Ch:) /hree 4ixed 5 *eel Factorial and Fractional Factorial designs (Ch:6) /ag,chi approach (S,%7er lain).
" | DoE Montgomery
Fifth Edition
Douglas C. Montgomery ARIZONA STATE NI!ERSIT"
JOHN WILEY & SONS. INC
2 | DoE Montgomery
Fifth Edition
Douglas C. Montgomery ARIZONA STATE NI!ERSIT"
JOHN WILEY & SONS. INC
2 | DoE Montgomery
Pengantar oleh Dr. Suparman IA, MS.
S!"t!m# Mo$el
3 | DoE Montgomery
Experiments it# a Sin4le Fa)tor$ T#e /nalsis o 6arian)e -n 7#apter 2 we dis)ussed met#ods or )omparin4 two )onditions or treatments8 For example9 t#e Portland )ement tension bond experiment in:ol:ed two dierent mortar ormulations8 /not#er wa to des)ribe t#is experiment is as a sin4le;a)tor experiment wit# two le:els o t#e a)tor9 w#ere t#e a)tor is mortal ormulation and t#e two le:els are two dierent ormulation met#ods8 an experiments o t#is tpe in:ol:e more t#an two le:els o t#e a)tor8 -n t#is )#apter we present met#ods or t#e desi4n and analsis o sin4le;a)tor experiments wit a le:els o t#e a)tor (or a treatments8 e will assume t#at t#e experiment #as been )ompletel randomi
E / produ)t de:elopment en4ineer is interested in in:esti4atin4 t#e tensile stren4t# o a new snt#eti) iber t#at will be used to make )lot# or men?s s#irts8 T#e en4ineer knows rom pre:ious experien)e t#at t#e stren4t# is ae)ted b t#e wei4#t per)ent o )otton used in t#e blend o materials or t#e iber8 Furt#ermore9 s#e suspe)ts t#at in)reasin4 t#e )otton )ontent will in)rease t#e stren4t#9 at least initiall8 S#e also knows t#at )otton )ontent s#ould ran4e between about "0 and !0 per)ent i t#e inal produ)t is to #a:e ot#er @ualit )#ara)teristi)s t#at are desired (su)# as t#e abilit to take a permanent;press inis#in4 treatment8 T#e en4ineer de)ides to test spe)imens at i:e le:els o )otton wei4#t per)ent$ "59 209 259 309 and 35 per)ent8 S#e also de)ides to test i:e spe)imens at ea)# le:el o )otton )ontent8 T#is is an example o a sin4le;a)tor experiment wit# a A 5 levels o t#e a)tor and n A 5 replicates 8 T#e 25 runs s#ould be made in random order8 To illustrate #ow t#e run order ma be randomi
! | DoE Montgomery
" & "" "& 2"
Experiment Run ,umber 2 3 ! B C "2 "3 "! "B " "C 22 23 2!
5 "0 "5 20 25
,ow we sele)t a rondom number between " and 258 Suppose t#is number is 8 T#en t#e number obser:ation (20 per)ent )otton is run irst8 T#is pro)ess would be repeated until all 25 obser:ations #a:e been assi4ned a position in t#e test se@uen)e8 " an )omputer sotware pa)ka4es or assistin4 experiments in sele)tin4 and )onstru)tin4 a desi4n randomi
Run ,umber
7otton ei4#t Per)enta4e
" 2 3 ! 5 & B C "0 "" "2 "3 "! "5 "& "B " "C 20 2" 22 23 2! 25
" "0 23 "B 5 "! & "5 20 C ! "2 B " 2! 2" "" 2 "3 22 "& 25 "C 3
20 30 20 35 30 "5 25 20 25 30 20 "5 25 20 "5 35 35 25 "5 25 35 30 35 30 "5
T#is randomi
T#e onl restri)tion on randomi
7H/PTER 3 E=PER-E,TS -TH / S-,D>E F/7T'R$ THE /,/>S-S 'F 6/R-/,7E
Tabel 383 T#e /nalsisi o 6arian)e Table or t#e Sin4le;Fa)tor9 Fixed Ee)ts odel Sour)e o 6ariation
5 | DoE Montgomery
Sum o S@uares
.e4rees o
ean S@uare
Fa
Freedom SS Treatmens
1eetween treatments
a
An
( i8
a
; 88 2
; "
F0 A
MS Treatment
MS E
iA" Error (wit#in treatment
SSt A SS T ; MS Treatment
a
Total
SS T A
N ; a
a
(8 ij
MS Treatmenn
MS E
N - "
; 88 2
iA" *A"
independentl distributed )#i;s@uare random :ariables8 T#ereore9 i t#e null #pot#esis o no dieren)e in treatment means is true9 t#e ratio F0 A
MS Treatmenn G(a ; "
MS Treatmenn
A
SS G( N - a
MS E
E
is distributed as F wit# a ; " and N ; a de4rees o reedom8 E@uation 3;B is t#e best statitc or t#e #p#otesis o no dieren)es in treatment means8 From t#e expe)ted mean s@uares we see t#at9 in 4eneral MS in an unbiased estimator o 2 8 /lso9 under t#e null #pot#esis9 MS Treatmenn is an unbiased estimator o 28 Howe:er9 i t#e null #pot#esis is alse9 t#e expe)ted :alue9 MS Treatmenn is 4reater t#an 2 8 T#ereore9 under t#e alternati:e #pot#esis9 t#e expe)ted :alue o t#e numerator o t#e test statisti) (E@uation 3;B is 4reater t#an t#e expe)ted :alue o t#e denominator9 and we s#ould re*e)t 0 on :alues o t#e test statisti) t#at are too lar4e8 T#is implies an uppertail9 one;tail )riti)al re4ion8 T#ereore9 we s#ould re*e)t 0 and )on)lude t#at t#ere are dieren)es in t#e treatment means i E
F 0 I FJ8a ; "9 ,;a
w#ere F 0 is )omputed rom E@uation 3;B8 /lternati:el9 we )ould use t#e P;:alue approa)# or de)ision makin48 7omputin4 ormulas or t#e sums o s@uares ma be obtained b rewritin4 and simpliin4 t#e deinitions o MS Treatmenn and SS in E@uation 3;&8 T#is ields T
a SS T A
a
8 ;
288
iA" *A"
N
ij
and & | DoE Montgomery
(3;
SS Treatment A
" a
2 8 ;
n iA"
288
(3;C
i
N
T#e error sum o s@uares is obtained b subtra)tion as SS E A SS T ; SS Treatment
(3;"0
T#e test pro)edure is summari
3;". T#e tensile stren4t# o Portland )ement is bein4 studied8 Four dierent mxin4 te)#ni@ues )an be used e)onomi)all8 T#e ollowin4 data #a:e been )olle)ted$ ixin4 Te)#ni@ue
" 2 3 !
Tensile Stren4#t (lbGn 2
3"2C 3200 200 2&00
3000 3300 2C00 2B00
2&5 2CB5 2C5 2&00
2C0 3"50 3050 2B&5
(a Test t#e #pot#esis t#at mixin4 te)#ni@ues ae)t t#e stren4t# o t#e )omnet8 +se J A 09058 (b 7onstru)t a 4rap#i)al displa as des)ribed in Se)tion 3;5;3 to )ompare t#e mean tensile stren4t# or t#e our mixin4 te)#ni@ues8 #at are ou )on)lusionK () +se t#e Fis#er >S. met#od wit# J A 0905 to make )omparison between pairs o means8 (d 7onstru)t normal probabilit plot o t#e residuals8 #at )on)lusion would ou means8 (e Plot t#e residuals :ersus t#e predi)ted tensile stren4t#8 7ommnet on t#e plot8 ( Prepare a s)atter plot o t#e results to aid t#e interpretation o t#e results o t#is experiments8 3828 (a Rework part (b o Problem 3;" usin4 .un)anLs multiple ran4e test wit# J A 09058 .oes t#is make an dieren)e in our )on)lusionK (b Rework part (b o problem 3;" usin4 Turke?s test wit# J A 09058 .o ou 4et t#e same )on)lusions orm Turke?s test t#at ou did rom t#e 4rap#i)al pro)edure andGor .un)anLs multiple ran4e testK () Explain t#e dieren)e between t#e Turke and .un)an proedurs8 B | DoE Montgomery
3;38 Re)onsider t#e experiment in Problem 3;"8 Find a C5 per)ent )oniden)e inter:al on t#e mean tensile sren4#t o t#e Portland )ement produ)ed b ea)# o t#e our mixin4 te)#ni@ues8 /lso ind a C5 per)ent )oniden)e inter:al on t#e dieren)e in means o te)#ni@ues " and 38 .oes t#is aid ou in interpretin4 t#e results o t#e experimentK 3;!8 /nd in experiment was run to determine w#et#er our spe)ii) irin4 temperatures ae)t t#e densit o a )ertain tpe o bri)k8 T#e experiment led to t#e ollowin4 data$ Temperature
"00 "25 "50 "B5
.ensit
2"8 2"8B 2"8C 2"8C
2"8C 2"8! 2"8 2"8B
2"8B 2"85 2"8 2"8
2"8& 2"8! 2"8& 2"8!
2"8B 2"85
(a .oes t#e irin4 temperature ae)t t#e densit o t#e bri)kK +se J A 09058 (b -s it appropriate to )ompare t#e means usin4 .un)an?s multiple ran4e test (or example in t#is experimentK () /nalE F/7T'R$ THE /,/>S-S 'F 6/R-/,7E
3;58 Rework part (d o Problem 3;! usin4 t#e Fis#er >S. et#od8 #at )on)lusions )an ou drawK Explain )areull #ow ou modiied t#e te)#ni@ues to a))ount or une@ual sample si
7oatin4 Tpe
" 2 3 | DoE Montgomery
7ondu)ti:it
"!3 "52 "3!
"!" "!C "3&
"50 "3B "32
"!& "!3 "2B
!
3;B8 3;8
"2C
"2B
"2C
(a -s t#ere a dieren)e in )ondu)ti:it due to )oatin4 tpeK +se J A 09058 (b Estimate t#e o:erall mean and t#e treatment ee)ts8 () 7ompute a Cr per)ent )oniden)e inter:al estimate o t#e mean o )oatin4 tpe !8 7ompare a CC per)ent )oniden)e inter:al estimate o t#e mean dieren)e between )oatin4 tpes " and !8 (d Test all pairs o means usin4 t#e Fis#er >S. met#od wit# J A 09058 (e +se t#e 4rap#i)al met#od dis)ussed in S)etion 3;5;3 to )ompare t#e means8 #i)# )oatin4 tpe produ)es t#e #i4#est )ondu)ti:it8 ( /ssumin4 t#at )oatin4 tpe ! is )urrentl in use9 w#at are our re)ommendations to t#e manua)turerK e wis# to minimie:el
"0 "5 20 25
3;C8
"32
7ompressi:e Stren4#t
"530 "&"0 "5&0 "500
"530 "&50 "B30 "!C0
"!!0 "500 "530 "5"0
(a -s t#ere an dieren)e in )ompressi:e stren4t# due t#e roddin4 le:elK +se J A 09058 (b Find t#e % ;:alue or t#is F statisti) in part (a8 () /nal
C | DoE Montgomery
Radon;enri)#ed water was used in t#e experiment8 T#e data rom t#e experiment are s#owa in t#e ollowin4 table$ 'rii)e .iameter
083B 085" 08B" "802 "8!0 "8CC
Radon Released (%
0 B5 B! &B &2 &0
3 B5 B3 B2 &2 &"
3 BC B& B! &B &!
5 BC BB B! &C &&
(a .oes t#e si
" 2 3
Response Time
C 20 &
"2 2" 5
"0 23
"B "&
"5 30 B
(a Test t#e #pot#esis t#at t#e t#ree )ir)uit tpes #a:e t#e same response time8 +se J A 090"8 (b +se Turke?s test to )ompare pairs o treatment menas8 +se J A 090"8 () +se t#e 4rap#i)al pro)edure in Se)tion 3;5;3 to )ompare t#e treatment mean8 #at )on)lusion )an ou draw K How do t#e )ompare wit# t#e )on)lusion rom part (bK (d 7onstru)t a set ort#o4onal )ontrasts9 assumin4 t#at at t#e outset o t#e experiment u suspe)ted t#e reponse time o )ir)uit tpe 2 to be dierent rom t#e ot#er two8
"0 | DoE Montgomery
(e - ou were t#e desi4n en4ineer and ou wis#ed to minimi
"B8&
>ie (in # at 35 k6 >oad "8C "&83 "B8! 208"
2"8&
2
"&8C
"583
"8&
"B8"
"C85
2083
3
2"8!
238&
"C8!
"85
2085
2283
!
"C83
2"8""
"&8C
"B85
"83
"C8
(a -s t#ere an indi)ation t#at t#e luids dierK +se J A 09058 (b #i)# luid would ou sele)t9 4i:en t#at t#e ob*e)ti:e is lon4 lieK () /nal
" 2 3 !
Response Time
"C 0 !B C5
20 &" 2& !&
"C B3 25 3
30 5& 35 B
0 50 CB
(a -s t#e amount o noise present t#e same or all our desi4nK +se J A 09058 (b /nalow noise is best8 3;"38 Four )#emists are asked to determine t#e per)enta4e o met#l al)o#ol in a )ertain )#emi)al )ompound8 Ea)# )#emist makes t#ree determinations9 and t#e results are t#e ollowin4$ 7#emist "
Per)enta4e o et#l /l)o#ol !8CC !80! !83
2
58"5
58"3
!8
3
!8B2
!8!
58"&
"" | DoE Montgomery
!
!820
!8"0
!855
(a .o )#emists der si4nii)antlK +se J A 09058 (b /nal
eeks o >ie 1rand 2 B& 0 B5 ! 2
1rand 3 "0 "00 C& C "00
(a /re t#e li:es o t#ese brands o batteries dierent K (b /nal
7atalst " 582 5B82 58! 558
"2 | DoE Montgomery
2 5&83 5!85 5B80 5583
3 508" 5!82 558!
! 528C !C8C 5080 5"8B
5!8C
(a .o t#e our )atalsts #a:e t#e same ee)t on t#e )on)entrationK (b /nal
""0 " 0 !C5 B
Failure Time (minutes "5B "C! 2 ! "25& 52B& B0!0 530B 5 2C
"B " !355 "08050 2
(a .o all i:e materials #a:e #e same ee)t on mean ailure timeK (b Plot t#e residuals :ersus t#e predi)ted response8 7onstru)t a normal probabilit plot o t#e residuals8 #at inormation is )on:eed t#ese plotK () 1ased on our answer to part (b )ondu)t anot#er analsis o t#e ailure time data and draw appropriate )on)lusions8 3;"B8 / semi)ondu)tor manua)turer #as de:eloped t#ere dierent met#ods or redu)in4 parti)le )ounts o waers8 /ll t#ree met#ods are tested on i:e waers and t#e ater;treatment parti)le )ount obtained8 T#e data are s#own below$ etod "
"0
7oun 2"
3"
!
"
2
&2
!0
2!
30
35
3
53
2B
"20
CB
&
(a .o all met#ods #a:e #e same ee)t on mean parti)le )ountK (b Plot t#e residuals :ersus t#e predi)ted response8 7onstru)t a normal probabilit plot o t#e residuals8 /re t#ere potential )on)erns about t#e :alidit o t#e assumptionsK () 1ased on our answer to part (b )ondu)t anot#er analsis o t#e parti)le )ount data and draw appropriate )on)lusions8 "3 | DoE Montgomery
3;"8
3;"C8 3;208
3;2"8
3;228
3;238
3;2!
3;258
3;2&8
3;2B8
7onsider testin4 t#e e@ualit o t#e means o two normal population9 w#ere t#e :arian)es are unknown but are assumed to be e@ual8 T#e appropriate test pro)edure is t#e pooled t test8 S#ow t#at t#e pooled t test is e@ui:alent to t#e sin4le;a)tor analsis o :arian)e8 a a ( 2 S#ow t#at t#e :arian)e o t#e linear )ombination M M * y + is n * i&' i i i i , i&' -n a ixed ee)ts experiment9 suppose t#at t#ere are n obser:ation or ea)# o our ( ( ( treatments8>et ' ++ ( + ) be sin4le;de4ree;o reedom )omponents or t#e ( ( ( ort#o4onal )ontrasts8 Pro:e t#at SS treatments A ' N ( N ) 8 +se 1arlet?s test to determine i t#e assumption o e@ual :arian)e is satisied in Problem 3;"!8 +se J A 09058 .id ou rea)# t#e same )on)lusion re4ardin4 e@ualit o :arian)es b examinin4 residual plotsK +se t#e modiied >e:ene test to determine i t#e assumption o e@ual :arian)e is satisied in Problem 3;"!8 +se J A 09058 .id ou rea)# t#e same )on)lusion re4ardin4 t#e e@ualit o :arian)es b examinin4 residual plotsK Reer to Problem 3;"08 - we wis# to dete)t a maximum dieren)e in mean response times o "0 millise)onds wit# a probabilit o at least 08C08 w#at sample si
"! | DoE Montgomery
(b How would our answer )#an4e i a reasonable estimate o t#e experimental error :arian)e were 2 A !CK8 () 7an ou draw an )on)lusions about t#e sensiti:it o our aswer in t#is parti)ular situation about #ow our estimate o a 8 ae)t t#e de)ision about sample si
"
2
Produ)tion Run 3
080C
08"0
08"3
080
080B
"2
080&
080C
08"2
080B
08"2
"!
08""
080
080
0805
080&
"&
08"C
08"3
08"5
0820
08""
Feed Rate (inGmin "0
!
5
(a .oes eed rate #a:e an ee)t on t#e standard de:iation o t#is )riti)al dimensionK (b +se t#e residuals rom t#is experiment to in:esti4ate model ade@ua)8 /re t#ere an problems wit# experimental :aliditK "5 | DoE Montgomery
3;3"8 7onsider t#e data s#own in Problem 3;"08 (a rite out t#e least s@uares normal e@uations or t#is problem9 and sol:e t#em or Q and 0 i usin4 t#e usual )onstraint (3i A"0 i A 0 Estimate 0 i ; 0 28 (b Sol:e t#e e@uations in (a usin4 t#e )onstraint 0 2 A 08 /re t#e estimators 0 i and . t#e same as ou ound in (aK #K ,ow estimate 0 i ; 0 2 and )ompare our answer wit# t#at or (a8 #at statement )an ou make about estimatin4 )ontrasts in t#e 0 i K () Estimate . N 0 i9 2 0 i ; 0 2 ; 0 39 and . N 0 i9N 0 2 usin4 t#e two solutions to t#e normal e@uations8 7ompare t#e result obtained in ea)# )ase8 3;328 /ppl t#e 4eneral re4ression si4nii)an)e test to t#e experiment in Example 3;"8 S#ow t#at t#e pro)edure ields t#e same results as t#e usual analsis o :arian)e8 3;338 +se t#e ruskal;allis test or t#e experiment in Problem 3;""8 7ompare t#e )on)lusions obtained wit# t#ose rom t#e usual analsis o :arian)eK 3;3!8 +se t#e ruskal;allis test or t#e experiment in Problem 3;"28 /re t#e results )omparable to t#ose ound b t#e usual analsis o :arian)eK 3;358 7onsider t#e experiment in Example 3;"8 Suppose t#at t#e lar4est obser:ation on tensile stren4t# is in)orre)tl re)orded as 508 #at ee)t does t#is #a:e on t#e usual analsis o :arian)eK #at ee)t does it #a:e on t#e ruskal;allis test8
"& | DoE Montgomery
Randomiatin S@uares9 and Related .esi4ns
!;" THE R/,.'-E. 7'P>ETE 1>'7 .ES-D, -n an experiment9 :ariabilit arisin4 rom a nuisan)e a)tor )an ee)t t#e results8 Denerall9 we deine a nuisance factor as a desi4n a)tor t#at probabl #as an ee)t on t#e response9 but we are not interested in t#at ee)t8 Sometimes a nuisan)e a)tor is un$no%n an& unctrolle& t#at is9 we don?t know t#at t#e a)tor exist and it ma e:en be )#an4in4 le:els w#ile we are )ondu)tin4 t#e experiment8 an&omi'ation is t#e desi4n te)#ni@ue used to 4uard a4ainst su)# a Ulurkin4L nuisan)e a)tor8 -n ot#er )ases9 t#e nuisan)e a)tor is $no%n but unctrolle& 8 - we )an a least obser:e t#e :alue t#at t#e nuisan)e a)tor takes on at ea)# run o t#e experiment9 we )an )ompensate or it in t#e statisti)al analsis b usin4 t#e analysis of covariance 89 a te)#ni@ue we will dis)uss in 7#apter "!8 #en t#e nuisan)e sour)e o :ariabilit is $no%n an& controllable 9 a desi4n te)#ni@ue )alled bloc$ing )an be used to sstemati)all eliminate its ee)t on t#e statisti)al )omparisons amon4 treatments8 1lo)kin4 is an extremel important desi4n te)#ni@ue9 used extensi:el in industrial experimentation9 and is t#e sub*e)t o t#is )#apter8 To illustrate t#e 4eneral idea9 suppose we wis# to determine w#et#er or not aour dierent tips produ)e dierent readin4s on #ardness testin4 ma)#ine8 /n experiment su)# as t#is mi4#t be part a 4au4e )apa)ibilit stud8 T#e ma)#ine operates b pressin4 t#e tip into a metal test )oupon9 and rom t#e dept# o t#e resultin4 depression9 t#e #ardness o t#e )oupon )an be determined8 T#e experimenter #as de)ided to obtain our obser:ation or ea)# tip8 T#ere is onl one a)tor ;tip tpe; and a )ompletel randomi
"B | DoE Montgomery
results8 T#us9 "& dierent metal test )oupon would be re@uired in t#is experiment9 one or ea)# run in t#e desi4n8 T#ere is potentiall serious problem wit# a )ompletel randomi
"
2
3
!
