06 reducing the size of experiments

8/18/2019 06 Reducing the Size of Experiments

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*Training *Support *Consultancy Copyright © Dr. K. Kumar. All rights reserved.

REDUCING

THE SIZE OFEXPERIMENTS


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The number of runs required by a full factorial design

increases geometrcally as the number of factorsincreases.Factorial designs with two or three factorsmake eciant use of resources by using all the data toestimate the average eects and interactions.This is

one of the strengths of factorial designs.However, asthe number of factors increases, an increasingproportion of the data is used to estimate higherorderinteractions. These interactions are usually negligibleand therefore are of little interest to the e!primenter.

For e!ample, for a "#$factorial design,estimates offour main eects ,si! two %factor interaction can bemade from the design.&nly '( of the ') estimatesavailable are likly to be of interest. Fractional factorial

designs are an important class of e!perimental designsthat allow the si*e of factorial e! eriments to be ke t

Introduction


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The notation used for the e!perimental patterns inthe fractional factorial design diers slightly from

that used for full factorial designs.+hereas"#indicates a full factorial pattern using sevenfactors in '"- runs,"#$indicates a ''/ fraction ofa "# pattern, using seven factor in eight runs. Thenotation for describing fractional factorial patternscomes from the followings0

''/1"#2"#"#$2"#$

Figure /.' shows two tabular displays for fractional

factorial pattern involving seven factors. Thesepatterns are 3'4a'- fraction of a"#pattern 3"#54in '/ runs and 3"4a ''/ fraction of a"#pattern3"#$4in - runs.The letters 6 through 7

indicate the seven factors and the symbols % and 8indicate the two level of each factor.


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The darkaned squares indicate the appropriate teststhat would be included in the design matri! for thefractional factorial design.

The choice to use a full factorial design or a speci9cfractional factorial design depends on the level ofcurrent knowledge of the process or oroduct.6 highlevel of knowledge is indicated when all the factors in

the e!periment are kown to have a substantial eecton the responce. +hen there is a high level ofknowledge,the aim of the study is usually to obtaindetailed information on the eects of the factors andtheir interactions. This is best done with a full

factorial design.6 moderate level of knowledge is indicted bya storngbelief that most of,but probably not all, the factorshave a substantial eect on the responce.Fractional

factorial designs for these cases will be disscussed in


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6 low level of knowledge is indicated by a belif thatonly a small proportion of the factors has a substantialeect on the responce, and it is not known which onesthey are.Fractional factorial designs that are useful forscreening out the unimportant factor will also bedisscussed in the chapter. The analysis data from

fractional factorial design follows the method used fora full factorial design, given in chap. )0 The run chart is used to screen for special causes ofvariation from nuisance variables.

:ects are ne!t estimated from the design matri!.7eomatric displays. 3e.g.,cubes4 are used to identifyspecial causes.;esponse plots are prepared for the important factorsor interaction.


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>oth of these approches lead to unsatisfactory loss ofinformation about one og the four factors. How should

the eight test be choosen? =ntuitively, it seemsdesirable that the eight tests be choosen so that eachof the columns in the design matri! has four minusesand four pluses.This selection would provide a balanceto the design. =t turns out that this balance can beachieved for '$ of the ') columns in the design matri!in Table /.'. This balance is obtained by giving up theability to estimate one of the ') eects representedby the column headings of the design matri!.



The least important estimate obtained from a full"#$ design is the four factor interaction. Thisestimate would usually be readily given up to reduce

the si*e of the e!periment. Fortunately, there is asimple way to choose the eight tests to provide thedesired balance. The eight tests are chosen byselecting only the rows in the design matri! in which

the sign in the column headed '"5$ is plus 3or,alternatively, minus4. This results in the designmatri! in table /.". from this design matri!, it is seenthat the desired balance, four minuses and fourpulses in each column, has been obtained.=t would be fortunate if the only information lost byreducing the pattern in this manner were theestimate of the fourfactor interaction and someprecision of the remaining estimates. This is not the

case.



From table /." it is seen that columns '$ and "5 areidentical, which means that the estimates of the twoeects will be identical.

The estimate provided by either column is actually anestimate of the sum of the '$ and the "5interactions. These two interactions are said to beconfounded, or confused, with each other.

Further inspection of Table /." reveals other pairs ofcolumns that are identical and therefore produceestimate that are actually sums of two eects. Theconfounding pattern for a @ fraction of a "#$ design

is

' 8 "5$ '" 8 5$" 8 '5$ '5 8 "$

5 8 '"$ '$ 8 "5$ 8 '"5



:stimates of the singlefactor eects can be obtainedthat are confounded only by threefactorinteractions. 6 more serious confounding patterne!ists for the twofactor interactions.6nother Austi9cation for using this pattern is that anythree of the four factors form a full factorial design. =fany one of the factors has a negligible eect on the

response, then the other three can be analy*ed as afull "#5 factorial design in the manner laid out inchap. ). this analysis is performed under theassumption that the factor having a negligible

average eect also does not interact with any of theother factors. The lack of interaction cannot bedetermined by analysis of the data but is anassumption that must be made by e!perts in thesubAect matter.



:!ample /.' illustrate analysis of a fractional factorialdesign.

Ea!"le #$% T&e Solenoid E"eri!ent as a

Design

Suppose that the solenoid experiment in Chapter 5were run as a design rather than as a full . Theexperiment would then have included 8 tests rather

than 16 tests. Table 6.3 contains the design matrixassuming a design was run. The heading of thedesign matrix give the confounding pattern for thedesign. The propert that an three factors form a full

factorial pattern is given in parentheses below thetitle. The standard deviations of !ow of the foursolenoids at each of the eight factor combinationsthat would have been run if the pattern had onl

been a " fraction are listed. The run order from thefull factorial experiment is also provided.