C83
C8!
C8&
"080
2
C8!
C83
C8
C8C
3
C82
C8!
C85
C8B
!
C8B
C8&
"080
"082
#appen i t#e are taken rom in4ots t#at are produ)ed in dierent #eats9 t#e experimental units (t#e )oupons will )ontribute to t#e :ariabilit in t#e #ardness data8 /s a result9 t#e experimental error will rele)t 1oth random error and :ariabilit between )oupons8 e )ould like to make t#e experimental error as small as possible t#at is9 we would like to remo:e t#e :ariabilit between )oupons rom t#e experimental error8 / desi4n t#at would a))omplis# t#is re@uire t#e experimenter to test ea)# tip on)e on ea)# o our )oupons8 T#is desi4n9 s#own in Table !;"9 is )alled ran&omi'e& complete bloc$ &esign ()"D*8 T#e obser:ed reponse is t#e Ro)kwell 7 s)ale #ardness minus !08 T#e word U)ompleteL indi)ates t#at ea)# blo)k ()oupon )ontains all t#e treatments (tips8 1 usin4 t#is desi4n9 t#e blo)k9 or )oupons9 rom a more #omo4eneous experimental unit on w#i)# to )ompare t#e tips8 Ee)ti:el9 t#is desi4n strate4 impro:es t#e a))ura) o t#e )omparisons aon4 tips b eliminatin4 t#e :ariabilit amon4 t#e )oupons8 it#in a blo)k9 t#e order in w#i)# t#e our tips are tested is randoml determined8 ,oti)e t#e similarit o t#is desi4n problem to t#e one o Se)tion 2;5 in w#i)# t#e paired t ;test was dis)ussed8 T#e randomi
)atalst eed rate on t#e :is)osit o a polmer8 S#e knows t#at t#ere are se:eral a)tors9 su)# as raw material sour)e9 temperature9 operator9 and raw material to test t#e )atalst eed rate a)tor in blo)ks9 w#ere ea)# blo)k )onsists o some )ombination o t#ese un)ontrollable a)tors8 -n ee)t9 s#e is usin4 t#e blo)ks to test t#e robustness o #er pro)ess :ariable (eed rate to )onditions s#e )annot easil )ontrol8 For more dis)ussion o t#is9 see 7oleman and ont4omer ("CC38 E=/P>E !;" 7onsider t#e #ardness testin4 experiment des)ribed in S)etion !;"8 T#ere are our tips and our a:ailable metal )oupons8 Ea)# tip is tested on)e on ea)# )oupon9 resultin4 in a randomi
Tabel !;3 Randomi
7oupon 1lo)k 2 3
!
Tpe o Tip "
C83
C8!
C8&
"080
2
C8!
C83
C8
C8C
3
C82
C8!
C85
C8B
!
C8B
C8&
"080
"082
Tabel !;! Randomi
Tpe o Tip "
"
7oupon (1lo)k 2 3
;2
;"
"
5
2
;"
;2
3
!
3
;3
;"
0
2
!
2
"
5
B
"C | DoE Montgomery
!
Table !;5 /nalsis o 6arian)e or t#e Hardness Testin4 Experiment Sour)e o 6ariation
Sum o S@uares
.e4rees o Freedom
ean S@uare
F 0
% ;6alue
Treatments (tpe o tip
3850
3
"283
"!8!!
08000C
1lo)ks ()oupons
2850
3
2B850
Error
800
C
08C
"2C800
"5
Total
Table !;& -n)orre)t /nalsis o t#e Hardness Testin4 Experiment as a 7ompletel Randomi
Sum o S@uares
.e4rees o Freedom
ean S@uare
F 0
Tpe o tip
3850
3
"283
"8B0
Error
C0850
"2
B85!
Total
"2C800
"5
1e)ause F 0905838"2 A 38!C9 t#e #pot#esis o e@ual mean #ardness measurements orm t#e our be re*e)ted8 T#us9 t#e randomi
- t#e treatments in an R71. are ixed9 and t#e analsis indi)ates a si4nii)ant dieren)e in treatment means9 t#e experimenter is usuall interested in multiple )omparisons to dis)o:er whi*h treatment means dier8 /n o t#e multiple )omparison pro)edures 20 | DoE Montgomery
dis)ussed in 7#apter 3 (Se)tion 3;5 ma be used or t#is purpose8 -n t#e ormulas o Se)tion 3;59 simpl repla)e t#e number o repli)ates in t#e sin4le;a)tor )ompletel randomiS. pro)edure8 ,oti)e t#at i we use J A 08059 we would )on)lude t#at .2 A .38 ,ow be)ause 38 Y "8 Y 28 (t#at is9 t#e means 28 and 38 span some o remainin4 means9 we )an immediatel )on)lude t#at ." A .2 A .38 Furt#ermore9 .! is dierent rom all t#ree ot#er means8 T#us we )on)lude t#at tip tpe ! produ)es a mean #ardness t#at is si4nii)antl #i4#er t#an t#e mean #ardness readin4s t#e ot#er t#ree o tips8 e )an also use t#e 4rap#i)al pro)edure o 7#apter 3 (Se)tion 3;58" to )ompare tip
esponse+ ,ar&ness in oc$%ell ) !/ for Selecte& 0actorial Mo&el
/nalsis o :arian)e table WPartial sum o s@uaresX Sour)e
Sum o S@uares
ean S@uare
F 6alue
1lo)k
082
.F 3
odel
083
3
08"3
"!8!!
08000C
2,)3
)
2,')
'4,44
2,2225
0800
C
8CE;003
"82C
"5
/ Residual 7or Total Std8 .e:8 ean 7868 PRESS
080C! C8&3 08C 0825
082B
R;S@uared /d* R;S@uared Pred R;S@uared /de@ Pre)ision
Treatment eans (/d*usted9 i ,e)essar ";/"
Estimate& Mean
Stan&ar& Error
C85B
08!B
2" | DoE Montgomery
Prob I F
0820 08BB0& 08!5&3 "58&35
si4nii)ant
2;/2
C8&0
08!B
3;/3
C8!5
08!B
!;/!
C8
08!B
-2 ,E #4 S6E DES47
-n Se)tion !;" we introdu)ed t#e randomiatin letters A+ B+ !+ D9 and E #en)e t#e name >atin s@uare8 e see t#at bot# bat)#es o raw material (rows and operators ()olumns are ort#o4onal to treatments8 T#e >atin s@uare desi4n is used to eliminate two nuisan)e sour)es o :ariabilit t#at is9 it sstemati)all allows blo)kin4 in two dire)tions8 T#us9 t#e rows and )olumns a)tuall represent two restri*tions on randomi6ation 8 -n 4eneral9 a >atin s@uare or p a)tors9 or a p = p >atin s@uare9 is a s@uare )ontainin4 p rows and p )olumns8 Table !;C >atin S@uare .esi4n or t#e Ro)ket Propellant Problem 1at)#es o Raw aterial "
" A A 2!
2 B A 20
'perators 3 ! A "C
2
B A "B
! A 2!
D A 30
E A 2B
A A 3&
3
! A "
D A 3
E A 2&
A A 2B
B A 2"
!
D A 2&
E A 3"
A A 2&
B A 23
! A 22
5
E A 22
A A 30
B A 20
! A 2C
D A 3"
22 | DoE Montgomery
! D A 2!
5 E A 2!
Table !;"2 /nalsis o :arian)e or t#e Ro)ket Propellant Experiment Sour)e o 6ariation
Sum o S@uares
.e4rees o Freedom
ean S@uare
F 0
% ;6alue
Formulations
330800
3
2850
B8B3
080025
1at)#es o raw material
&800
!
"B800
'perators
"50800
!
3B850
Error
"2800
"2
"08&B
Total
&B&800
2!
T#e totals or t#e treatmens (>atin letters are #attin #etter
reatment otal
8"8 A " 828 A ;2! B 838 A ;"3 ! 8!8 A 2! D 858 A 5 E T#e sum o s@uares resultin4 rom t#e ormulations is )omputed rom t#ese total as A
SS Formulations A
A
" p
p
2 8 ; iA"
2888
,,j
N
"2 N (;2!2 N (;"32 N 2!2 N 52 5
;
("02 25
A 330800
T#e error sum o s@uare is ound b subta)tion $ SS E A SS T
; SS 1at)#es ; SS 'perators ; SS Formulations A &B&800 ; &800 ; "50800 ; 330800 A "2800
T#e analsis o :arian)e is summariatin S@uare and ,umber o >atin S@uares o 6arian)e Si
Si
3x3 A B ! B ! A !AB
!x!
5x5
A B ! A B ! D E D B A E ! D B ! D ! D A E B A D E B A ! ! D A E ! D B A B D A B !
,umber o standard s@uares Total number o >atin s@uares
&x&
BxB
pxp
A B ! D E F A B ! D E F G
AB! ,,,,,, %
B ! F A D E B ! D E F G A
B!D ,,,,,,, A
!FBEAD
!DE ,,,,,,,,B
!DEFGAB
D E A B F ! D E F G A B !
,
E A D F ! B
E F G A B ! D
,
F D E ! B A
F G A B ! D E
%AB ,,,,,7% - '8
G A B ! D E F
"
5&
C!0
"&8C!2800
-
"2
"&"920
"95"9200
&"9!BC9!"C9000
pZ( p ; "Z x
!
(number o standard s@uares
5B& a
Some o t#e inormation in t#is table is ound in Statisti)al or 1iolo4i)al and edi)al Resear)# ! t# edition9 b R8/8 Fis#er and F ates9 'li:er and 1od9 Edinbur48 "C538 >ittle is known about t#e properties o >atin s@uare la r4er t#an B x B
-3 ,E 7E)!-#4 S6E DES47
7onsider a p x p >atin s@uare9 and superimpose on it a se)ond p x p >atin s@uare in w#i)# t#e treatments are denoted Dreek letters8 - t#e two s@uares w#en superimposed #a:e t#e propert t#at ea)# Dreek letters appears on)e and onl on)e wit# ea)# >atin letter9 t#e two >atin s@uares are said to be orthogonal+ and t#e desi4n obtained is )alled a 7raeco#atin s8uare. /n example o a ! x ! Drae)o;>atin s@uare is s#own in Table !;"8 T#e Drae)o;>atin s@uare desi4n )an be used to )ontrol sstemati)all t#ree sour)es o extraneous :ariabilit9 t#at is9 to blo)k in three dire)tions8 T#e desi4n allows in:esti4ation o our a)tors (rows9 )olumns9 >atin letters9 and Dreek letters9 ea)# at p le:els in onl p2 runs8 Drae)o;>atin s@uares exist or all p [ 3 exe)pt p A &8 Table !;" ! x ! Drae)o;>atin S@uare .esi4n 7olumn Row "
"
2
3
!
A9
B;
!<
D:
2
B:
A<
D;
!9
3
!;
D9
A:
B<
!
D<
!:
B9
A;
Table !;"C8 /nalsis o 6arian)e or a Drae)o;>atin S@uare .esi4n 2! | DoE Montgomery
Sour)e o 6ariation >atin letter treatments Dreek letter treatments
Sum o S@uares " p 28 j88 SS = A p jA"
;
" p 28> 88 ; SS G A p > A"
Rows 7olumns Error Total
SS ?ows A
" p 28i 88 ;
p ; "
2888
N p ; "
2888
N
p iA"
N p ; " 2888
p iA"
N
SS E 7 b substra)tion 8
28ij>l ; i j > l
p ; "
2888
" p 28i 88 ; SS !ol$mns A
SS T A
.e4rees o Freedom
( p - ) 7p ; " p( ; "
2888
N
T#e statisti)al model or t#e Dra)eo;>atin s@uare desi4n is i A "9 2 + ,,,,+p j & "9 2 + ,,,,+p yij>l & . @\i N ] j @ ω> + ^l @ єij>l > & "9 2 ,,,,+p l &"9 29 ,,,,+p
(!;25
#ere yij>l is t#e obser:ation in row i and )olumn l or >atin letter j and Dreek letter > 9 i is t#e ee)t o t#e it# row9 0 j is t#e ee)t o >atin letter treatment j9 ω> is t#e ee)t o Dreek letter treatment > 9 l is t#e ee)t o )olumn l 9 and C ij>l is an ,-. (092 random error )omponent8 'nl two o t#e our subs)ripts are ne))esarr to )ompletel identi an obser:ation8 T#e analsis o :arian)es is :er similar to t#at o >atin s@uare8 1e)ause t#e Dreek letters appear exa)tl on)e in ea)# row and )olumn and exa)tl on)e wit# ea)# >atin letter9 t#e a)tor represented bt t#e Dreek letters is ort#o4onal to rows9 )olumns9 and >atin letters treatments8 T#ereore9 a sum o s@uares due to t#e 4reek letter a)tor ma be )omputed rom t#e Dreek letters totals and t#e experimental error is urt#er dedu)ed b t#is smount8 T#e )omputational details are illustrated in Table !;"C8 T#e null #pot#eses o e@ual rows9 )olumn9 >atin letter9 and Dreek latter treatments would be tested b di:idin4 t#e )orrespondin4 mean s@uare b mean s@uare error8 T#e re*e)tion re4ion is t#e upper;tail point o t#e F p - '+ 7p - )8 7p - '8 distribution8 25 | DoE Montgomery
E9MP#E - ...................................................................................................................
Suppose t#at in t#e ro)ket propellant experiment o Example !;3 an additional a)tor9 test assemblies9 )ould be o importan)e8 >et t#ere be i:e test assemblies denoted b t#e Dreek letters J9 _9 `9 9 and є t#e resultin4 5 x 5 Drae)o;>atin s@uare desi4n is s#own in Table !;20 on t#e a)in4 pa4e8 ,oti)e t#at9 be)ause t#e totals or bat)#es o raw material (rows9 operators ()oloumns and ormulations (>atin letters are identi)al to t#ose in Example !;39 we #a:e SS 1at)#es A &800 SS 'petarors A "50800 and SS Formulations A 330800 Table !;C >atin S@uare .esi4n or t#e Ro)ket Propellant Problem 1at)#es o Raw aterial " 2 3 ! 5
" A9 A ;" B; A ; ! ` A ;B D A " E єA ;3
2 B< A ;5 !: A ;" Dє A "3 E9 A & A_A 5
,,,l
;"
"
3 ! є A D9 A E _A A `A B: A
'perators ! ;& D; A ;" 5 E< A 2 " A: A 2 " Bє A ;2 ;5 !9 A !
;!
5 E: A Aє A B9 A !; A D< A
;" "" ;! ;3 &
C
5
i,,, ;"! C 5 3 B "0; 8888
T#e totals or t#e test assemblies (Dreek letters are #attin #etter
reatment otal
88"8 A 8828 A 8838 A 88!8 A 8858 A
9 ; < : є
"0 ;& ;3 ;! "3
T#us9 t#e sum o s@uare due to t#e test assemblies is "
SS /ssemblies A
p
2 88 ;
p > A"
A
" 5
2888
,,>
N
W("02 N (;&2 N (;32 N (;!2 N "32X ;
("02 25
A &2800
T#e )omplete analsis o :arian)e is summari
reedom rom "2 (in t#e >atin s@uare desi4n o Example !;3 to 8 T#us9 our estimate o error #as ewer de4rees o reedom9 and t#e test ma be less sensiti:e8 888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888 Table !;2" /nalsis o :arian)e or t#e Ro)ket Propellant Experiment Sum o S@uares
.e4rees o Freedom
ean S@uare
F 0
% ;6alue
330800
3
2850
"0800
080033
1at)#es o raw material
&800
!
"B800
'perators
"50800
!
3B850
Test assemblies
&2800
!
"5850
&&800
825
&B&800
2!
Sour)e o 6ariation Formulations
Error Total
T#e )on)ept o ort#o4onal pairs >atin s@ares ormin4 a Drae)o;>atin s@uare )an be extended somew#at8 / p x p hypers$are is a desi4n in w#i)# t#ree or more ort#o4onal p x p >atin s@uare are superimposed8 -n 4eneral9 up to p N " a)tors )ould be studied i a )omplete set o p ; " ort#o4onal >atin s@uares is a:ailable8 Su)# a desi4n would utili
-n )ertain experiments usin4 randomi
k a
k
treatments appear to4et#er an e@ual number o times8 Suppose t#at t#ere are a treatment and t#at ea)# blo)k )an #old exa)tl > (> c a treatments8 / balan)ed in)omplete blo)k desi4n ma be )onstru)ted b takin4 ( blo)k and assi4nin4 a dierent )ombination o treatments to ea)# blo)k8 Fre@uentl9 #owe:er9 balan)e )an be obtained wit# ewer t#an( blo)k8 Tables o 1-1.s are 4i:en in Fis#er and ates ("C539 .a:ies ("C5&9 and 7o)#ran and 7ox ("C5B8 /s an example9 suppose t#at a )#emi)al en4ineer t#inks t#at t#e time o rea)tion or a )#emi)al pro)ess is a un)tion o t#e tpe o )atalst emploed8 Four )atalst are raw material9 loadin4 t#e pilot plant9 applin4 ea)# )atalst in a separate run o t#e pilot plant9 and obser:in4 t#e rea)tion time8 1e)ause :ariations in t#e bat)#es o raw material ma ae)t t#e perorman)e o t#e )atalsts9 t#e en4ineer de)ides to use bat)#es o raw material as blo)k8 Howe:er9 ea)# bat)# is onl l ar4e enou4# to permit t#ree )atalsts
Table !;228 1alan)ed -n)omplete 1lo)k .esi4n or 7atalst Experiment Treatment (7atalst " 2 3 !
" B3 ; B3 B5
,i
22"
1lo)k (1at)# o Raw aterial 2 3 ! B! ; B" B5 &B B" B5 & ; ; B2 B5 22!
20B
2"
i, 2" 2"! 2"& 222 B0 A 888
Table !;23 /nalsts o 6arian)e or t#e 1alan)ed -n)omplete 1lo)k .esi4n Sour)e o .e4rees o 6ariation Freedom Sum o S@uares ean S@uare Treatmens (ad*usted 1lo)ks
>
SS Treatments (ad*usted
2
"
a ; "
a " k
F 2
2 ;"
;
2 | DoE Montgomery
SS Treatments (ad*usted a ; "
888 N
1 ; "
SS 1lo)ks 1 ; "
F 0 A MS E
SS E (b substra)tion 2
Error
8888
2
88ij ;
Total
SS E
N - a ; 1 N "
N - a ; 1 N " N ; "
N
w#ere i is t#e ad*usted total or t#e it# treatment9 w#i)# is )omputed as " i A
p
nij yij
i A " 9 2 888888889 a
> jA"
(!;30
wit# nij A " i tretament i appears in blo)k j and nij A 0 ot#erwise8 T#e ad*usted treatment totals will alwas sum to
(!;3"
and #as N ; a ; 1 N " de4rees o reedom8 T#e appropriate statisti) or testin4 t#e e@ualit o t#e treatment ee)ts is SS Treatments(ad*usted F 0 A MS E
T#e analsis o :arian)e is summariE !;58 888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888
7onsider t#e data in Table !;22 or t#e )atalst experiment8 T#is is a 1-1. wit# a A !9 1 A !9 > A 39 r A 39 A 29 and N A "28 T#e analsis o t#is data is a ollows8 T#e total sum o s@uares is ( SS T A ij ; 2888 "2
i j l
(B02
A &39"5& ;
"2
A "800
T#e blo)k sum o s@uares is ound rom E@uation !;2 as " ! (
,j
2C | DoE Montgomery
SS /ssemblies A
3 jA"
"
A
;
2888
"2
W(22"2 N (20B2 N (22!2 N (2"2X ;
3
(B02 "2
A 55800
Table !;2! /nalsis o :arian)e or Example !;5 Sour)e o 6ariation Treatmens (ad*usted 1lo)ks Error Total
Sum o S@uares 228B5
.e4rees o Freedom 3
ean S@uare B85
55800 3825 "800
3 5 ""
; 08&5
F 0
% ;6alue
""8&&
080"0B
To )ompute t#e treatment sum o s@uares ad*usted or :lo)ks9 we irst determine t#e ad*usted treatment total usin4 E@uation !;30 as
' A (2" ; (22" N 22! N 2" A
;CG3 ( A (2"! ; (20B N 22! N 2" A ;BG3 ) A (2"& ; (22" N 20B N 22! A ;!G3 4 A (222 ; (22" N 20B N 2" A 20G3 T#e ad*usted sum o s@uare or treatments is )omputed rom @uation !;2C as 2
SS Treatments (ad*usted A
> " iA"
a 3W(;CG32 N (;BG32 N ("!G32 N (20G32X A
(2 (!
T#e error sum o s@uare is obtained b substra)tion as SS E & SS T - SS Treatments (ad*usted - SS 1lo)ks
A "800 ; 228B5 ; 55800 A 3825
30 | DoE Montgomery
A 228B5
T#e analsis o :arian)e is s#own in Table !;2!8 1e)ause t#r P;:alue is small8 we )on)lude t#at t#e )atalst emploed #as a si4nii)ant ee)t on t#e time o rea)tion8 88888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888 Table !;25 /nalsis o :arian)e or Example !;59 -n)ludin4 1ot# Treatment and 1lo)ks Sour)e o 6ariation Treatmens (ad*usted Treatmens (unad*usted 1lo)ks (unad*usted 1lo)ks (ad*usted Error Total
Sum o S@uares 228B5 ""8&B 55800 &&80 3825 "800
.e4rees o Freedom 3 3 3 3 5 ""
ean S@uare B85
22803 08&5
F 0
% ;6alue
""8&&
080"0B
338C0
0800"0
!;5 PR'1>ES !;"8 / )#emist wis#es to test t#e ee)t o our )#emi)al a4ents on t#e stren4t# o a parti)ular tpe o )lot#8 1e)ause t#ere mi4#t be :ariabilit rom one bolt to anot#er9 t#e )#emist de)ides to use a randomi
7#emi)al "
"
2
1olt 3
B3
&
B!
B"
&B
2
B3
&B
B5
B2
B0
3
B5
&
B
B3
&
!
B3
B"
B5
B5
&C
3" | DoE Montgomery
!
5
!;28 T#ree dierent was#in4 solutions are bein4 )ompared to stud t#eir ee)ti:eness in retardin4 ba)teria 4rowt# in 5;4allon milk )ontainers8 T#e analsis is done in a laborator9 and onl t#ree trials )an be run on an da8 1e)ause das )ould represent a potential sour)e o :ariabilit9 t#e experimenter de)ides to use a randomi
"
2
3
!
"3
22
"
3C
2
"&
2!
"B
!!
3
5
!
"
22
!;38 Plot t#e mean tensile stren4t# obser:ed or ea)# )#emi)al tpe in Problem !;" and )ompare t#em to an appropriatel s)aled t distribution8 #at )on)lusions would ou draw rom t#is displaK !;!8 Plot t#e a:era4e ba)teria )ounts or ea)# solution in Problem !;2 and )ompare t#em to a s)aled t distribution8 #at )on)lusions )an ou drawK !;58 /n arti)le in t#e Fire Saet ournal (UT#e Ee)t o ,o<
32 | DoE Montgomery
""8B3 08B 085 08C3 "8"! 08CB
et Elux 6elo)it (mGs "!8B3 "&85C 208!3 238!& 080 08" 08B5 08BB 085 08C2 08& 08" 08C2 08C5 08C 08C 08CB 08C 08 08& 08& 08B 08B& 08B&
28B! 08B 083 083 083 08B5
(a .oes no<
" !8C3(0805 !85(080! !83(080C !8C(0803
2 !8&(0805 !8C"(0802 !8(08"3 !8BB(080!
Time Period 3 ! !8B5(0805 !8BC(0803 !8C0(08"" !8C!(0805
!8C5(080& !85(0805 !8B5(08"5 !8&(0805
5
&
!8BC(0803 !8B5(0803 !82(080 !8BC(0803
!8(0805 !85(0802 !8C0(08"2 !8B&(0802
(a /nal
33 | DoE Montgomery
Ea)# urna)e )an be run at our dierent atirrin4 rates8 / randomi
Furna)e " "! "! "B
2 ! 5 & C
3 5 & C 3
! ! C 2 &
(a -s t#ere an e:iden)e t#at stirrin4 rate ae)ts 4rain se
!;"38 /n industrial en4ineer is )ondu)tin4 an experiment on ee o)us time8 #e is interested in t#e ee)t o t#e distan)e o t#e ob*e)t rom t#e ee on t#e o)us time8 Four dierent distan)es are o interest8 He #as i:e sub*e)ts a:ailable or t#e experiment8 1e)ause t#ere ma be dieren)es amon4 indi:iduals9 #e de)ides to )ondu)t t#e experiment in arandomi
" "0 B 5 &
2 & & 3 !
Sub*e)t 3 & & 3 !
! & " 2 2
5 & & 5 3
!;"!8 T#e ee)t o i:e dierent in4redients ( A+ B+ !+ D+ E on t#e rea)tion time o a )#emi)al pro)ess is bein4 studied8 Ea)# bat)# o new material is onl lar4e enou4# to permit i:e runs to be made8 Furt#ermore9 ea)# run re@uires approximatel "f #ours9 so onl i:e runs )an be made in one da8 T#e experimenter de)ided to run t#e experiment as a >atin s@uare so t#at da and bat)# ee)ts ma be sstemati)all )ontrolled8 S#e obtains t#e data t#at ollow8 /nal
1at)# " 2 3 ! 5
" A A ! A "" B A ! D A & E A !
2 B A B E A 2 A A C ! A D A 2
.a 3 D A " A A B ! A "0 E A & B A 3
! ! A B D A 3 E A " B A & A A
5 E A 3 B A D A 5 A A "0 ! A !
!;"58 /n industrial en4ineer is in:esti4atin4 t#e ee)t o our assembl met#ods ( A+ B+ !+ D+ on t#e assembl time or a )olor tele:ision )omponent8 Four operators are sele)ted or t#e stud8 Furt#ermore9 t#e en4ineer knows t#at ea)# assembl met#od produ)es su)# ati4ue t#at t#e time re@uired or t#e last assembl ma be 4reater t#an t#e time re@uired or t#e irst9 re4ardless o t#e met#od8 T#at is9 a trend de:elops in t#e re@uired assembl time8 To a))ount or t#is sour)e o
35 | DoE Montgomery
:ariabilit9 t#e en4ineer uses t#e >atin s@uare desi4n s#own below8 /nal
'perator " ! A "0 BAB AA5 D A "0
2 D A "! ! A " B A "0 A A "0
3 A A B D A "" ! A "" B A "2
! B A A A D A C ! A "!
!;"&8 Suppose t#at in Problem !;"! t#e obser:ation rom ea)# bat)# 3 on da ! is missin48 estimate t#e missin4 :alue rom e@uation !;2!9 and perorm t#e analsis usin4 t#e :alue8 !;"B8 7onsider a p x p >atin s@uare wit# rows (9i9 )olumns ( ; > 9 and treatments (0 j ixed8 'btain least s@uares estimates o #e model parameters 9 +i ; > + and 0 j8 !;"8 dri:e t#e missin4 :alue ormula (E@uation !;2! or t#e >atin s@uare desi4n8 !;"C8 Design involving several Latin squares 8 WSee 7o)#ran and 7ox ("C5B9 o#n ("CB"X8 T#e p x p >atin s@uare )ontains onl p obser:ations or ea)# treatment8 To obtain more repli)ations t#e experimenter ma use se:eral s@uares9 sa n8 -t is immaterial w#et#er t#e s@uares used are t#e same or dierent8 T#e appropriate model is
i A "9 2 + ,,,,+p j & "9 2 + ,,,,+p yij>h & . @ h N Ji7h8 @ ] j + ; >7h8 @ 708 jh @ єij>l
> & "9 2 ,,,,,+p h & "9 29,,,,+n
#ere ij>l is t#e obser:ation on treatment j in row i and )olumn > o t#e ht# s@uare8 ,ote t#at Ji7h8 and ; >7h8 are t#e row and )olumn ee)ts in t#e ht# s@uare9 h is t#e ee)t o t#e ht# s@uare9 and 708 jh is t#e intera)tin4 between treatment and s@uares8 (a set up t#e normal e@uations or t#is model9 and sol:e or estimates o t#e model parameters8 /ssume t#at appropriate side )onditions on t#e parameters are h h A 09 i Ji7h8 A 09 and > ; >7h8 A 0 or ea)# h9 j] j A 09 j 708 jh A 0 or ea)# h and h 708 jh & 0 or ea)# j, 3& | DoE Montgomery
(b rite down t#e analss o :arian)e table or t#is desi4n8 !;208 .is)uss #ow t#e operatin4 )#ara)teristi)s )ur:es in t#e /ppendix ma be used wit# t#e >atin s@uare desi4n8 !;2"8 Suppose t#at in Problem !;"! t#e data taken on da 5 were in)orre)tl analatin s@uare t#at ollow was used8 /nal
1at)#es " 2 3 ! 5
" A9 A B` A ! є A D; A E: A
/)id )on)entrations 3 !
2 2& " 20 "5 "0
B_ A "
! ` A "!
!: A
Dє A "
2" D9 A "2 E< A "5 Aє A 2!
E _ A"& A: A 22 B9 A "B
D: A
"& E9 A "" A< A 25 Bє A "! ! ; A "B
5 E є A A ; A B : A !9 A D< A
"3 2" "3 "B "!
!;238 Suppose t#at in Problem !;"5 t#e en4ineer suspe)ts t#at t#e workpla)es used b t#e aour operators ma represent an additional sour)e o :ariation8 / ourt# a)tor9 workpla)e (J9 _9 `9 9 ma be introdu)ed and anot#er experiment )ondu)ted9 ieldin4 t#e Dra)eo;>atin s@uare t#at ollows9 /nal
!;2!8 5 or
" 2 3 !
'perator " !; A "" B9 A A: A C D` A C
2 B` A "0 !: & "2 D9 A "" A; &
3 D: A "! A< A "0 B; A B !9 A "
! A9 A D; A "2 !< A "5 B; A &
7onstru)t a 5 x #pers@uare studin4 t#e ee)t o i:e a)tors8 Ex#ibit
t#e anals o :arian)e table or t#is desi4n8 !;258 7onsider t#e data in Problem !;"5 and !;238 Suppressin4 t#e Dreek letters in !; 239 anal
3B | DoE Montgomery
!;2&8 7onsider t#e randomi
7ar 3 "!
2 "B "!
"! "2 "3 ""
"3 "" "0
"" "2
! "3 "3 "2 "2
5 "2 "0 C
!;28 7onstru)t a set o ort#o4onal )ontrasts or t#e data in Problem !;2B8 7ompute t#e sum o s@uares or ea)# )ontrast8 !;2C8 Se:en dierent #ardwood )on)entrations are bein4 studied to determine t#eir ee)t on t#e stren4t# o t#e paper produ)ed8 Howe:er9 t#e pilot plant )an onl produ)e t#ree runs ea)# da8 /s das ma dier9 t#e analst uses t#e balan)ed in)omplete blo)k desi4n t#at ollows8 /nal
" ""! "2&
2 "20 "3B
"!"
3
.as !
5 "20
&
B ""B
""C ""B "2C
"!5
"3! "!C "50
"20 "3&
"!3 ""
"23 "30
"2B
!8308 /nali
2
"
3 | DoE Montgomery
!;3"8 Pro:e t#e > i&' (a is t#e ad*usted sum o s@uares or treatment in a 1-1.8 !;328 /n experimenter wis#es to )ompare our treatment in blo)ks o two runs8 Find a 1-1. or t#is experiment wit# six blo)k8 !;338 /n experimenter wis#es to )ompare ei4#t treatments in blo)ks o our runs8 Fin a 1-1. wit# "! blo)ks and A 3 !;3!8 Perorm t#e interblo)k analsis or t#e desi4n in Problem !;2B8 !;358 Perorm t#e interblo)k analsis or t#e desi4n in Problem !;2C8 !;3&8 6eri t#at a 1-1. wit# t#e parameters a A 9 r A 9 > A ! and 1 A "& does not exist8 !;3BS#ow t#at t#e :arian)e o t#e intrablo)k estimators (]i is > (a ; " 2 G (a28 !;38 Extended incomplete block designs 8 '))asionall t#e blo)k si c 2a8 /n extended in)omplete blo)k desi4n )onsists o a sin4le repli)ate o ea)# treatment in ea)# blo)k alon4 wit# an in)omplete blo)k desi4n wit# > g A > ; a8 -n t#e balan:ed )ase9 t#e in)omplete blo)k desi4n will #a:e parameters > g A > ; a+ r g A r ; 1 and g 8 rite out t#e tatisti)al analsis8 ( int $ -n t#e extended in)omplete blo)k desi4n9 we #a:e A 2r ; 1 N g8 Table 5;! >ie .ata (in #ours or t#e 1atter .esi4n Experiment aterial Tpe
Temperature (oF "5 "30 B! "50 "5C "3 "&
"55 "0 " "2& ""0 "&0 "B3
3! 0 "3& "0& "B! "50
!0 B5 "22 ""5 "20 "3C "2C"
20 2 25 5 C& 2
B0 5 B0 !5 "0! &0 BB0
CC "30 "50" 3BCC A 888
T#e sum o s@uares are )omputed as ollows $ a
SS T A
1
n
2
i,, ;
i&' j&' l>&'
2 888
a1n
A ("302 N ("552 N (B!2 N 888 N (&02 ; " a SS aterial A
2 ;
2888
,,j
1n iA"
3C | DoE Montgomery
"2
(3BCC2 3&
A BB9&!&8CB
(B02 A W(CC2 N ("3002 N ("50"2 ; A "09&38B2 (3(! 3& "
" 1 SS Temperature
2
',, ;
A
an jA"
2
888 a1n
(3BCC2 A W("B32 N ("2C"2 N (BB02 ; A 3C9""8B2 (3(! 3& "
" SS -ntera)tion
A
a
1
2
',, ;
n iA" j&'
2
888
;
SS aterial ; SS Treatment
a1n
(3BCC2 A W(53C2 N (22C2 N 8888 N (3!22 ; A "09&38B2 ! 3& ; 3C9""8B2 A C&"38B "
and SS E & SS T - SS aterial - SS Temperature ; SS -ntera)tion
A BB9&!&8CB ; "09&38B2 ; 3C9""8B2 ; C&"38B A "92308B5 T#e analsis o :arian)e is s#own in Table 5;5 on t#e next pa4e8 1e)ause F 2,2A,(H A 28B39 we )on)lude t#at t#ere is a si4nii)ant intera)tion between material tpes and temperature8 Furt#ermore F 2,2,(,(H A 39359 so t#e main ee)t o material tpe and temperature are also si4nii)ant8 table 5;5 also s#ows t#e % ;:alues or t#e test statisti)s8 To assist in interpretin4 t#e results o t#is experiment9 it is #elpul to )onstru)t a 4rap# o t#e a:era4e response at ea)# treatment )ombination8 T#is 4rap# is s#own in Fi4ure 5;C8 at t#e bottom t#e nex pa4e8 T#e si4nii)ant intera)tion is indi)ated b t#e la)k o parallelism o t#e lines8 -n 4eneral9 lon4er lie is attained at low temperature9 re4ardless o material tpe8 7#anin4 rom low to intermediate temperature9 batter lie Table 5;5 /nalsis o 6arian)e or 1atter >ie .ata Sour)e o 6ariation aterial tpes Temperature -ntera)tion Error
Sum o S@uares
.e4rees o Freedom
"09&8B2 3C9""8B2 C9&"38B "92308B5
2 2 ! 2B
!0 | DoE Montgomery
ean S@uare 593!"8& "C95CC83& 29!038!! &B582"
F 2
% ;:alue
B8C" 28CB 385&
080020 08000" 080"&
Total
BB9&!&8CB
35
wit# material tpe 3 a)tuall in*reases9 w#ereas it de)rease or tpes " and 28 From intermediate to #i4# temperature9 batter lie de)rease or material tpes 2 and 3 and is essentiall un)#an4ed or tpe "8 aterial tpe 3 seems ti 4i:e t#e best results i we want less loss o ee)ti:e lie as t#e temperature )#an4es8 esponse+ #ife
in ;ours
!/ for Selecte& 0actorial Mo&el
/nalsis o :arian)e table WPartial sum o s@uaresX Sum o S@uares 5C!"&822
.F
A B AB
'2I3),H( )5''3,H(
( (
=a*> of Fit %$re error
2,222 '3()2,H
Sour)e odel
Residual
7or Total Std8 .e:8 ean 7868 PRESS
C&"38B "2308B5 BB&!&8CB
258C "05853 2!8&2 32!"0822
! 2B 0 (H
35
ean S@uare B!2B803
F 6alue ""800
Prob I F c08000"
)4',3I '55,)I (42),44
H,5' (3,5H ),I
2,222( J2,222' 2,2'3I
&B582"
si4nii)ant
IH,('
R;S@uared /d* R;S@uared Pred R;S@uared /de@ Pre)ision
08B&52 08&C5& 0852& 8"B
E9MP#E 5-3 8888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888 ;e Soft Drin$ "ottling Problem
/ sot drink bottler is interested in obtainin4 more uniorm ill #ei4#es in t#e bottles produ)ed b t#e manua)turin4 pro)ess8 T#e illin4 ma)#ine t#eoreti)all ills ea)# bottle to t#e )orre)t tar4et #ei4#t9 but in pra)ti)e9 t#ere is :ariation around t#is tar4et9 and t#e bottler would like to understand better t#e sour)es o t#is :ariabilit and e:entuall redu)e it8 T#e pro)ess en4ineer )an )ontrol t#ree :ariables durin4 t#e illin4 pro)ess t#e per)ent )arbonation ( A9 t#e operatin4 pressure in t#e iller ( B9 and t#e bottles produ)ed per minute or t#e line speed (! 8 T#e pressure and speed are eas to )ontrol9 but t#e per)ent )arbonation is more dii)ult to )ontrol durin4 a)tual manua)turin4 be)ause it :aries wit# produ)t temperature8 Howe:er9 or purpose o an experiment9 t#e en4ineer )an )ontrol )arbonation at t#ree le:els$ "09 "29 and "! per)ent8 S#e )#ooses two le:els or pressure (25 and 30 psi and two le:els or line speed (200 and 250 bpm8 S#e de)ides to !" | DoE Montgomery 2
"
run two repli)ates o a a)torial desi4n in t#ese t#ree a)tors9 wit# all 2! runs taken in random order8 T#e response :ariable obser:ed is t#e a:era4e de:iation rom t#e tar4et ill #ei4#t obser:ed in a produ)tion run o bottles at ea)# set o )onditions8 T#e data t#at resulted rom t#is experiment are s#own in Table 5;"38 Positi:e de:iation are ill #ei4#ts abo:e t#e tar4et9 w#ereas ne4ati:e de:iations are ill #ei4#ts below t#e tar4et8 t#e )ir)led numbers in Table 5;"3 are t#e t#ree;wa )ell total ij>, 8 T#e total )orre)ted sum o s@uare is ound rom E@uation 5;2B as a
1
*
n
2
2
SST A ij>l ; i &'
j &'
> &' l&'
2
A 5B" ;
(B5
!2 | DoE Montgomery
2!
8888
a1*n
A 33&8&25
5
!3 | DoE Montgomery
F/7T'R-/> .ES-D,
Table 5;"38 Fill Hei4# .e:iation .ata Example 5;3 'peratin4 apressure 25 psi line Speed (! 200 250 ; ";" ;! 0 23 " "
Per)ent 7arbonation ( A "0 "2 "!
30 psi line Speed (! 200 250 ;";" "2 0 " 2 5 & "" 3 5
;3 ;" 0 " 3 C !
B"3 &
B "& C
"0 2" ""
&
"5
20
3!
B x ! Total ,j>,
,j,
2" A xB Total ij,,,
"0 "2 "!
25 ;5 ! 22
;! 20 5C
B5 A y,,,
2! B x ! Total
i,>,
B A
i,,,
!
30 " "& 3B
200 250 ;5 " & "! 25 3!
A
"0 "2 "!
Table 5;"! /nalsis o 6arian)e or Example 5;3 Sour)e o 6ariation Per)enta4e o )arbonation ( A 'peratin4 pressure ( B >ine speed (! AB A! B! AB!
Error Total
!! | DoE Montgomery
Sum o S@uares
.e4rees o Freedom
2528B50 !583B5 2280!2 58250 0853 "80!2 "803 8500 33&825
2 " " 2 2 " 2 "2 23
ean S@uare "2&83B5 !583B5 2280!2 28&25 082C2 "80!2 085!2 08B0
F 2
% ;:alue
"B8!"2 &!805C 3"8"" 38B0& 08!"2 "8!B" 08B&5
c08000" c08000" 08000" 08055 08&B"3 082!5 08!&B
esponse+ #ife
in ;ours
!/ for esponse Surface e&uce& )ubic Mo&el
/nalsis o :arian)e table WPartial sum o s@uaresX Source
odel
Sum of S8uares
5C!"&822
D0
B!2B803
' ( ' ( (
)524(,IH )4',3I HI,2I ''H,4 )I45,)
IH,('
A B A( AB A( B
)524(,IH '2I3),H( HI,2I ()',23 H(53,I5
"2308B5
2B
=a*> of Fit %$re error
2,222 '3()2,H
2 (H
Residual
7or Total Std8 .e:8 ean 7868 PRESS
BB&!&8CB 258C "05853 2!8&2 32!"0822
Mean S8uare
&B582"
35 R;S@uared /d* R;S@uared Pred R;S@uared /de@ Pre)ision
0 /alue
Prob < 0
H,3( H,5' 2,'' ',H' ,42
J2,222' 2,22(2 2,H)53 2,'55' 2,2'2I
""800
c08000"
Si4nii)ant
08B&52 08&C5& 0852& 8"B
E=/P>E 5;5 8888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888 T#e ee)ti:e lie o a )uttin4 tool installed a numeri)all )ontrolled ma)#ine is t#ou4#t to be ae)ted b t#e )uttin4 speed and t#e tool an4le8 T#ree speeds and t#ree an4les are sele)ted9 and a a)torial experiment wit# two repli)ates is perormed8 T#e )oded data are s#own in Table 5;"&8 T#e )ir)led numbers in t#e )ell are )ell totals ( ij,8 7ondensed .esi4n;Expert output or t#is example is s#own in Table "B8 T#e terms A and A( are t#e linear and @uadrati) ee)ts o tool an4le9 and B and B( are t#e linear and @uadrati) ee)ts o speed8 T#e terms AB A( B9 AB( + and A( B( represent linear x linear9 @uadrati) x linear9 linear x @uadrati)9 an4 @uadrati) x components o t#e two;a)tor intera)tion8 /lt#ou4# t#ere are some lar4e % ;:alue9 all model terms #a:e been retained to ensure #ierar)#8 T#e predi)tion e@uation expressed in doded a)tors uses t#e le:els ;"9 09 and N" or A and B to represent t#e low9 middle9 and #i4# le:els o t#ese a)tors9 respe)ti:el8 Fi4ure 5;"C (pa4e 20& presents a counter plot o t#e sura)e 4enerated b t#e predi)tion e@uation or tool lie8 Examination o t#is response sura)e indi)ates t#at maximum tool lie is a)#e:ed at )uttin4 speeds around "50 rpm and tool an4les o 25o8 T#e t#ree;dimensional response sura)e plot in Fi4ure 5;20 (pa4e 20& pro:ides essentiall t#e sam inormation9 but it pro:ides a dierent and sometimes more useul perpe)ti:e o t#e tool lie response sura)e8 Exploration o response sura)es is a :er important aspe)t o experiment desi4n9 w#i)# we will dis)uss in more detail in 7#apter ""8 !5 | DoE Montgomery
Table 5;"& .ata or Tool >ie Experiment Tool /n4le (de4rees
7uttin4 Speed (inGmm "50 "B5 ;3;3 2 5 0 3 " ! ! "0 3 & 5 "" 0 ;" & ;"
"25 ;2;3 ;" 0 2 2 ;";" 0
"5 20 25 8,j,,
;2
esponse+ #ife
"2
i,,, ;" "& C 2! A 888
"!
in ;ours
!/ for esponse Surface e&uce& )ubic Mo&el
/nalsis o :arian)e table WPartial sum o s@uaresX Sum of S8uares
Source
odel
A B A( B( AB A( B AB( A( B(
Residual
=a*> of Fit %$re error
7or Total Std8 .e:8 ean 7868 PRESS
"""8000 45,22 'I,22 2,222 ',)) 3,22
28&B
4(,IH 3,222 '),22 2,222
"2!800 "820 "833 C08"! 52800
!& | DoE Montgomery
D0
Mean S8uare
"38B
' ' ' ' ' ' ' ' 5 2 5
45,22 'I,22 2,222 ',)) 3,22 (,IH 4(,IH
800 "8!!
0 /alue
C8&"
)),5( '',23 2,222 2,5( ,4 ',3 (5,4 ,4
',44
"B R;S@uared /d* R;S@uared Pred R;S@uared /de@ Pre)ision
08C52 08020 0850& 823B
Prob < 0
c0800"3
2,222) 2,2233 ',2222 2,24)' 2,(2H) 2,2224 2,24)'
Si4nii)ant
Table 5;C -ntensit >e:el at Tar4et Rete)tion 'perators (blo)ks Filter Tpe Dround )lutter >ow medium Hi4#
"
2
"
2
C0 "02 ""!
& B C3
3
!
"
2
"
2
"
2
C& "0& ""2
! C0 C"
"00 "05 "0
C2 CB C5
C2 C& C
" 0 3
Table 5;20 /nalsis o 6arian)e or Example 5;& Sour)e o 6ariation Dround )lutter (D Filter tpe (F DF 1lo)ks Error Total
Sum o S@uares
.e4rees o Freedom
33585 "0&&8&B BB80 !028"B "&&833 20!B83
2 " 2 3 "5 23
ean S@uare "&B8BC "0&&8&B 385! "3!80& ""80C
F 2
% ;:alue
"58"3 C&8"C 38!
080003 c08000" 0805B3
+sual manner8 T#e sum o suares due to blo)ks is ound rom t#e operator total ( ,,,> "
SS 1lo)ks A a1
n
2
,,,>
2,,, a1n
> A"
"
(22B2
A W(5B22 N (5BC2 N (5CB2 N (5302 X; (3(2 (3(2(! A !028"B T#e )omplete analsis o :aian)e or t#is experiment is summari
operator?s abilit to dete)t t#e tar4et9 and t#ere is some e:iden)e o mild intera)tion between t#ese a)tors8 88888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888 -n t#e )ase o two randomi ;a)tor a)torial desi4n exa)tl e@uals t#e number o restri)tion le:els9 t#at is9 i p A a1 ,,,,m9 t#en t#e a)torial desi4n ma be run in a p x p >atin s@uare8 For example9 )onsider a modii)ation o #e radar tar4et dete)tion experiment o Example 5;&8 T#e a)tors in t#is experiment are ilter tpe (two le:els and 4round )lutter (t#ree le:els9 and operators are )onsidered as blo)ks8 Suppose now t#at be)ause o t#e setup time re@uired9 onl six runs )an be made per da8 T#us9 das be)ome a se)ond randomiatin s@uare desi4n9 as s#own in Table 5;2" on t#e next pa4e8 -n t#is table we #a:e used t#e lower)ase letter f i and g j f ' g ( represents ilter tpe " and medium 4round )lutter8 ,ote t#at now six operators are re@uired9 rat#er t#an our as in t#e ori4inal experiment9 so t#e number o treatment )ombinations in t#e 3 x 2 a)torial desi4n exa)tl e@uals t#e number o restri)tion le:els8 Furt#ermore9 in t#is desi4n9 ea)# operator would be used onl on)e on ea)# da8 T#e >atin letters A+ B+ !+ D+ E and F represent t#e 3 x 2 A & a)torial treatment )ombinations as ollows$ A A f ' g '9 B A f ' g ( 9 ! A f ' g ) 9 D A f ( g '9 E A f ( g ( and F A f ( g ) 8 T#e i:e de4rees o reedom between t#e six >atin letters )orrespond to t#e main ee)ts o ilter tpe (one de4ree o reedom9 4round )lutter (two de4rees o reedom9 Table 5;2" Radar .ete)tion Experiment Run in a & x & >atin S@uare 'perator "
2
3
"
A 7f ' g ' A C0
B7 f ' g ( A "0&
!7 f ' g ) A"0
D ( f ( g ' A "
F7 f ( g ) A C0
E7 f ( g ( A
2
!7 f ' g ) A""!
A 7f ' g ' A C&
B7 f ' g ( A"05 F7 f ( g ) A 3
E7 f ( g ( A &
D ( f ( g ' A !
3
B7 f ' g ( A "02
E7 f ( g ( A C0
F7 f ( g ) A C5
B7 f ' g ( A "00
!7 f ' g ) A"0!
!
E7 f ( g ( A B
D ( f ( g ' A !
A 7f ' g ' A "00 B7 f ' g ( AC&
D ( f ( g ' A 5
F7 f ( g ) A C"
5
F7 f ( g ) A C3
!7 f ' g ) A""2
D ( f ( g ' A C2
E7 f ( g ( A 0
!7 f ' g ) A""0
B7 f ' g ( A C
&
D ( f ( g ' A &
F7 f ( g ) A C"
E7 f ( g ( A CB
!7 f ' g ) AC
A 7f ' g ' A C0
A 7f ' g ' A C2
.a
!
A 7f ' g ' A C2
5
&
and t#eir intera)tion (two de4rees o reedom8 T#e linear statisti)al model or t#is desi4n is i A "9 29 88889& j & "9 2 + 3
! | DoE Montgomery
yij>l & . @ Ji @ ] j + ; > @ 70;8 j> @ i
N
єij>l
> & "9 2
(5;3
l & "9 29,,,,+&
w#ere ] j and ; > 9 are ee)ts o 4round )lutter and ilter tpe9 repe)ti:el9 and Ji and i represent t#e randomiow edium Hi4#
Filter Tpe " 5&0 &0B &!& ""3
y ,,>,
Filter Tpe 2 5"2 52 5!3 "53
Furt#ermore9 t#e row and )olumn totals are Row ( y ,j>l 5&3 5& 5& 5&5 7olumns ( y i j>, 5B2 5BC 5CB
5&! 530
Filter Tpe 3 "0B2 ""35 ""C 33C& A y ,,,,
55B
T#e analsis o :arian)e is summariatin S@uare Sour)e o 6ariation Dround )lutter (D Filter tpe (F DF .as (rows 'perators ()olumns Error Total
Sum o S@uares 5B"850 "!&C8!! "2&8B3 !933 !2800
.e4rees o Freedom 2 " 2 5 5
Deneral Formula or .e4rees o Freedom a ; " 1;" (a ; " (1 ; " a1 ; " a1 ; "
"C8000 2BC800
20 35
(a1 ; " (a1 ; 2 (a12 ; "
!C | DoE Montgomery
ean S@uare 258B5 "!&C8!! &383B 08B 58&0 C8C0
F 2
% ;:alue
28& c08000" "!8!3 c08000" &8!0 0800B"
5;B PR'1>ES 5;" T#e ield o )#emi)al pro)ess is bein4 studied8 T#e two most important :ariables are t#ou4#t to be t#e pressure and t#e temperature8 T#ree le:els o ea)# a)tor are sele)ted9 and a a)torial experiment wit# two repli)ates is perormed8 T#e ield data ollow$ Temperature (o7 "50 "&0 "B0
200 C08! C082 C08" C083 C085 C08B
Pressure (psi4 2"5 C08B C08& C085 C08& C08 C08C
230 C082 C08! C08C C08" C08! C08"
(a /nal
0825 CC "0! C&
0825
C2 & &
C "0!
CC "0 C5
"0! ""0 CC
0830
CC C "02
"0! CC C5
"0 ""0 CC
""! """ "0B
Feed Rate (inGmin 0820
08"5 B! &! &0
(a /nal
5;38 For t#e data in Problem 5;29 )ompute a C5 per)ent )oniden)e inter:al estimate o t#e mean dieren)e in response or eed rates o 0920 and 0825 inGmin8 5;!8 /n arti)le in "nd$strial $ality !ontrol ("C5&9 pp8 5; des)ribes an experiment to in:esti4ate t#e ee)t o t#e tpe o 4lass and t#e tpe o p#osp#or on t#e bri4#tness o an tele:ision tube8 T#e response :ariable is t#e )urrent ne)essar (in mi)roamps to obtain a spe)iied bri4#tness le:el8 T#e data are as ollow$
Dlass Tpe " 2
" 20 2C0 25
P#osp#or Tpe 2 300 3"0 2C5
3 2C0 25 2C0
230 235 2!0
2&0 2!0 235
220 225 230
(a -s t#ere an indi)ation t#at eit#er a)tor inluen)es bri4#tnessK +se J A 08058 (b .o t#e two a)tors itera)tK +se J A 08058 () /naleone (Statisti* and Experimental Design in Engineering and the %hysi*al S*ien*es9 ale9 "CBB des)ribe an experiment to in:esti4ate warpin4 o )opper plates8 T#e two a)tors studied were t#e temperatures and t#e )opper )ontent o t#e plates8 T#e response :ariable was a measure o t#e amount o warpin48 T#e data were as ollows$
Temperature ( 7 20 B5 "00 "25 o
!0 "B820 "28C "&8"2 2"8"B
7opper 7ontent (% &0 0 "&82" 2!822 "8"3 "B8"2 "82" 25823 2382" 23822
"00 282B 2B83" 30823 2C83"
(a -s t#ere an indi)ation t#at eit#er a)tor ae)ts t#e amount o warpin4K -s t#ere an intera)tion between t#e a)torK +se J A 08058 (b /nal
t#e dierent le:els o )opper )ontent on warpin48 - low warpin4 is desirable9 w#at le:el o )opper )ontent would ou spe)iK (d Suppose t#e temperature )annot be easil )ontrolled in t#e en:ironment in w#i)# t#e )opper plate are to be used8 .oes t#is )#an4e our answer or part ()K 5;&8 T#e a)tors t#at inluen)e t#e breakin4 stren4t# o a snt#eti) iber re bein4 studied8 Four produ)tion ma)#ines and t#ree operators are )#oses and a a)torial experiment is run usin4 iber rom t#e sama produ)tion bat)#8 T#e results are as ollows$ a)#ine 'perator "
" "0C ""0
2 ""0 ""5
3 "0 "0C
! ""0 "0
2
""0 ""2
""0 """
""" "0C
""! ""2
3
""& ""!
""2 ""5
""! ""C
"20 ""B
(a /nal
.rill Speed "25
080"5 28B0 28B
Feed Rate 08030 080!5 28!5 28&0 28!C 28B2
200
283 28&
285 280
52 | DoE Montgomery
28& 28B
080&0 28B5 28& 28C! 28
5;8 /n experiment is )ondu)ted to stud t#e inluen)e o operatin4 temperature and t#ree tpes o a)e;plate 4lass in t#e li4#t output o an os)illos)ope tube8 T#e ollowin4 data are )olle)ted$
Dlass Tpe "
2 3
"00 50 5& 5B0
Temperature "25 "0C0 "0B "05
"50 "3C2 "30 "3&
550 530 5BC
"0B0 "035 "000
"32 "3"2 "2CC
5!& 5B5 5CC
"0!5 "053 "0&&
&B C0! C
(a +se J A 0805 in t#e analsis8 -s t#ere a si4nii)ant intera)tion ee)tK .oes 4lass tpe or temperature ee)t t#e reponseK #at )on)lusions )an ou drawK (b Fit an appropriate model relatin4 li4#t output to 4lass tpe and temperature8 () /nal
Position "
53 | DoE Montgomery
"00 5B0 5&5 53
Temperature (o7 "25 "0&3 "00 "0!3
"50 5&5 5"0 5C0
52 5!B 52"
2
C "02& "00!
52& 53 532
Suppose we assume t#at no intera)tion exists8 rite down t#e statisti)al model8 7ondu)t t#e analsis o :arian)e and test #pot#esis on t#e main ee)ts8 #at )on)lusions )an be drawnK 7ommnet on t#e model?s ade@ua)8 5;"28 .eri:e t#e expe)ted mean s@uares or a two;a)tor analsis o :arian)e wit# one obser:ation per )ell9 assumin4 t#at bot# a)tors are ixed8 5;"38 7onsider t#e ollowin4 data rom a two;a)tor a)torial experiment8 /nal
Row Fa)tor " 2 3
080"5 3& " 30
7olumn Fa)tor 08030 080!5 3C 3& 20 22 3B 33
080&0 32 20 3!
5;"!8 T#e s#ear stren4t# o an ad#esi:e is t#ou4#t to be ae)ted b t#e appli)ation pressure and temperature8 / a)torial experiment is perormed in w#i)# bot# a)tors are assumed to be ixed8 /nal
yij> & . @
] j @ ; j + <> @ 70;8 j> @ i N
єij>
"9 29 ,,,,+a j & "9 2 +,,,,,+1 > & "9 2 ,,,,,+*
,oti)e t#at t#ere is onl on)e repli)ate8 /ssumin4 all t#e a)tors are ixed9 write down t#e analsis o :arian)e table9 in)ludin4 t#e expe)ted mean s@uares8 #at would ou use as t#e Uexperimental errorL to test #pot#esisK 5;"&8 T#e per)enta4e o #ardwood )on)entration in raw pulp9 t#e :at preesure9 and t#e )ookin4 time o t#e pulp are bein4 in:esti4ated or t#eir ee)ts on t#e stren4t# o paper8 T#ere le:els o #ardwood )on)entration9 t#ree le:els o pressure9 and two )ookin4 times are sele)ted8 / a)torial experiment wit# two repli)ates is )obdu)ted9 and t#e ollowin4 data are obtained8 5! | DoE Montgomery
Per)enta4e o Hardwood 7on)entration
7ookin4 Time 380 Hours Pressure !00 500 &50 "C&8& "CB8B "CC8 "C&80 "C&80 "CC8
7ookin4 Time 380 Hours Pressure !00 500 &50 "C8! "CC8& 2008& "C8& 2008! 2008C
!
"C85 "CB82
"C&80 "C&8C
"C8! "CB8&
"CB85 "C8"
"C8B "C80
"CC8& "CC80
"CB85 "C&8&
"C58& "C&82
"CB8! "C8"
"CB8& "C8!
"CB80 "CB8
"C85 "CC8
2
(a /nal
!0
55 | DoE Montgomery
" 23 2! 25
300 'perator 2 2B 2 2&
3 3" 32 2C
" 2! 23 2
3&
3!
33
3B
o
350o 'perator 2 3 3& 35 3!
3 3! 3& 3C 3!
50 &0
35 3&
3 3C
3! 35
3C 35
3 3&
3& 3"
2 2! 2B
35 35 3!
2& 2B 25
2& 2C 25
3& 3B 3!
2 2& 2!
5;"8 -n Problem 5;"9 suppose t#at #e wis# to re*e)t t#e null #pot#esis wit# a #i4# probabilit i t#e dieren)e in t#e true mean ield at an two pressure is as 4reat as 0858 - a reasonable prior estimate o t#e standard de:iation o ield is 08"9 #ow man repli)ates sould be runK 3;"C8 T#e ield o a )#emi)al pro)ess is bein4 studied8 T#e two a)tors o interest are temperature and pressure8 T#ree le:els o ea)# a)tor are sele)ted #owe:er9 onl nine runs )an be made in one da8 T#e experimenter runs a )omplete repli)ate o t#e desi4n on ea)# da8 T#e data are s#own in t#e ollowin4 table8 /nal
Temperature >ow edium Hi4#
250 &83 85 C8"
.a " Pressure 2&0 !80 B83 C082
2B0 58 C80 C"83
250 &8" C8! C"8B
.a 2 Pressure 2&0 582 C8C C382
2B0 B83 C083 C38B
58208 7onsider t#e dta in Problem 5;58 /nal
5& | DoE Montgomery
/ir 282C 28!B 28! 28"2
En:ironment H2' 280& 2805 2823 28"03
Salt H2' "8C0 "8C3 "8B5 280&
28&5
3820
38"0
28& 280& 283
38" 38C& 38&!
382! 38C 382!
282! 28B" 28" 280
""800 ""800 C80& ""830
C8C& "080" C83& "08!0
(a /nal
/nneal Temperature ( o7 C00 C50 "000 !8&0 "08"5 ""80" !8!0 20820 "085 3820 3850
C83 "0802
"08" "08&0
(a -s t#ere e:iden)e (wit# J A 0805 indi)atin4 t#at eit#er polsili)on dopin4 le:el or anneal temperature ae)t base )urrentK (b Prepare 4rap#i)al displas to assist in interpretation o t#is experiment8 () /nal
y & ; 0 @ ; ' x'@ ; ( x( @ ; ( x2 @ ; '( x x( @ є
5B | DoE Montgomery
T#e 2> Fa)torial .esi4n =-1. 4!D6)4!
Fa)torial desi4n are widel used in experiments in:ol:in4 se:eral a)tors w#ere it is ne)essar to stud t#e *oint ee)t o a)tors on a response8 7#apter 5 presented 4eneral met#ods or t#e analsis o a)torial desi4n8 Howe:er9 t#ere are se:eral spe)ial )ases o t#e 4eneral a)torial desi4n t#at are important be)ause t#e are widel used in resear)# work and also be)ause t#e orm t#e basis o ot#er desi4n o )onsiderable pra)ti)al :alue8 T#e most important o t#ese spe)ial )ases is t#at o > a)tors9 ea)# at onl two le:els9 T#ese le:els ma be @uantitati:e9 su)# as two :alues o temperature9 pressure9 or time or t#e ma be @ualitati:e9 su)# as two ma)#ines9 two operators9 t#e U#i4nL and UlowL le:els o a a)tor9 or per#aps t#e presen)e and absen)e o a a)tor8 / )omplete repli)ate o su)# a desi4n re@uires 2 x 2 x 2 888 x 2 A 2 > obser:ations and is )alled a 2k factorial &esign.
T#is )#apter o)uses on t#is extremel important )lass o desi4n8 T#rou4#out t#is )#apter we assume t#at (" t#e a)tors are ixed9 (2 t#e desi4n are )ompletel randomi desi4n is parti)ularl useul in t#e earl sta4es o experimental work9 w#en t#ere are likel to be man a)tors to be in:esti4ated8 -t pro:ides t#e smallest number o 5 | DoE Montgomery
runs wit# w#i)# > a)tors )an be studied in a )omplete a)torial desi4n8 7onse@uentl9 t#ese desi4ns are widel used in factor screening experiments. 1e)ause t#ere are onl two le:els or ea)# a)tor9 we assume t#at t#e response is approximatel linear o:er t#e ran4e o t#e a)tor le:els )#oosen8 -n man a)tor s)reenin4 experiments9 w#en we are *ust startin4 to stud t#e pro)ess or sstem9 t#is is oten a reasonable assumption8 -n Se)tion &;&9 we will present a simple met#od or )#ekin4 t#is assumption9 and dis)uss w#at a)tion to take i it is :iolated8
&;2 THE 22 .ES-D, T#is irst desi4n in t#e 2> series is one wit# onl two a)tors9 sa A and B9 ea)# run at two le:els8 T#is desi4n is )alled a 2k factorial &esign. T#e le:els o t#e a)tors ma be arbitraril )alled UlowL and U#i4#L8 /s an example9 )onsider an in:esti4ation into t#e ee)t o t#e )on)entration o t#e rea)tant and t#e amount o t#e )atalst on t#e )on:ersion (ield in a )#emi)al pro)ess8 >et t#e rea)tant )on)entration be a)tor A9 and let t#e two le:els o interest be "5 and 25 per)ent8 T#e )atalst is a)tor B9 wit# t#e #i4# le:el denotin4 t#e use o 2 pounds o t#e )atalst and t#e low le:el denotin4 t#e use o onl " pound8 T#e experiment is repli)ated t#ree times9 and t#e data are as ollows$ Fa)tor A
B
; N ; N
; ; N N
Treatment 7ombination A low9 B low A #i4#9 B low A low9 B #i4# A #i4#9 B #i4#
2 3& " 3"
Repi)ate -25 32 "C 30
--2B 32 23 2C
Total 0 "00 &0 C0
T#e treatment )ombinations in t#is desi4n are s#own 4rap#i)all in Fi4ure &;" 1 )on:ention9 we denote t#e ee)t o a a)tor b a )apital >atin letter8 T#us U AL reers to t#e ee)t o a)tor A+ L BL reers to t#e ee)t o a)tor B9 and U ABL reers to t#e AB UNL respe)ti:el9 on t#e A and B axes8 at#us9 ;on t#e A axis represents t#e low le:el o )on)entration ("5%9 w#ereas N represents t#e #i4# le:el (25%9 and ; on t#e B axis represents t#e low le:el o )atalst9 w#ereas N denotes t#e #i4# le:el8 Table &;" /nalsis o 6arian)e or Experiment in Fi4ure &;"8 Sour)e o 6ariation
5C | DoE Montgomery
Sum o
.e4rees o
ean
S@uares 20833 B5800 833 3"83! 323800
A B AB
Error Total
Freedom " " " ""
S@uare 20833 B5800 833 38C2
F 2
% ;:alue
538"5 "C8"3 28"3
08000" 08002! 08"2&
b9 ab8 T#is is reered to as stan&ar& or&er (or ates? order9 or .r8 Frank ates8 +sin4 t#is standard order9 we see t#at t#e )ontrast )oei)ients used in estimatin4 t#e ee)t are
Ee)t AK BK ABK
(" ;" ;" N"
a
1
a1
N" ;" ;"
;" N" ;"
N" N" N"
,ote t#at t#e )ontrast )oei)ients or estimatin4 t#e intera)tion ee)t are *ust t#e produ)t o t#e )orrespondin4 )oei)ients or #e two main ee)ts8 T#e )ontrast )oei)ient is alwas eit#er N " or ; "9 and a table of plus an& minus signs su)# as in Table &;2 )an be used to determine t#e proper si4n or ea)# treatment )ombination8 T#e )olumn #eadin4s in Table &;2 are t#e main ee)ts ( A and B9 t#e AB intera)tion9 and " 9 w#i)# represents t#e total or a:era4e o t#e entire experiment8 ,oti)e t#at t#e )olumn )orrespondin4 to " #as onl plus si4n8 T#e row desi4nators are t#e treatment )ombinations8 To ind t#e )ontrast or estimatin4 an ee)t9 simpl treatment )ombination and add8 For example9 to estimate A9 t#e )ontrast is ; (" N a - 1 N a1+ w#i)# a4rees wit# E@uation &;"8 E9MP#E =-2 .................................................................................................................... a. Single eplicate of t;e 2 Design
/ )#emi)al produ)t is produ)ed in a pressure :essel8 / a)torial experiment is )arried out in t#e plot plant to stud t#e a)tors t#ou4#t to inluen)e t#e iltration rate o t#is produ)t8 T#e our a)tors are temperature ( A9 pressure ( B9 )on)entration o ormal de#de (! 9 and stirrin4 rate ( D8 Ea)# a)tor is present at two le:els8 T#e desi4n matrix and t#e response data obtained rom a sin4le repli)ate o t#e 2! experiment are s#own in Table &; "0 and Fi4ure &;"08 T#e "& Runs are made in random order8 T#e pro)ess en4ineer is interested in maximi
)on)entration as mu)# as possible but #as been unable to do as be)ause it alwas results in lower iltration rates8 e will be4in t#e analsis o t#is data b )onstru)tin4 a normal probabilit plot o t#e ee)t estimates8 T#e table o plus and minus si4ns or t#e )ontrast )onstants or t#e
Table &;"08 Pilot Plant Filtration rate Experiment Run ,umber
# G l a 4 ( e t a r n o i t a r t l i e 4 a r e : /
" 2 3 ! 5 & B C "0 "" "2 "3 "! "5 "&
Fa)tor A
B
!
D
; N ; N ; N ; N ; N ; N ; N ; N
; ; N N ; ; N N ; ; N N ; ; N N
; ; ; ; N N N N ; ; ; ; N N N N
; ; ; ; ; ; ; ; N N N N N N N N
Run >abel
Filtration Rate (4alG# !5 B" ! &5 & &0 0 &5 !3 "00 !5 "0! B5 & B0 C&
(" a 1 a1 * a* 1* a1* d ad 1d a1d *d a*d 1*d a1*d
C0 C0
0
0
B0
B0
&0
&0
50
&" | DoE Montgomery 50 ;
; N A
N !
;
N D
(a ain ee)t plot
A! intera)tion
! A ;
"00
AD intera)tion
.AN
C0 0
! A N
"00
.A ;
B0
C0
&0
0
50
B0
!0 ;
&0
A
(1 -ntera)tion plot Fi4ure &;"2 ain ee)t and intera)tion plot or Example &;2 50
Ta1le I-') Analysis of Larian*e for the %ilot %lant Filtration ?ate Experimen A+ !+ and .
Sour)e o 6ariation
!0
;
A ! D A! AD !D A!D
Error
&2 | DoE Montgomery
Sum o S@uares
.e4rees o Freedom
"B085& A 3C080& 5585& "3"!80& ""0585& 580& "085& "BC852
" " " " " " "
N
ean S@uare "B085& 3C080& 5585& "3"!80& ""0585& 580& "085& 228!!
F 2
% ;:alue
383& "B83 38"3 585& !C82B c" c"
c08000" c08000" c08000" c08000" c08000"
N
Total
5B308C!
"5
edure9 see 1ox and eer ("C& and ers and ont4omer ("CC58 /lso9 in order or t#e model residuals to properl )en:e inormation about dispersion ee)ts9 t#e location mo&el must be )orre)tl spe)iied8 Reer to t#e supplemental text material or t#is )#apter or more details and an example8 E9MP#E =-= ................................................................................................................... Duplicate Measurements on t;e esponse
/ teamo en4ineers at a semi)ondu)tor manua)turer ran a 2! a)torial in a :erti)al oxidation urna)e8 Pour waers are Usta)kedL in t#e urna)e9 and t#e response :ariable o interest is t#e oide t#i)kness on t#e waers8 T#e our desi4n a)tors are temperature ( A9 time ( B9 pressure (! 9 and 4as low ( D8 T#e experiment is )ondu)ted b loadin4 our waers in to t#e urna)e9 settin4 t#e pro)ess :ariables to t#e test )onditions re@uired b t#e experimental desi4n9 pro)essin4 t#e waers9 and t#e measurin4 t#e oxide t#i)kness on all our waers8 table &;"& presents t#e desi4n and t#e resultin4 t#i)kness measurements8 -n t#is table9 t#e our )olumns labeled UT#i)knessL )ontain t#e oxide t#i)kness measurements on ea)# indi:idual waer9 and t#e last two )olumns )ontain t#e sample a:era4e and sample :aian)e o t#e t#i)kness measurement on t#e our waers in ea)# run8 T#e proper anals o #is experiment is to )onsider t#e indi:idual waer t#i)kness measurements as &uplicate measurements9 and not as repli)ates8 - t#e were rall repli)ates9 ea)# waer would #a:e been pro)essed indi:iduall on a sin4le run o t#e urna)e8 Howe:er9 be)ause all our waers were pro)essed to4et#er9 #e re)ei:ed t#e treatment a)tors (t#at is9 t#e le:els o t#e desi4n :ariables sim$ltaneo$sly9 so t#ere is mu)# less :ariabilit in t#e indi:idual waer t#i)kness measurements t#an would #a:e been obser:ed i ea)# waer was a repli)ate8 T#ereore9 t#e average o t#e t#i)kness measurements in t#e )orre)t response :ariable to initiall )onsider8 Table &;"B (pa4e 2&& presents t#e ee)t estimates or t#is experiment9 usin4 t#e a:era4e oxide t#i)kness as t#e response :ariable8 ,ote t#at a)tor A and B and t#e AB intera)tion #a:e lar4e ee)ts t#at to4et#er a))ount or nearl C0 per)ent o t#e :ariabilit8 Table &;"& T#e 'xide T#i)kness Experiment Standard 'rder
Run 'rder
A
B
!
D
" 2 3 ! 5
"0 B 3 C &
;" " ;" " ;"
;" ;" " " ;"
;" ;" ;" ;" "
;" ;" ;" ;" ;"
&3 | DoE Montgomery
3B !"5 30 !50 3B5
T#i)kness 3B& 3BC !"& !"& 3BC 32 !!& !!C 3B" 3B3
3BC !"B 33 !!B 3&C
3B !"& 3" !! 3B2
s2
& B C "0 "" "2 "3 "! "5 "&
2 5 ! "2 "& " "! "5 "" "3
" ;" " ;" " ;" " ;" " ;" "
;" " " ;" ;" " " ;" ;" " "
" " " ;" ;" ;" ;" " " " "
;" ;" ;" " " " " " " " "
3C" 3! !2& 3" 5"& 3B" !!5 3BB 3C" 3B5 !30
3C0 35 !33 3" !20 3B2 !! 3BB 3C" 3B& !30
3 3& !30 3B5 !"2 3B" !!3 3BC 3& 3B& !2
3C" 35 !3" 33 !"2 3B0 !! 3BC !00 3BB !2
3C0 35 !30 30 !"5 3B" !!& 3B 3C2 3B& !2C
Table &;" /nalsis o 6arian)e (From desi4n;Expert or t#e /:era4e 'xide T#i)kness Source
odel
A B ! AB A!
Residual 7or Total Std8 .e:8 ean 7868 PRESS
Sum of S8uares
"0BB!83" H4)5,2I ')'4,2I 4)2,I '')5,2I 4',I
"B&8"2 "0C508!!
!820 3CC8"C "805 !508
D0
5 " " " " " "0 "5
Mean S8uare
2"5!8& B!3C80& "3"!80& 4)2,I '')5,2I 4',I
"B8&"
R;S@uared /d* R;S@uared Pred R;S@uared /de@ Pre)ision
0 /alue
Prob < 0
4((,)H H4,I' (4,4 I4,IH (,I4
J2,222
"22835
c08000
c08000 08000& c08000 080005
si4nii)ant
08C3C 08CB5C 08C5 2B8C&B
Table &;"C /nalsis o 6arian)e (From desi4n;Expert o t#e -ndi:idual waer 'xide T#i)kness response Source
Sum of S8uares
odel
!30"8B5
A B ! D AB A! AD B! BD !D ABD
(5HI,( (I,( 'H((,( 4(,( 4I,( '32I,( (2,( (42,( (42,( (2,( ')(,(
&! | DoE Montgomery
D0
"5 " " " " " " " " " " "
Mean S8uare
2C208"2 (5HI,( (I,( 'H((,( 4(,( 4I,( '32I,( (2,( (42,( (42,( (2,( ')(,(
0 /alue
!B&8B5 !58"& 58"& 2"8" &8C0 B!38 2C!8C0 383" 3C822 383" 2"85C 083B
Prob < 0
c08000"
J2,222' J2,222' J2,222' 2,2'' J2,222' J2,222' 2,2H) J2,222' J2,222' 2,2H) J2,222'
AB! A!D B!D AB!D
Residual
=a*> of Fit %$re error
7or Total
=->
" " " " ! 0 ! &3
(,( 2,( I,( 2,( (54,22 2,222
2C!800 !!0C58B5
(,( 2,( I,( 2,( I,'(
080!" "802 080!"
2,4)H 2,342H 2,)'H 2,342H
I,')
P!"#EMS hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
&;"8 /n en4ineer is interested in t#e ee)t o )uttin4 speed ( A8+ tool 4eometr ( B9 and )uttin4 an4le (! on t#e lie (in #ours o a ma)#ine tool8 Two le:els o ea)# a)tor are )#oosen9 and t#ree repli)ates o a 2 3 a)torial desi4n are run8 T#e results ollow$ A
B
!
; N ; N ; N ; N
; ; N N ; ; N N
; ; ; ; N N N N
Treatment 7ombination (" a 1 a1 * a* 1* a1*
22 32 35 55 !! !0 &0 3C
Repli)ate -3" !3 3! !B !5 3B 50 !"
--25 2C 50 !& 3 3& 5! !B
(a Estimate t#e a)tor ee)ts8 #i)# ee)ts appear to be lar4eK (b +se t#e analsis o :arian)e to )onirm our )on)lusions or part (a8 () rite down a re4ression model or predi)tin4 tool lie (in #ours based on t#e results o t#is experiment8 (d /nal
&;38 Find t#e standard error o t#e a)tor ee)ts and approximate C5 per)ent )oniden)e limits or t#e a)tor ee)ts in Problem &;"8 .o t#e results o t#is analsis a4ree wit# t#e )on)lusions rom t#e analsis o :arian)eK &;!8 Plot t#e a)tor ee)ts rom Problem &;" on a 4rap# relati:e to an appropriatel sealed t distribution8 .oes t#e 4rap#i)al displa ade@uatel identi: t#e important a)torsK 7ompute t#e )on)lusion rom t#is plot wit# t#e results rom t#e analsis o :arian)eK &;58 / router is used tu )ut lo)atin4 not)#es on a printed )ir)uit broad8 T#e :ibration le:el at t#e sura)e o t#e board as it is )ut is )onsidered to be a ma*or sour)e o dimensional :ariation in t#e not)#es8 Two a)tors are t#ou4#t to inluen)e :ibration$ bit si
A
B
; N ; N
; ; N N
!
Treatment 7ombination (" a 1 a1
"82 2B82 "58C !"80
Repli)ate ---"8C "28C 2!80 228! "!85 "58" 3!38C 3&83
-6 "!8! 2285 "!82 3C8C
(a /nal
(e #at )on)lusions would ou draw about t#e appropriate operatin4 )onditions or t#is pro)essK &;B8 /n experiment was perormed to impro:e t#e ield o a )#emi)al pro)ess8 Four a)tors were sele)ted9 and two repli)ates o a )ompletel randomi
Treatment 7ombination ("
Repli)ate -C0 C3 B! B " 5 3 0 BB B " 0 2 B3 B0
a 1 a1 * a* 1* a1*
Treatment 7ombination
Repli)ate C B2 B 5 CC BC B 0
d ad 1d a1d *d a*d 1*d a1*d
-C5 B& 3 & C0 B5 ! 0
&;8 / ba)teriolo4ist is interested in t#e ee)t o two dierent )ulture media and two dierent times on t#e 4rowt# o a parti)ular order8 /nal
7ulture edium "
"2
"
&B | DoE Montgomery
2" 23 20 3B 3 35
2 22 2 2 3C 3 3&
25 2! 2C 3" 2C 30
2& 25 2B 3! 33 35
&;C8 /n industrial en4ineer emploed b a be:era4e bottler is interested in t#e ee)ts o two dierent tpes o 32;oun)e bottles on t#e time to deli:er "2;bottle )ases o t#e produ)t8 T#e two bottle tpes are 4lass an plasti)8 Two worker are used to perorm a task )onsistin4 o mo:in4 !0 )ases o t#e produ)t 50 eet on astandard tpe o #and tru)k and sta)kin4 t#e )ases in a displa8 Four repli)ates o a 2 2 a)torial desi4n are perormed9 and t#e times obser:ed are listed in t#e ollowin4 table8 /nal
1ootle Tpe
orker "
2
Dlass
58"2 !8C
!8C 5800
&8&5 58!C
&82! 5855
Plasti)
!8C5 !82B
!8!3 !825
582 !8B5
!8C" !8B"
&;"08 -n Problem &;C9 t#e en4ineer was also interested ati4ue dieren)es resultin4 rom t#e two tpes o bottles8 /s a measure o t#e amount o eort re@uired9 #e measured t#e ele:ation o t#e #eart rate (pulse indu)ed b t#e task8 T#e results ollow9 anal
orker "
Dlass Plasti)
3C 5 !! !2
2 !5 35 35 2"
20 "& "3 "&
"3 "" "0 "5
&;""8 7al)ulate approximate C5 per)ent )oniden)e limits or t#e a)tor ee)ts in Problem &;"08 .o t#e results o t#is analsis a4ree wit# t#e analsis o :arian:e perormed in Problem &;"0K
& | DoE Montgomery
&;"28 /n arti)le in t#e ATT Te*hni*al #o$rnal (ar)#G/pril "C&9 6ol8 &59 pp83C;50 des)ribes t#e appli)ation o two le:el a)torial desi4ns to inte4rated )ir)uit manua)turin48 / basi) pro)essin4 step is to 4row an epitaxial laer on polis#ed sili)on waers8 T#e waers mounted on a suspe)tor are positioned inside a bell *ar9 and )#emi)al :apore are introdu)ed8 T#e suspe)tor is rotated and #eat is applied untuil t#e epitaxil laer is t#i)k enou4#8 /n experiment was run usin4 two a)tors$ arseni) low rate ( A and deposition time ( B8 From repli)ates were run and t#e epitaxil laer t#i)kness was measured (in m8 T#e data are s#ow below$
A
B
; N ; N
; ; N ;
"!803B "380 "!82" "!8
Repli)ate ---"&8"&5 "38CB2 "38&0 "!8032 "!8B5B "!8!3 "!8C2" "!8!25
-6 "38C0B A "38C"! "!8B B "!8C32
Fa)tor >e:els >ow (; Hi4# (N 55% 5C% S#ort >on4 ("0 min ("5 min
(a Estimate t#e a)tor ee)ts8 (b 7ondu)t an analsis o :arian)e9 #i)# a)tors are importantK () rite down an re4ression e@uation t#at )ould be used to predi)t epitaxial laer t#i)kness o:er t#e re4ion o arseni) low rate and deposition time used in t#is experiment8 (d /nal
determine t#e ee)t o our a)tors on )ra)ks8 T#e our a)tors are pourin4 temperature ( A9 titanium )ontent ( B9 #eat treatment met#od (! 9 and amount o 4rain reiner used ( D8 Two repli)ates o a! desi4n are run9 and t#e len4t# o )ra)k (in mm x "0;2 indu)ed in a sample )oupon sub*e)ted to a standard test is measured8 T#e data are s#own in t#e ollowin4 table$
A
; N ; N ; N ; N ; N ; N ; N ; N
B
; ; N N ; ; N N ; ; N N ; ; N N
!
; ; ; ; N N N N ; ; ; ; N N N N
D
; ; ; ; ; ; ; ; N N N N N N N N
Treatmen 7ombination (" a 1 a1 * a* 1* a1* d ad 1d a1d *d a*d 1*d a1*d
Repli)ate B803B "!8B0B ""835 "B82B3 "08!03 !83& C83&0 "38!!0 85" "&8&B "38B& "C82! ""8"& &8"25 "28"C0 "58&53
- &83B0 "582"C "280C "B8"3 "08"5" !80C C8253 "28C23 8C5" "B8052 "38&5 "C8&3C "2833B 58C0! "08C35 "58053
(a Estimate t#e a)tor ee)ts8 #i)# a)tor ee)ts appear to be lar4eK (b 7ondu)t an analsis o :arian)e9 .o an o t#e a)tors )ra)kin4K +se J A 08058
B0 | DoE Montgomery
() rite down an re4ression model t#at )an be used to predi)t )a)k len4t# as a un)tion o a si4nii)ant main ee)t and intera)tion ou #a:e identiied in part (b8 (d /nale:els
B" | DoE Montgomery
Run /)tual ,umber Run A 'rder " "3 ; 2 N 3 "2 ; ! C N 5 ! ; & "5 N B "& ; 3 N C " ; "0 "! N "" 5 ; "2 "0 N "3 "" ; "! 2 N "5 B ; "& & N
B
!
D
Et)# Rate (Gmin
; ; N N ; ; N N ; ; N N ; ; N N
; ; ; ; N N N N ; ; ; ; N N N N
; ; ; ; ; ; ; ; N N N N N N N N
550 &&C &0! &50 &33 &!2 &0" &35 "03B B!C "052 & "0B5 &0 "0&3 B2C
>ow ( ; A ()m 080 B (mTorr !50 ! (S77 "25 D( 2B5
Hi4# ( N "20 550 200 325
(a Estimate t#e a)tor ee)ts8 7onseder a normal probabilit plot o t#e a)tor ee)ts8 #i)# ee)t appear lar4eK (b 7ondu)t an analsis o :arian)e to )onirm our indin4s or part (a8 () #at is a re4ression model relatin4 et)# rate to t#e si4nii)ant pro)ess :ariablesK (d /nal desi4n wit# > c ! and )ondu)t t#e analsis o :arian)e8 ( .rawa 4rap#s to interpret an si4nii)ant intera)tions8 (4 Plot t#e residuals :ersus t#e a)tual run order8 #at problems mi4#t be rele:ated b t#is plotK &;"C8 Contunuation of Problem 6!% 8 7onsider t#e re4ression model obtained in part (a o Problem &;"8 (a 7onstru)t )ontour plots o t#e et)# rate usin4 t#e model8 (b Suppose t#at it was ne)essar to operate t#is pro)ess at an et)# rate 00 Gmin8 #at settin4s o t#e pro)ess :ariables would ou re)ommendK &;208 7onsider t#e sin4le repli)ate o t#e 2! desi4n in Example &;28 Suppose we #ad arbitraril de)ided to anal
&;2"8/n experiment was run in a semi)ondu)tor abri)ation plant in an eort to in)rease ield8 Fi:e a)tors9 ea)# at two le:els9 were studied8 T#e a)tors (and le:els were A A aperture settin4 (small9 lar4e9 B A exposure time (20% below nominal9 20% abo:e nominal9 ! A de:elopment time (30 s9 !0 s9 D A mask dimension (small9 lar4e9 and E A et)# time ( "!95 min9 "595 min8 T#e unrepli)ated 25 desi4n s#own below was run8 (" A B a A C 1 A 3! a1 A 55 a A "& a* A 20 1* A !0 a1* A &0
dA ad A "0 1d A 32 a1d A 50 *d A " a*d A 2" 1*d A !" a1*d A &"
e A ae A "2 1e A 35 a1e A 52 *e A "5 a*e A 22 1*e A !5 a1*e A &5
de& I ade A "0 1de A 30 a1de A 53 *de A "5 a*de A 20 1*de A !" a1*de A &3
(a 7onstru)t a normal probabilit plot o t#e ee)t estimate8 #i)# ee)ts appear to be lar4eK (b 7ondu)t an analsis o :arian)e to )onirm our indin4s or part (a8 () rite down t#e re4ression model iled to t#e si4nii)ant pro)ess :ariables8 (d Plot t#e residuals on normal probabilit paper8 -s t#e plot satisa)torK (e Plot t#e residuals :ersus t#e predi)ted ield and :ersus ea)# t#e i:e a)tors8 7omment on t#e plots8 ( -nterpret an si4nii)ant intera)tions8 (4 #at are our re)ommendation re4ardin4 pro)ess operatin4 )onditionsK (# Pro*e)t t#e 25 desi4n in t#is Problem into a 2 > desi4n in t#e important a)tors8 Sket)# t#e desi4n and s#ow t#e a:era4e and ran4e o ield at ea)# run8 .os t#is sket)# aid in interpretin4 t#e results o t#is experimentK &;228 Contunuation of Problem 6"!# Suppose t#at t#e experimenter #ad run our )enter points in addition to t#e 32 trials in t#e ori4inal experiment8 T#e ield obtained at t#e )enter point runs were &9 B!9 B&9 and B08 (a Reanale:els
B3 | DoE Montgomery
Run /)tual ,umber Run A 'rder " 5 ; 2 C N 3 ; ! "3 N 5 3 ; & B N B "! ; " N C & ; "0 "" N "" 2 ; "2 "5 N "3 ! ; "! "& N "5 "0 ; "& "2 N
B
!
D
ield (lbs
; ; N N ; ; N N ; ; N N ; ; N N
; ; ; ; N N N N ; ; ; ; N N N N
; ; ; ; ; ; ; ; N N N N N N N N
"2 " "3 "& "B "5 20 "5 "0 25 "3 2! "C 2" "B 23
>ow ( ; A (# B (% ! (psi D(o7
285 "! &0 225
Hi4# ( N 3 " 0 250
(a 7onstru)t a normal probabilit plot o t#e ee)t estimate8 #i)# a)tors appear to be lar4e ee)tsK (b 7ondu)t an analsis o :arian)e usin4 t#e normal probabilit plot in part (a or 4uidan)e in ormin4 an error term8 #at are our )on)lusionsK () rite down t#e re4ression model relatin4 iled to t#e important pro)ess :ariables8 (d /nal
T#ere are man dierent was to bake brownies8 T#e purpose o t#is experiments was to determine #ow t#e pan material9 t#e brand o brownie mix9 and t#e stirrin4 met#od ae)t t#e s)rumptiousness o brownies8 T#e a)tor le:els were8 Fa)tor A A pan material B A stirrin4 met#od ! A brand o mix
>ow ( ; Dlass Spoon Expensi:e
Hi4# ( N /luminum ixer 7#eap
T#e reponse :ariable was s)rumptiousness9 a sub*e)ti:e measure deri:ed rom a @uestionnaire 4i:en to t#e sub*e)ts w#o sampled ea)# bat)# o brownies8 (T#e @uestionnaire dealt wit# su)# issues as taste9 appearan)e9 )onsisten)9 aroma9 and so ort#8 T#e desi4n matrix and t#e reponse data are s#own below$
1rownie 1at)# " 2 3 ! 5 & B
/ ; N ; N ; N ; N
1 ; ; N N ; ; N N
7 ; ; ; ; N N N N
" "" "5 C "& "0 "2 "0 "5
2 C "0 "2 "B "" "3 "2 "2
Test Panel Results 3 ! 5 & "0 "0 "2 "0 "& "! "2 C "" "" "" "" "5 "2 "3 "3 "5 "! "3 "3 "3 "3 "0 B B "5 "3 "2 "2
B & "" "" C "! "B C
C "5 "2 "" "! C "3 "!
(a /nal
B5 | DoE Montgomery
B
!
D
ole)ular ei4#t 6is)osit
Fa)tor >e:els >ow ( ;
Hi4# ( N
" 2 3 ! 5 & B C "0 "" "2 "3 "! "5 "& "B " "C 20
" C "3 3 "" "! "B & B 2 "0 ! "C "5 20 " 5 "& "2
; N ; N ; N ; N ; N ; N ; N ; N 0 0 0 0
; ; N N ; ; N N ; ; N N ; ; N N 0 0 0 0
; ; ; ; N N N N ; ; ; ; N N N N 0 0 0 0
; ; ; ; ; ; ; ; N N N N N N N N 0 0 0 0
2!00 2!"0 23"5 25"0 2&"5 2&25 2!00 2B50 2!00 23C0 2300 2520 2&25 2&30 2500 2B"0 25"5 2500 2!00 2!B5
"!00 "500 "520 "&30 "30 "525 "500 "&20 "!00 "525 "500 "500 "!20 "!C0 "500 "&00 "500 "!&0 "525 "500
A (o7 "00 B (% ! ! (min 20 D (psi &0
(a 7onsider onl t#e mole)ular wei4#t response8 Plot t#e ee)t estimates on a normal probabilit s)ale8 #at ee)ts appear importantK (b +se o analsis o :arian)e to )onirm t#e results rom part (a8 -s t#ere indi)ation o )ur:atureK () rite down a re4ression model to predi)t mole)ular wei4#t as a un)tion o t#e important :ariables8 (d /nal
B& | DoE Montgomery
"20 30 B5
&;2C8 ' missing value in a " k factorial 8 -t is not unusual to ind t#at one o t#e obser:ations in a 2> desi4n is missin4 due to ault measurin4 e@uipment9 a sploiled test9 or some ot#er reason8 - t#e desi4n is repli)ated n times (n I " some o t#e te)#ni@ue dis)ussed in 7#apter 5 )an be emploed8 Howe:er9 or an unrepli)ated a)torial (n A " some ot#er met#od must be used8 'ne lo4i)al approa)# is to estimate t#e missin4 :alue wit# a number t#at makes t#e #i4#est; order intera)tion )ontrast
Run " 2 3 ! 5 & B C "0 "" "2 "3 "! "5 "&
A
B
!
D
; N ; N ; N ; N ; N ; N ; N ; N
; ; N N ; ; N N ; ; N N ; ; N N
; ; ; ; N N N N ; ; ; ; N N N N
; ; ; ; ; ; ; ; N N N N N N N N
Sura)e Rou4#ness 08003!0 08003&2 080030" 0800"2 080020 08002C0 0800252 0800"&0 080033& 08003!! 080030 0800"! 08002&C 08002! 0800253 0800"&3
(a Estimate t#e a)tor ee)ts8 Plot t#e ee)t estimates on a normal probabilit plot and sele)t a tentati:e models8 (b Fit t#e model identiied in part (a and anal
BB | DoE Montgomery
(d Fit a model in terms o a )oded :ariables t#at )an be used to predi)t t#e sura)e rou4#ness8 7on:ert t#is predi)tion e@uation into a model in t#e natural :ariables8 &;3"8 Resisti:it on a sili)on waer is inluen)ed b se:eral a)tors8 T#e results o a 2! a)torial experiment perormed durin4 a )riti)al pro)essin4 step is s#own in t#e ollowin4 table$ Run " 2 3 ! 5 & B C "0 "" "2 "3 "! "5 "&
A
B
!
D
; N ; N ; N ; N ; N ; N ; N ; N
; ; N N ; ; N N ; ; N N ; ; N N
; ; ; ; N N N N ; ; ; ; N N N N
; ; ; ; ; ; ; ; N N N N N N N N
Sura)e Rou4#ness "8C2 ""82 "80C 58B5 28"38 C853 "803 5835 "8&0 ""8B3 "8"& !8& 28"& C8"" "80B 5830
(a Estimate t#e a)tor ee)ts8 Plot t#e ee)t estimates on a normal probabilit plot and sele)t a tentati:e models8 (b Fit t#e model identiied in part (a and anal a)torial desi4n is used to make predi)tions at points o interest in t#e desi4n spa)e8 (a Find t#e :arian)e o t#e predi)ted response at a point xi + x( + ,,,,,,,,,x> in t#e desi4n spa)e8 Hint$ Remember t#at t#e x?s are )oded :ariables9 and assume a 2> desi4n wit# an e@ual number o repli)ates n at ea)# desi4n point so t#at t#e j
B | DoE Montgomery
:arian)e o a re4ression )oei)ient _ is a 2 G (n2> and t#at t#e )o:arian)e between an pair o re4ression )oei)ient is in desi4n spa)e8 &;3!8 Hierar)#ial models8 Se:eral times we #a:e used t#e #ierar)# prin)iple in sele)tin4 a model t#at is9 we #a:e in)luded nonsi4nii)ant lower;order terms in a model be)ause t#e were a)tors in:ol:ed in si4nii)ant #i4#er;order terms8 Hierar)# is )ertainl not an absolute prin)iple t#at must be ollowed in all )ases8 To illustrate9 )onsider t#e model resultin4 rom Problem &;"9 w#i)# re@uired t#at a nonsi4nii)ant main ee)t be in)luded to a)#ie:e #ierar)#8 +sin4 t#e data rom Problem &;"8 (a Fit# bot# t#e #ierar)#ial and t#e non#ierar)#i)al model8 (b 7al)ulate t#e PRESS statisti)9 t#e d*usted ?29 and t#e mean s@uare error or bot# models8 () Find C5 per)ent )oniden)e inter:al on t#e estimate o t#e mean response at a )ube )orner ( xi +A x( A x) A O "8 Hint$ +se t#e results o problem &;338 (d 1ased on t#e analses ou #a:e )ondu)ted9 w#i)# model do ou preerK
BC | DoE Montgomery
Two;>e:el Fra)tional Fa)torial .esi4n
;" -,TR'.+7T-', /s t#e number o a)tors in a 2> a)torial desi4n in)reases9 t#e number o runs re@uired or a )omplete repli)ates o t#e desi4n rapidl out4rows t#e resour)es o most experimenters8 For example9 a )omplete o t#e 2& desi4n re@uires &! runs8 -n t#is desi4n onl & o t#e &3 de4rees o reedom )orrespond to main ee)ts9 and onl "5 de4rees o reedom )orrespond to two;a)tors intera)tions8 T#is remainin4 !2 de4rees o reedom are asso)iated wit# t#ree;a)tor and #i4#er intera)tions8 - t#e experimenter )an reasonabl assume t#at )ertain #i4#;order intera)tions are nebli4ible9 inormation on t#e main ee)ts and low;order intera)tions ma be obtained b runnin4 onl a ra)tion o t#e )omplete a)torial experiment8 ;ese fractional factorial &esigns are amon4 t#e most widel used tpes o desi4n or produ)t and pro)ess desi4n and or pro)ess impro:ement8 / ma*or use o ra)tional is in screening experiments. T#ese are experiments in w#i)# man a)tors are )onsidered and t#e ob*e)ti:e is to identi t#ose a)tors (i an t#at #a:e lar4e ee)ts8 S)reenin4 experiments are usuall perormed in t#e earl sta4es o a pro*e)t w#en it is likel t#at man o t#e a)tors initiall )onsidered #a:e little or no ee)t on t#e response8 T#e a)tors t#at are identiied as important are t#en in:esti4ated more t#orou4#l in subse@uent experiments8 T#e su))esull use o rational a)torial desi4n is based on t#ree ke ideas$ "8 T&e sparsit( of effects principle 8 #en t#ere are se:eral :ariables9 t#e sstem or pro)ess is likel to be dri:en b some o t#e main ee)ts and low;order intera)tions8 28 T&e pro)ection propert( 8 Fra)tional a)torial desi4n )an be pro*e)ted into stron4er (lar4er desi4n in t#e subset a a si4nii)ant a)tors8
0 | DoE Montgomery
38 *equential experimentation 8 -t is possible to )ombine t#e runs o two (or more ra)tional a)torials to assemble se@uentiall a lar4er desi4n to estimate t#e a)tor ee)ts and intera)tion o interest8 E9MP#E ?-1
7onsider t#e iltration rate experiment in Example &;28 T#e ori4inal desi4n9 s#own in Table &;"09 is a sin4le repli)ate o t#e 2 ! desi4n8 -n t#at example9 we ound t#at t#e main ee)ts A9 ! 9 and D and t#e intera)tions A! and AD were dierent eom
Ta1le 3-), The -6 (
Design with the Defining ?elation " & AB!D
1asi) .esi4n Run " 2 3 ! 5 & B
A
B
!
D & AB!
; N ; N ; N ; N
; ; N N ; ; N N
; ; ; ; N N N N
; N N ; N ; ; N
Treatment 7ombination (" ad 1d a1 *d a* 1* a1*d
Filtration Rate !5 "00 !5 &5 B5 &0 0 C&
E=/P>E ;2 /8 25;" a)tors in a manua)turin4 pro)ess or an inte4rated )ir)uit were in:esti4ated in a Fi:e a)tors in a manua)turin4 pro)ess o impro:in4 t#e pro)ess ield8 T#e i:e a)tors were A A aperture settin4 -small9 lar4e9 B A exposure time (20 per)ent below nominal9 20 per)ent abo:e nominal9 ! A de:elop time (30 s9 !5 s9 D A mask dimension (small9 lar4e9 and E A et)# time ("!9 5 min9 "595 min8 T#e )onstru)tion o t#e 2 5;" desi4n is " | DoE Montgomery
s#own in Table ;58 ,oti)e9 t#at t#e desi4n was )onsrtru)ted b writin4 down t#e basi) desi4n #a:in4 "& runs ( a2 ! desi4n in A+ B+ ! 9 and D9 sele)tin4 AB!DE as t#e 4enerator9 and t#e settin4 t#e le:els o t#e it# a)tor E & AB!D8 Fi4ure ;5 (on pa4e 3"2 4i:es a pi)torial representation o t#e desi4n8 T#e deinin4 relation or t#e desi4n is " & AB!D8 7onse@uentl9 e:er main ee)t is aliased wit# a our;a)tor intera)tion (For example9 / A N B!DE and e:er two a)tor intera)tion is aliased wit# a t#ree ; a)tors intera)tion (For example AB AB N !DE 8 T#us9 t#e desi4n is o resolution 68 e would expe)t t#is 2 5;" desi4n to pro:ide ex)ellent inormation )on)ernin4 t#e main ee)ts and two;a)tors intera)tions8 Table ;& (on pa4e 3"2 )ontains t#e ee)t estimates9 sums o s@uares9 and model re4ression )oei)ients or t#e "5 ee)ts rom t#is experiment8 Fi4ure ;& (on pa4e 3"3 presents a normal probabilit plot t#e ee)t estimates rom t#is experiment8 T#e main ee)ts o /9 19 and 79 and t#e /1 intera)tion are lar4e8 Remenber t#at9 be)ause o aliasin49 t#ese ee)ts are rarel A N B!DE+ B N A!DE+ ! N ABDE+ and AB N !DE, Howe:er9 be)ause it seems plausible t#at t#ree a)tors and #i4#er intera)tions are ne4li4ible9 we eel sae in )on)ludin4 t#at onl A+ B+ ! 9 and AB are important ee)ts8 Table ;B (pa4e 3"3 summari
1asi) .esi4n Run " 2 3 ! 5 & B C "0 "" "2 "3
A
B
!
D
E& AB!D
; N ; N ; N ; N ; N ; N ;
; ; N N ; ; N N ; ; N N ;
; ; ; ; N N N N ; ; ; ; N
; ; ; ; ; ; ; ; N N N N N
N ; ; N ; N N ; ; N N ; N
2 | DoE Montgomery
Treatment 7ombination e a 1 a1e * a*e 1*e a1* d ade 1de a1d *de
ield C 3! 52 "& 22 !5 &0 & "0 30 50 "5
"! "5 "&
N ; N
; N N
N N N
N N N
; ; N
2" !! &3
a*d 1*d a1*de
Table ;B /nalsis o 6arian)e or Example ;2 Sour)e o 6ariation A (/perture B (Exposure time ! (.e:elop time AB
Error Total
Sum o S@uares
.e4rees o Freedom
!C580&25 !5C080&25 !B380&25 "C80&25 28"B5 5BB58!3B5
" " " " "" "5
ean S@uare !C580&25 !5C080&25 !B380&25 "C80&25 285&25
F 2
% ;:alue
"C3820 "BC"82! "!8&" B38B
c08000" c08000" c08000" c08000"
; PR'1>E 8"8 Suppose t#at in t#e )#emi)al pro)ess de:elopment experiment des)ribed in Problem &;B9 it was onl possible tu run a one;#al r)tion o t#e 2 ! desi4n8 7onstru)t #e desi4n and perorm #e statisti)al analsis9 usin4 t#e data rom repli)ate "8 828 Suppose t#at in Problem &;"59 onl a one;#al ra)tion o t#e 2! desi4n )ould be run8 7onstru)t t#e desi4n an perorm t#e analsis9 usin4 t#e data rom repli)ate "8 838 7onsider t#e plasma et)# experiment des)ribed in Problem &;"8 Suppose t#at onl a one;#al ra)tion9 o t#e desi4n )ould be run8 Set up t#e desi4n and analar4e ,umber o 6ariables9 U in Experiments in "nd$stryK Design+ Analysis and "nterpretaion of ?es$lts+ b R8.8 Snee9 >818 Hare and 818 Trout9 Editors9 /S79 "C5 des)ribe an experiment in w#i)# a 2 5;" desi4n wit# " & AB!DE was used to in:esti4ate t#e ee)ts o i:e a)tors on t#e )olor o )#emi)al produ)t8 T#e a)tors are A A sol:entGrea)tant9 B A )atalstK 3 | DoE Montgomery
rea)tant9 ! A temperature9 D A Rea)tant purit9 and E A rea)tant pH8 T#e results obtained were as ollows$ e A
;08&3 d A &8BC a A 285" ade A 58!B 1 A ;28& 1de A 38!5 a1e A "8&& a1d A 58& * A 280& *de A 5822 a*e A "822 a*d A !83 1*e A ;280C 1*d A !8308 a1* A "8C3 a1*d A !805 (a Prepare a normal probabilit plot o t#e ee)t8 #i)# ee)ts seem a)ti:eK (b 7al)ulate t#e residuals8 7onstru)t a normal probabilit plot o t#e residuals and plot t#e residuals :ersus t#e itted :alues8 7omment on t#e plots8 () - an a)tors are ne4li4ible9 )ollapse t#e 25;" desi4n into a ull a)torial in t#e a)ti:e a)tors8 7omment on t#e resultin4 desi4n9 and interpret t#e results8 8B8 /n arti)le b 88 Pi4natiello9 R8 and 8 S8 Ramber4 in t#e ournal o ualit Te)#nolo4 (:ol8 "B9 "C59 pp8 "C;20& des)ribes t#e use o a repli)ated ra)tional to in:esti4ate t#e ee)t o i:e a)tors on t#e ree #ei4#t o lea sprin4s used in an automoti:e appli)ations8 T#e a)tors are A A urna)e temperature9 B A #eatin4 time9 ! A transer time9 D A #old down time9 and E A @uen)# oil temperature8 T#e data are s#own below$ A
B
!
; N ; N ; N ; N ; N ; N ; N ; N
; ; N N ; ; N N ; ; N N ; ; N N
; ; ; ; N N N N ; ; ; ; N N N N
! | DoE Montgomery
D
; N N ; N ; ; N ; N N ; N ; ; N
E
; ; ; ; ; ; ; ; N N N N N N N N
B8B 8"5 B850 B85C B85! B8&C B85& B85& B850 B8 B850 B8&3 B832 B85& B8" B8"
Free Hei4#t B8B 8" B85& B85& 800 C80C B852 B8" B825 B8 B85& B8B5 B8!! B8&C B8" B850
B8" B8 B850 B8B5 B85! B8&C B85& B85& B850 B8 B850 B8&3 B832 B85& B8" B8"
(a rite out t#e alias stru)ture or t#is desi4n8 #at is t#e resolution o t#is desi4nK (b /nal
!;" -6
8"28 /nal
8208 7onstru)t a 25;" desi4n8 S#ow #ow t#e desi4n ma be run in two blo)ks o ei4#t obser:ations ea)#8 /re an main ee)ts or two;a)tors intera)tion )onounded wit# blo)kK 82"8 7onstru)t a 2B;2 desi4n8 S#ow #ow t#e desi4n ma be run in two blo)ks o ei4#t obser:ations ea)#8 /re an main ee)ts or two;a)tors intera)tion )onounded wit# blo)kK 8228 ,rregular fractions of t&e " k -.o&n /!01!23# 7onsider a 2! desi4n8 e must estimate t#e our main ee)ts and t#e six two;a)tors intera)tions9 but t#e ull 2! a)torial )annot be run8 T#e lar4est possible blo)k si
!;" -6
Run " 2 3 ! 5 & B C "0 "" "2 "3 "! "5 "&
>amination >amination >aminatin4 Firin4 Temperature Time Pressure Temperature (o7 (s (tn (o7 55 "0 5 "50 B5 "0 5 "50 55 25 5 "50 B5 25 5 "50 55 "0 "0 "50 B5 "0 "0 "50 55 25 "0 "50 B5 25 "0 "50 55 "0 5 "&20 B5 "0 5 "&20 55 25 5 "&20 B5 25 5 "&20 55 "0 "0 "&20 B5 "0 "0 "&20 55 25 "0 "&20 B5 25 "0 "&20
Firin4 7)le Time (# "B85 2C 2C "B85 2C "B85 "B85 2C "B85 2C 2C "B85 2C "B85 "B85 2C
Firin4 .ew Point (o7 20 2& 20 2& 2& 20 2& 20 2& 20 2& 20 20 2& 20 2&
Ea)# run was repli)ated our times9 and a)amber measurement was taken on t#e substrate t#e data re s#ow beloe$ Run " 2 3 ! 5 & B C "0 "" "2 "3
7amber or Repli)ate (inGin " 2 3 ! 080"&B 080"2 080"!C 080"5 0800&2 0800&& 0800!! 080020 0800!" 0800!3 0800!2 080050 0800B3 0800" 08003C 080030 0800!B 0800!B 0800!0 0800C 0802"C 08025 080"!B 0802C& 080"2" 0800C0 0800C2 0800& 080255 080250 08022& 080"&C 080032 080023 0800BB 0800&C 0800B 080"5 0800&0 0800!5 0800!3 08002B 08002 08002 080"& 080"3B 080"5 080"5C 080""0 0800& 080"0" 080"5
| DoE Montgomery
Total ("0;! inGin &2C "C2 "B& 223 223 C20 3C C00 20" 3!" "2& &!0 !55
ean ("0;! inGin "5B825 !800 !!800 558B5 558B5 230800 CB825 225800 50825 5825 3"850 "&0800 ""38B5
Standar .e:iation 2!8!" 208CB& !803 258025 228!"0 &38&3C "&802C 3C8!2 2&8B25 5083!" B8&" 20803 3"8"2
"! "5 "&
0800&5 080"0C 080"2& 0800B" 080"55 080"5 080"!5 080"!5 0800C2 080"2! 080""0 080"33
3B" &03 !&0
C28B5 "508B5 ""5800
2C85" &8B5 "B8!5
Table ;30 .ata or Problem ;25 A
B
Run
6olume
1at)#
" 2 3 ! 5 & B C "0 "" "2 "3 "! "5 "& "B " "C 20 2" 22 23 2! 25 2& 2B 2 2C 30 3" 32
5 5 3 3 3 5 3 5 5 3 3 3 5 3 5 5 3 3 5 3 5 3 5 3 5 3 3 5 5 5 5 3
1at)# 2 1at)# " 1at)# " 1at)# 2 1at)# " 1at)# " 1at)# " 1at)# 2 1at)# " 1at)# " 1at)# 2 1at)# " 1at)# " 1at)# " 1at)# 2 1at)# 2 1at)# 2 1at)# " 1at)# 2 1at)# 2 1at)# " 1at)# 2 1at)# " 1at)# 2 1at)# " 1at)# 2 1at)# " 1at)# 2 1at)# " 1at)# 2 1at)# 2 1at)# 2
!
Time9 s "! & & "! "! & & "! "! "! "! & & & "! & "! "! & & "! & "! & "! & "! & & & "! "!
D
E
Speed
/))
B350 B350 &&50 B350 B350 &&50 B350 &&50 &&50 &&50 &&50 B350 &&50 &&50 B350 B350 B350 &&50 B350 B350 &&50 &&50 B350 B350 B350 &&50 B350 &&50 B350 &&50 &&50 &&50
5 5 5 20 5 20 5 20 5 5 20 20 5 20 20 5 5 20 20 5 20 5 20 20 5 20 20 5 20 20 5 5
F
7o:er >et ' ' ' ' ' ' 'n ' ' 'n 'n ' 'n 'n 'n 'n 'n ' ' ' 'n 'n ' 'n 'n ' 'n ' 'n 'n 'n '
!53" !!!& !!52 !3"& !30B !!B0 !!C& !5!2 !&2" !&53 !!0 !22" !&20 !!55 !255 !!C0 !5"! !!C! !2C3 !53! !!&0 !&50 !23" !225 !3" !533 !"C! !&&& !"0 !!&5 !&53 !&3
Resist T#i)kness 7enter Ri4#t /:48 !53" !!&! !!C0 !32 !2C5 !!C2 !502 !5!B !&!3 !&B0 !!& !233 !&!" !!0 !2 !53! !55" !503 !30& !5!5 !!5B !& !2!! !22 !3C" !52" !230 !&C5 !2B3 !!C& !&5 !B"2
!5"5 !!2 !!52 !30 !2C !!C5 !!2 !53 !&"3 !&!5 !!B0 !2"B !&"C !!&& !2!3 !523 !5!0 !!C& !302 !5"2 !!3& !&5& !230 !20 !3B& !5"" !"B2 !&B2 !"CB !!&3 !&&5 !5BB
!5258B !!!& !!&!8B !3"B83 !2CB !!58B !!C383 !5!283 !&258B !&5& !!B8B !2238B !&2&8B !!&B !2&2 !5"58B !535 !!CB8B !30083 !53083 !!5" !&&!8B !235 !22083 !328B !52"8B !"C8B !&BB8B !"C&8B !!B!8B !&&B8B !&C08B
Ran4e "& 3& 3 20 " 25 20 C 30 25 "& "& 22 25 !5 !! 3B C "3 33 2! 3 "! 20 "5 22 5 2C 33 33 32 35
(d (d .o an an o t#e pro) pro)es esss b:ar b:aria iabl bles es ae ae)t )t t#e t#e :ari :ariab abiilit lit in )amb )amber er measurementK (e - it is important to redu)e )amber as mu)# as possible9 w#at re)ommendations would ou makeK &;258 / spin )oater is used to apll p#otoresist to a bare sili)on waer8 T#is operation usuall o))urs earl in t#e semi)ondu)tor manua)turin4 pro)ess9 and t#e a:era4e )oatin4 t#i)kness and t#e :ariabilit in t#e )oatin4 t#i)kness #as an important
C | DoE Montgomery
impa impa)t )t on down downst strea ream m manu manua)t a)turi urin4 n4 step steps8 s8 Six Six :ari :ariab able less are are used used in t#e t#e experiment8 T#e :ariables and t#eir #i4# and low le:els are as ollow$
Fa)tor Final spin speed /))eleration rate 6olume o resist applied Time o spin resist bat)# :ariation Ex#aust pressure
>ow >e:el B350 rpm 5 3 )) "! s 1at)# " 7o:er o
Hi4# >e:el &50 rpm 20 5 )) &s 1at)# 2 7o:er on
T#e experimenter de)ides to use a 2 &;" desi4n and to make t#ree readin4s on resist t#i)kness on ea)# test waer8 T#e data are s#own in Table ;308 (a 6eri t#at t#at t#is t#is is a 2;" desi4n8 .is)uss t#e alias relations#ip in t#is desi4n8 (b #at a)tors a)tors appear to ae)t ae)t a:era4e resist resist t#i)knessK t#i)knessK () 1e)ause 1e)ause t#e :olume :olume o resist resist applie appliedd #as little little ee)t ee)t on a:era4e a:era4e t#i)kne t#i)kness9 ss9 does t#is #a:e an important pra)ti)al impli)ations or t#e pro)ess an4ineersK (d Pro*e)t t#is t#is desi4n into a smaller smaller desi4n in:ol:in4 in:ol:in4 onl t#e si4nii)ant a)tors8 a)tors8 Drap#i)all displa t#e results8 .oes t#is aid in interpretationK (e +se t#e ran4e o resist t#i)kne t#i)kness ss a response :ariable8 :ariable8 -s t#ere an indi)atio indi)ationn t#at an o #ese a)tors ae)t t#e :ariabilit in resist t#i)knessK ( #ere would would ou re)ommended re)ommended t#at t#e t#e pro)ess en4ineers en4ineers run t#e t#e pro)essK ;2&8 Harr and ud Peterson;,edr (two riends o t#e aout#or own a :ineard and wine winer r in ,ewb ,ewber er49 49 're4 're4on on88 T# T#e e 4row 4row se:e se:era rall :ari :ariet etes es o 4rap 4rapes es and and manua)ture wine8 #arr and ud #a:e used a)torial desi4ns or pro)ess and produ)t de:elopment in t#e wine makin4 se4ment o t#eir bussines8 T#is problem des)ribes t#e experiment )ondu)ted or t#eir "C5 Pinot ,oir8 Ei4#t :ariables9 s#ow below9 were ori4inall studied in t#is experiment$ 6ariable A A Pinot ,oir )lone B A 'ak tpe ! A A /4e o barrel D A eastGskin )onta)t A Stems E A A 1arrel toast F A
C0 | DoE Montgomery
>ow >e:el Pommard /llier 'ld 7#ampa4ne ,one >i4#t
Hi4# >e:el adenswil Tron)ais ,ew ontra)#et /ll edium
G A #ole )lussier & Fermentation temperature
;!
,one >ow (B5oF max
"0% Hi4# (C2oF max
Harr and ud de)ided to use a -6 2 desi4n wit# "& runs8 T#e wine was taste; tested b a panel o experts on ar)# 9 "C&8 Ea)# expert ranked t#e "& samples o wine tasted8 wit# rank " bein4 t#e best8 T#e desi4n and t#e taste;test panel results are s#own in Table ;3" on pa4e 35&8 (a #at are t#e alias relations#ip in t#e desi4n sele)ted b Harr and udK (b +se t#e a:era4e ranks ( as a response :ariable8 /nal
7oded >e:el ;" "
C2 | DoE Montgomery
A,
Presure (bar !"5 550
B,
Temp8 (o7 25 C5
!, oisture
D, Flow
(% b wei4#t
(littersGmin
5 "5
!0 &0
E,
Part Si
Table ;3"8 .esi4n and Results or ine Testin4 Experiment 6ariable Run " 2 3 ! 5 & B C "0 "" "2 "3 "! "5 "&
C3 | DoE
A
B
!
D
E
F
G
; N ; N ; N ; N ; N ; N ; N ; N
; ; N N ; ; N N ; ; N N ; ; N N
; ; ; ; N N N N ; ; ; ; N N N N
; ; ; ; ; ; ; ; N N N N N N N N
; ; N N N N ; ; N N ; ; ; ; N N
; N ; N N ; N ; N ; N ; ; N ; N
; N N ; N ; ; N ; N N ; N ; ; N
; N N ; ; N N ; N ; ; N N ; ; N
HP, "2 "0 "! C "& & "5 " B "3 3 2 ! 5 ""
Panel Rankin4s P, 7/> .7 & "3 "0 B "! "! "3 "0 "" C B C "" "2 "5 "& 5 & 5 "& "& "5 2 3 3 "" ! B 3 "2 " 5 " "0 2 ! ! " 2 "5 C & "! "2 "3
Summar RD1 B C "5 "2 "0 "& 3 "! 2 & ! 5 " "" "3
Montgomery
A
B
!
D
E
y,,,
!"5 550 !"5 550 !"5 550 !"5 550 !"5 550 !"5 550 !"5 550
25 25 C5 C5 25 25 C5 C5 25 25 C5 C5 25 25
5 5 5 5 "5 "5 "5 "5 5 5 5 5 "5 "5
!0 !0 !0 !0 !0 !0 !0 !0 &0 &0 &0 &0 &0 &0
"82 !805 !805 "82 !805 "82 "82 !805 !805 "82 "82 !805 "82 !805
&3 2" 3& CC 2! && B" 5! 23 B! 0 33 &3 2"
C8& "08 "28& C82 C80 "580 580 "582 282 B80 8 28 !8& 28! C82 "28&
s
3805 38"" 280B "8BC "8!" "8B3 "822 08! 08! 2855 38C& "8BC 382C "852 !802 "8"!
A
B
!
D
E
y,,,
!"5 550 !"5 550 !"5 550 !"5 550 !"5 550 !"5 550 !"5 550 !"5 550
25 25 C5 C5 25 25 C5 C5 25 25 C5 C5 25 25 C5 C5
5 5 5 5 "5 "5 "5 "5 5 5 5 5 "5 "5 "5 "5
!0 !0 !0 !0 !0 !0 !0 !0 &0 &0 &0 &0 &0 &0 &0 &0
"82 !805 !805 "82 !805 "82 "82 !805 !805 "82 "82 !805 "82 !805 !805 "82
&3 2" 3& CC 2! && B" 5! 23 B! 0 33 &3 2" !! C&
(a #at tpe desi4n #as been usedK -denti t#e deinin4 relation and t#e alias relations#ip8 (b Estimate t#e a)tor ee)ts and use a normal probabilit plot to tentati:el identi t#e important a)tors8 () Perorm an appropriate statisti)al analsis to test t#e #pot#eses t#at t#e a)tors identiied in part (b abo:e #a:e a si4nii)ant ee)t on t#e iled o peanut oil8 (d Fit a model t#at )ould be used to predi)t peanut oil ield in terms o t#e a)tors t#at (e ou #a:e identiied as important8 ( /naliter -ntake aniold Poor Sandill Stud9L b .8 1a)knell (Fourt# Smposium on Ta4u)#i et#ods9 /meri)an Supplier -nstitute9 dearbon9 -9 "C&9 pp8 "20;"308 T#e purpose was determine w#i)# o "0 a)tors #as an ee)t on t#e proportion o dee)ti:e )astin4s8 T#e desi4n and t#e resultin4 proportion o nondee)ti:e )astin4s ; absor:ed on ea)# run are s#own below8 T#is is a resolution --- ra)tion C! | DoE Montgomery
wit# 4enerators E & !D+ F & BD+ G & B!+ & A!+ " & AB9 and O & AB! 8 /ssume t#at t#e number o )astin4s made at ea)# run in t#e desi4n is "0008 Run A " 2 3 ! 5 & B C "0 "" "2 "3 "! "5 "&
; N ; N ; N ; N ; N ; N ; N ; N
B
; ; N N ; ; N N ; ; N N ; ; N N
!
D
; ; ; ; N N N N ; ; ; ; N N N N
; ; ; ; ; ; ; ; N N N N N N N N
E
N N N N ; ; ; ; ; ; ; ; N N N N
F
G
"
#
N N ; ; N N ; ; ; ; N N ; ; N N
N N ; ; ; ; N N N N ; ; ; ; N N
N ; N ; ; N ; N N ; N ; ; N ; N
N ; ; N N ; ; N N ; ; N N ; ; N
N ; ; N N ; ; N N ; ; N N ; ; N
O
; N N ; N ; ; N ; N N ; N ; ; N
;
08C5 "8000 08CBB 08BB5 08C5 08C5 08"3 08C0& 08&BC 08B" "8000 08C& 08C5 08" 08!" 08C55
j
/r)sin % "83&! "85B" "8!"C "80BB "83&! "83&! "8"2! "825C 08C&C "80" "85B" "82!" "83&! "8"30 "8"&" "835B
F T?s odii)ation "83&3 "8555 "8!"B "80B& "83&3 "83&3 "8"23 "825C 08C& "803 "855& "82!2 "83&3 "8"30 "8"&0 "835&
(a Find t#e deinin4 relation and t#e alias relations#ip in t#is desi4n8 (b Estimate t#e a)tor ee)ts and use a normal probabilit plot to tentati:el identi t#e important a)tors8 () Fit an appropriate model usin4 t#e a)tors identiied in part (b abo:e9 (d Plot t#e residuals rom t#is model :ersus t#e predi)ted proportion o nondee)ti:e )astin4s8 /lso prepare a normal probabilit plot o t#e residuals8 7omment on t#e ade@ua) o t#ese plots8 (e -n part (d ou s#ould #a:e noti)ed an indi)ation t#at :arian)e o t#e response is not )onstant ()onsiderin4 t#at t#e response is a proportion9 ou s#ould #a:e j expe)ted t#is8 T#e pre:ious table also s#ows a inormation on %+ t#e ar)sin s@uare root9 t#at is a widel used /arian*e sta1ili6ing transformation or proportion data (reer to t#e dis)ussion o :arian)e stabili
j np G ( n N "
War)sin
N ar)sin
j
(np N " G ( n N "X G2
Rework part (a t#rou4# (d usin4 t#is transormation and )omment on t#e results (For an interestin4 is)ussion and analsis o t#is experiment9 reer to U/nalsis o Fa)torial Experimens wit# .ee)ts or .eeti:es as t#e Response9L b S8 1is4aard and H8T8 Fuller9 $ality Engineering 9 6ol8 B9 "CC!;C59 pp8 !2C; !!3 ;2C8 / "&;run ra)tional a)torial experiment in nine a)tors was )ondu)ted b 7#rsler otors En4nineerin4 and des)ribed in t#e arti)le U S#eet olded 7ompound Pro)ess -mpro:ementL b P8-8 Hsie# and .8E8 Doodwin (Fourt# Smposium on Ta4u)#i et#ods9 /meri)an Supplier -nstitute9 .earborn9 -9 "C&9 pp8 "3;2"8 T#e purpose was to redu)e te number o dee)ts in t#e inis# o s#eet;molded 4rill openin4 panels8 T#e desi4n and t#e resultin4 number o dee)ts9 * obser:ed on ea)# run9 is s#own below8 T#is is a resolution --- ra)tion wit# 4enerators E & BD+ F & B!D+ G & A!+ & A!D+ # & AB,
Run A " 2 3 ! 5 & B C "0 "" "2 "3 "! "5 "&
; N ; N ; N ; N ; N ; N ; N ; N
B
; ; N N ; ; N N ; ; N N ; ; N N
!
D
E
F
G
#
; ; ; ; N N N N ; ; ; ; N N N N
; ; ; ; ; ; ; ; N N N N N N N N
N N ; ; N N ; ; ; ; N N; ; ; N N
; ; N N N N ; ; N N ; ; ; ; N N
N ; N ; ; N ; N N ; N ; ; N ; N
; N ; N N ; N ; N ; N ; ; N ; N
N ; ; N N ; ; N N ; ; N N ; ; N
j
*
5& "B 2 ! 3 ! 50 2 " 0 3 "2 3 ! 0 0
/r)sin % B8! !8"2 "8!" 2800 "8B3 2800 B80B "8!" "800 0800 "8B3 38!& "8B3 2800 0800 0800
F T?s odii)ation
(a Find t#e deinin4 relation and t#e alias relations#ip in t#is desi4n8
C& | DoE Montgomery
B852 !8" "85B 28"2 "8B 28"2 B8"2 "85B "82" 0850 "8B 385! "8B 28"2 0850 0850
(b Estimate t#e a)tor ee)ts and use a normal probabilit plot to tentati:el identi t#e important a)tors8 () Fit an appropriate model usin4 t#e a)tors identiied in part (b abo:e9 (d Plot t#e residuals rom t#is model :ersus t#e predi)ted number o dee)ts8 /lso prepare a normal probabilit plot o t#e residuals8 7omment on t#e ade@ua) o t#ese plots8 (e -n part (d ou s#ould #a:e noti)ed an indi)ation t#at t#e :arian)e o t#e response is not )onstant ()onsiderin4 t#at t#e response is a )ount9 ou s#ould #a:e expe)ted t#is8 T#e pre:ious table also s#ows a inormation on *+ t#e s@uare root9 t#at is a widel used /arian*e sta1ili6ing transformation or )ount data (reer to t#e dis)ussion o :arian)e stabili
() N "XG2
Rework part (a t#rou4# (d usin4 t#is transormation and )omment on t#e results (For an interestin4 is)ussion and analsis o t#is experiment9 reer to U/nalsis o Fa)torial Experimens wit# .ee)ts or .eeti:es as t#e Response9L b S8 1is4aard and H8T8 Fuller9 $ality Engineering 9 6ol8 B9 "CC!;C59 pp8 !2C; !!3 ;308 /n experiment is run a semi)ondu)tor a)tor to in:esti4ate t#e ee)t o six &;2 a)tors on transistor 4ain8 T#e desi4n ele)ted is t#e -6 2 s#ow below$
Standard
CB | DoE Montgomery
Run
'rder " 2 3 ! 5 & B C "0 "" "2 "3 "! "5 "&
'rder 2 5 C 3 "! "" "0 "5 "3 " & "2 ! B "&
A
B
!
D
E
F
; N ; N ; N ; N ; N ; N ; N ; N
; ; N N ; ; N N ; ; N N ; ; N N
; ; ; ; N N N N ; ; ; ; N N N N
; ; ; ; ; ; ; ; N N N N N N N N
; N N ; N ; ; N ; N N ; N ; ; N
; ; N N N N ; ; N N ; ; ; ; N N
Dain "!55 "5"" "!B "5C& "!30 "!" "!5 "5!C "!5! "5"B "!B "5C& "!!& "!B3 "!" "5&3
(a +se a normal plot o t#e ee)ts to identi t#e si4nii)ant a)tors8 (b 7ondu)t appropriate tatisti)al test or t#e model identiid in part (a8 () /nal
Standard 'rder
C | DoE Montgomery
Run
" 2 3 ! 5 & B C "0 "" "2 "3 "! "5 "&
'rder A 5 ; B N ; 2 N "0 ; "2 N "& ; " N & ; C N "! ; "3 N "" ; 3 N "5 ; ! N
B
!
D
E
F
; ; N N ; ; N N ; ; N N ; ; N N
; ; ; ; N N N N ; ; ; ; N N N N
; ; ; ; ; ; ; ; N N N N N N N N
; N N ; N ; ; N ; N N ; N ; ; N
; ; N N N N ; ; N N ; ; ; ; N N
Dain B! "C0 "33 "2B ""5 "0" 5! "!! "2" " "35 "B0 "2& "B5 "2& "C3
(a Estimate t#e a)tor ee)t and plot t#em on a normal probabilit plot8 Sele)t a tentati:e model8 (b Perorm appropriate statisti)al tests on t#e model8 () /nal
CC | DoE Montgomery
B
!
D
E
y
; N N ; ; N ; ; N ; N N ; ; N
; ; ; N N N ; ; ; N N N ; ; ;
; ; ; ; ; ; N N N N N N ; ; ;
; ; ; ; ; ; ; ; ; ; ; ; N N N
"&833 "8!3 2B80B "&8C5 "!85 "C8"2 "8C& 2385& 2C8"5 "58B! 208B3 2"852 "585 2"803 2&8B
B
!
D
E
y
"& "B " "C 20 2" 22 23 2!
22 2" & 23 " 2! "B 2 "!
N ; N ; ; N ; N ;
; N N ; N N ; ; N
N N N ; ; ; N N N
; ; ; N N N N N N
N N N N N N N N N
"383C "8&3 "C80" "B8C& 208!C 2C83" "B8&2 "&803 2"9!2
(a /nal
"00 | DoE Montgomery
"0" | DoE Montgomery
T#ree ixed ; >e:el Fa)torial and Fra)tional Fa)torial desi4ns T#e two;le:els series o a)torial and ra)tional desi4ns dis)ussed in 7#apter &89 B9 and are widel used in industrial resear)# and de:elopment8 T#ere are some extensions and :ariations o t#ese desi4ns t#at are o))asionall useul9 su)# as t#e desi4ns or )ases w#ere all t#e a)tors are present at t#ree le:els8 T#ese 3> desi4n will be dis)ussed in t#is )#apter8 e will also )onsider )ases w#ere some a)tors #a:e two le:els and ot#er a)tors #a:e eit#er t#ree or our le:els8 @-1.2
;e 32 Design
T#e smplest desi4n in t#e 3> sstem is t#e 32 desi4n9 w#i)# #as two a)tors9 ea)# at t#ree le:els8 T#e treatment )ombination or t#is desi4n were s#own in Fi4ure C;"8 1e)ause t#ere are 32 A Ctreatment )ombinations8 T#e main ee)ts o A and B ea)# #a:e two de4rees o reedom9 and t#e AB intera)tion #as our de4rees o reedom8 - t#ere are n repli)ates9 t#ere will be n3 ; " total de4rees o reedom and 32 ( n ; " de4rees o reedom or error8 T#e sums o s@uares or A+ B9 and AB ma be )omputed b t#e usual met#ods or a)torial desi4ns dis)ussed in 7#apter 58 Ea)# main ee)t )an be represented b linear and @uadrati) )omponent9 ea)# wit# a sin4le de4ree o reedom9 as demonstrated in E@uation C;"8 ' )ourse9 t#is is onl meanin4ul i t#e a)tor is @uantitati:e8 T#e two;a)tor intera)tion AB ma be partitioned in two was8 T#e irst met#od )onsists o subdi:idin4 AB into t#e our sin4le;de4ree;o;reedom )omponents )orrespondin4 to AB =x=+ AB =x+ AB x=+ and AB x8 T#is )an be done b ittin4 t#e terms ( ' ( ( ; '( x' x( + ; '(( x' x( + ; '(( x( x( + and ; '(( x' (x 9 respe)ti:el9 as demonstrated in Examle 5;58 For t#e tool lie data9 t#is ields SS AB AB AB x =A 28&B and = x = A 8009 SS = x A !28&B9 SS A SS AB AB x A 8008 1e)ause t#is is an ort#o4onal partitionin4 o AB9 note t#at SS SS AB = x N= SS AB = x N SS AB x =N SS AB x A &"83!8 T#e se)ond met#od is based on ort;ogonal #atin s8uare 8 7onsider t#e totals o t#e treatment9 )ombinations or t#e data in Example 5;58 T#ese totals are s#ow in Fi4ure C;3 as t#e )ir)led numbers in #e s@uares8 T#e two a)tors / and 1 )orrespond to t#e rows and "02 | DoE Montgomery
)olumns9 respe)ti:el9 o a 3 x 3 >atin s@uare8 -n Fi4ure C;39 two parti)ular 3 x 3 >atin s@uares ar s#own superimposed on t#e )ell totals8
0 / t" a m r o F
0 ;3
" ?;3
2 5
0
S
2 S ;"
?
!
"0
""
? ;"
S
2
/ t" a m r o F
2
(a
0 ;3 S 2
" ? ;3 !
S
?
;" ?
2 5 "0 ;"
"" S
(1
Fi4ure C;38 Treatment )ombination totals rom Example 5;5 wit# two ort#o4onal >atin s@uare superimposed E9MP#E @-1.
/ ma)#ine is used to ill 5;4allon metal )ontainers wit# sot drink strup8 T#e :ariable o interest is t#e amount o srup loss due to rot#in48 T#reee a)tors are t#ou4#t to inluen)e ort#in4$ t#e no<
" Pressure (in psi (! "0
"03 | DoE Montgomery
"00 ;35 ;25
"20 ;!5 ;&0
"!0 ;!0 "5
,o<
3 "00 ;3C ;35
"20 ;55 ;&B
"!0 "5 ;30
"5
""0 B5 ! 5
20
;"0 30 ;!0 ;30
0 5! 3" 3&
55 "20 ;23 ;5
;55 ;!! ;&! ;&2
""0 !! ;20 ;3"
C0 ""3 ;30 ;55
;2 ;2& ;&" ;52
""0 "35 5! !
Example C;" illustrates a situation w#ere t#e t#ree;le:el desi4n oten inds some appli)ation$ one or more o t#e a)tors is 8ualitativeA naturall takin4 on t#ree le:els9 and t#e remainin4 a)tors are 8uantitative. -n t#is example9 suppose t#at t#ere are onl t#ree no<E C;28 e illustrate t#e statisti)al analsis o t#e 3 2 desi4n )onounded in t#ree blo)ks b usin4 t#e ollowin4 data9 w#i)# )ome rom t#e sin4le repli)ate o t#e 32 desi4n s#own in Fi4ure C;&8 1lo)k " 1lo)k 2 1lo)k 3 00 A ! "0 A ; 2 0" A 5 "" A ; ! 2" A " "2 A ;5 22 A 0 02 A 20 A 0 1lo)k Totals A
0
B
0
Table C;! /nalsis o 6arian)e or .ata in Example C;2 Sour)e o 6ariation 1lo)ks ( AB2 A B
Total
Sum o S@uares "08C "3"85& 0822 28C "!585&
.e4rees o Freedom 2 2 2 2
e now look at a sli4#tl more )ompli)ated desi4n a 33 a)torial )onounded in t#ree blo)ks o nine runs ea)#8 T#e AB2! 2 )omponent o t#ree;a)tor intera)tion will be )onounded wit# blo)ks8 T#e deinin4 )ontrast is = & x' @ (x( @ (x)
"0! | DoE Montgomery
-t is eas to :eri t#at #e treatment )ombinations 0009 0"29 and "0" belon4 in t#e prin)ipal blo)k8 T#e remainin4 runs in t#e prin)ipal blo)k are 4enerated as ollows$ (" 000 (! "0" N "0" A 202 (B "0" N 02" A "22 (2 0"2 (5 0"2 N 0"2 A 02" ( 0"2 N 202 A 2"" (3 "0" (& "0" N 0"2 A ""0 (C 02" N 202 A 220 To ind t#e rns in anot#er blo)k9 note t#at t#e treatment )ombination 200 is not in t#e prin)ipal blo)k8 T#us9 #e elements o blo)k 2 are $ (" 200 N 000 A 200 (! 200 N 202 A "02 (B 200 N "22 A 022 (2 200 N 0"2 A 2"2 (5 200 N 0"2 A 22" ( 200 N 2"" A """ (3 200 N "0" A 00" (& 200 N ""0 A 0"0 (C 200 N 220 A "20 ,oti)e t#at t#ese rns all satis = A 2 (mod 38 T#e inal blo)k is ound b obser:in4 t#at "00 dose not belon4 in blo)k " or 28 usin4 "00 as abo:e ields8 (" "00 N 000 A "00 (! "00 N 202 A 002 (B "00 N "22 A 222 (2 "00 N 0"2 A ""2 (5 "00 N 0"2 A "2" ( "00 N 2"" A 0"" (3 "00 N "0" A 20" (& "00 N ""0 A 2"0 (C "00 N 220 A 020 T#e blo)ks are s#own in Fi4ure C;B 1lo)k 3 1lo)k " 1lo)k 2 000 0"2 "0" 202 02" ""0 "22 2"" 220
200 2"2 00" "02 22" 0"0 022 """ "20
"00 ""2 20" 002 "2" 2"0 222 0"" 020
(a /ssi4nment o t#e treatment )ombination to blo)k Table C;5 /nal
"05 | DoE Montgomery
.e4rees o Freedom 2 2 2
2 ! ! ! & 2&
! AB A! B!
Error ( AB! N AB(! ( N AB! 2 Total
T#e analsis o :arian)e or t#is desi4n is s#own in Table C;58 usin4 t#is )ounoundin4 s)#eme9 inormation on all t#e main ee)ts and two;a)tor intera)tions os :ailable8 T#e remainin4 )omponents o t#e t#ree;a)tor intera)tiaon ( AB!+ AB(! ( + AB! 2 are )ombined as an estimate o error8 T#e sum o s@uares or t#ose t#ree )omponents )ould be obtained b substra)tion8 -n 4eneral9 or t#e 3 > desi4n in t#ree blo)ks9 we would alwas sele)t a )omponent o t#e #i4#est;order intera)tion to )onound wit# blo)ks8 T#e remainin4 un)onounded )omponents o t#is intera)tion )ould be obtained b )omputin4 t#e > ;a)tor intera)tion in t#e usual and substa)tin4 rom t#is @ualit t#e sum o s@uares or blo)ks8 C;5 PR'1>ES 8"8 T#e ee)ts o de:eloper stren4t# ( A and de:elopment time ( B on t#e densit o p#oto4rap#i) plate ilm are bein4 studied8 T#ree stren4t#s and t#ree times are used9 and our repli)ates o a 32 a)torial experiment are run8 T#e data rom t#is experiment ollow8 /nal
C828 C838
0 5 ! B B
.e:elopment Time (minutes "0 "! " 2 " 3 2 ! ! 2 ! & & C 5 B B "0 "0 "0 "2 B B C
5 & "0 5 "0
7ompute t#e " and # )omponents o t#e two;a)tor intera)tion in Problem C;"8 /n experiment was perormed to stud t#e ee)t o t#ree dierent tpes o 32; oun)e bottles ( A and t#ree dierent s#el tpes ( B smot# permanent s#el:es9 end;aisle displas wit# 4rilled s#el:es9 and be:era4e )oolers on t#e time it takes to sto)k ten "2;bottles )ases on t#e s#el:es8 T#ree workers (a)tor ! were emploed in t#e experiment9 and two repli)ates o a 3 3 a)torial desi4n were
"0& | DoE Montgomery
run8 T#e obser:ed time data are s#own in t#e olloein4 table8 /nal
C8!8
7ooler 580 58! 58&B
Repli)ate -Permanent End /isle 383& !8"C 3852 !82& 38& !823B
7ooler 35823 !85 585
2
Plati)s 2;mm 4lass 3;mm 4lass
!80 !852 !8C&
!822 58"5 58"B
&82" &825 &803
38!0 !8!! !83C
!8B0 !8&5 !8B5
58 &820 &83
3
Plati)s 2;mm 4lass 3;mm 4lass
!80 !8 30 !8"B
38C! !853 !8&
58"! !8CC !85
38&5 !80! 38
!80 !80 !8!
!8!C !85C !8C0
/ edi)al resear)#er is studin4 t#e ee)t o lido)aine on t#e en<me le:el in t#e #eart mus)le o bea4le do4s8 T#ree dierent )ommer)ial brands o lido)aine ( A9 t#ree dosa4e le:els ( B9 and t#ree do4s (! are used in t#e experiment9 and two repli)ates o a 3 3 a)torial desi4n are run8 T#e obser:ed en<me le:els ollow8 /nal
C858 C8&8
1ottle tpes Plati)s 2;mm 4lass 3;mm 4lass
Repli)ate Permanent End /isle 38"5 !8"! !80B !83 !820 !82&
.osa4e Stren4#t
"
" 2 3
C& C! "0"
Repli)ate .o4 -! CC "0&
! C5 "05
Repli)ate -.o4 -5 CB "0!
--5 C C
--& C0 "03
2
" 2 3
5 C5 "0
! C ""!
& CB "0C
0 C3 ""0
2 CC "02
! C5 "00
3
" 2 3
! C5 "05
3 C5 "00
" C3 "0&
3 C2 "02
0 C """
BC C3 "0
7ompute t#e " and # )omponents o t#e two;a)tor intera)tion or Example "0;"8 /n experiment is run in a )#emi)al pro)ess usin4 a 3 2 a)torial desi4n8 T#e desi4n a)tors are temperature and pressure9 and t#e reponse :ariable is ield8 t#e data t#at result rom t#is experiment are s#own below$ Temperature9 o7 0
"0B | DoE Montgomery
"00 !B859 !9 BB
Pressure9 psi4 "20 &!8CB9 &C822
"!0 08C29 B28&0
C0 "00
5"8&9 29 !3 B"8"9 C28BB
8!B9 !823 C&85B9 8B2
C38C59 85! B&859 380!
(a /nal
C8"28 7onsider t#e data rom repli)ate - o Problem C;38 Suppose t#at onl a one;t#ird ra)tion o t#is desi4n wit# " A AB! is run8 7onstru)t t#e desi4n9 determine t#e alias atru)ture9 and anal
/ariable
7lone o Pinot ,oir 1err si
"0C | DoE Montgomery
#evels
adenswil Pommard Small9 lar4e 0oF9 5oF9 C0G0oF9 C0oF ,one9 "0% "0 das9 2" das /ssman#au9 7#ampa4ne Tro)ais9 /lliers
Harr and ud de)ided to use a "&;run two;le:el ra)tional desi4n9 treatin4 t#e our le:els o ermentation temperature as two;le:el :ariables8 /s in Problem ; 2&9 t#e used t#e rankin4s rom a taste;test panel as t#e response :ariable8 t#e desi4n and t#e resultin4 a:era4e ranks are s#own below$ Run
7lone
" 2 3 ! 5 & B C "0 "" "2 "3 "! "5 "&
; N ; N ; N ; N ; N ; N ; N ; N
1err Si
Ferm8 Temp ; ; ; ; ; ; ; ; N ; N ; N ; N ; ; N ; N ; N ; N N N N N N N N N
#ole 1err ; ; N N N N ; ; N N ; ; ; ; N N
a)er Time ; N ; N N ; N ; N ; N ; ; ; ; N
east Tpe ; N N ; N ; ; N ; N N ; N ; ; N
'ak Tpe ; N N ; ; N N ; N ; ; N N ; ; N
/:era4e Rank ! "0 & C "" " "5 5 "2 2 "& 3 "! B "3
N
(a .es)ribe t#e aliasin4 in t#is desi4n8 (b /nal
Heat Treatment Pro)ess
""0 | DoE Montgomery
a)#ine "
2
3
!