Figure /." contains a list of eects estimated fromthe design matri! in table /.5 and a plot of theseeects on a dot diagram. The substantial eect ofbobbin depth on the standard deviation of Cow isseen, as is the interaction of bobbin depth andarmature length. This interaction is now confoundedwith the



". Twofactor interactions are confounded with eachother. To allow further analysis of interactions, itwill take an assumption by e!perts in the subAect

that one of the two interactions added together issmall. Eethods to separate the confoundedinteractions by using additional tests are given in>o! et al. 3'-, p. $'54.

5.



Figure ).) illustrates the sides of the cube that are

used for each comparison. Dlearly, the structure of

factorial designs allows all the all data from the

e!periment to be used to study each factor. The

design and analysis of multifactor studies are

developed in the ne!t three e!amples.



8

8 8

8

8

8

8

8

8

3a4 design

3b4 design

3c4 design



$ %esign & 'anufacture of (lastici)er $s was thecase for the one*factor design+ careful planning of

the experiment is important in order to maximi)e theamount of information obtained for the expandedresources. To aid the planning of experiments withmultiple factors+ the planning from introduced in

Chap. 3 will be used. ,igure 5.6 contains the formused to summari)e an experiment performed toimprove the process of manufacturing a certain tpeof plastici)er.#ormall the product is made in batches+ and thereaction is allowed to continue until a certainviscosit is obtained. The unit manager desired thatthis reaction ta-e from to / h. The plant wasexperiencing other problems+ which led to the

reaction0s proceeding too fast. The fast reaction time

:!ample ).'



The purpose of the experiment was to 2nd acombination of the percentage of a -e ingredient and the reaction temperature that would result in areaction that proceeded at the desired rate.

The response variable was the reaction timenecessar to reach the desired viscosit. There were

two factors under stud4 the percentage of ingredient and the reaction temperature. The levels for theingredient were chosen at and 8 percent. Thetemperature levels were chosen at 15 and 1/5.

Several bac-ground variables were considered. Theexperiment was initiall conducted in the laborator+so con2rmation of the results in the plant would benecessar. The rate of heat*up was controlled b a

temperature programmer. 7ne laborator operator



9n analtic studies+ it is important to run studiesse:uentiall over a variet of conditions to build upthe degree of belief. ,or this experiment+ tworeplications of the experimental were run. $nade:uate amount of ingredient was obtained andmixed so that the four combination of the two factorsneeded for one replication of a pattern could be

performed using a homogeneous blend of ingredient . The second replication of the pattern was performed using a blend of ingredient from adierent shipment. The order of the four runs in each

replication was randomi)ed separatel+ using a tableof random permutations.



%$ O'(ecti)eFind a combination of the amount of ingredient G and thereaction temperature to increase the reaction time to hours in a batch process for the manufacture of plastici*er.

*$ +ac,ground In-or!ation The reaction in plant production has often been proceedingtoo fast, resulting in batches that were dicult to control

and of unacceptably high viscosity.

.$ E"eri!ental )aria'les/0$ Res"onse )aria'les

Measure!ent tec&ni1ue

Reaction ti!e toreac& desired )iscosit2

3isco!eter and cloc,

Figure )./ ocumentation form for :!ample$."



+$ Factor under stud24e)els

'. Iercentage of ingredient G". Temperature 34

C$ +ac,ground )aria'lesMet&od o- control

'.".5.$.

Jab e!periment

;ate of heatup&perator>lend variation in G

5$ Re"lication/ Two replications of the e!perimental pattern, resulting in

eight batches. :ach replication use a dierent blend ofingredient G.

$"K 3 4$-K 3 8 4') 3 4 '(3 8 4

Don9rmation on plant

batches Temperature programmer&ne lab technicianblocking



6$ Met&od o- rando!i7ation/;andomi*e the order of the four runs within each blend of Gusing a table of random permutations.

#$ Design !atri/ 3attach copy4

8$ Data collection -or!s/

9$ Planned !et&od o- statistical anal2sis/;un charts of dataL6nalysis of the square


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X T

: :

; :

< ;

; ;

Table ).' esign Eatri! forthe Ilastici*er :!periment

Iercentage of G- $"8 $-

Temperature, MD- ')

8 ')


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+atc&ID

Percentageo- X

Te!"erature

Reactionti!e

;eplication ' 3>lend ' of G4

' $" ') ).)

" $" ') .(

5 $- ') .($ $- ') '.)

;eplication " 3>lend " of G4

) $" ') /.)

/ $- ') '.(

$" ') .)

- $- ') -.(

Table )." ata Dollection form for the Ilastici*er:!periment


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Figure ). ;un chart for

:!ample ).'


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The 9rst step in the analysis of data from a factorialdesign is a run chart of the data. This chart appearsin Fig. ).. No obvious time trends or outlying valuesare seen, so analysis of the eects of the factors canbe performed. The data from a factorial design canbe analy*ed as a series of paired comparisons, onefactor at a time. The comparisons are carried out

under various conditions of the other factors, whichincreases the degree of belief in the results. The dataare displayed on the square in Fig. ).-.

:ach corner of the square contains two reactiontimes. The 9rst reaction time is the result from the9rst replication 3the 9rst blend of ingredient G4, andthe second is the result from the second replication.For each replication, there are two paired

comparisons of the eect of the temperature. The


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This is done quantitatively by subtracting thereaction time for the low temperature from thereaction time for the high temperature, with G held

constant at $- percent, i.e.,

+hen the concentration of G is $" percent, theeect of temperature is

8

.(,.)

).),/.)

'.),'.(

.(,-.(

') ')

$"

$-

Temp.

G 3K4

Figure ).-


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Di=erence inreaction ti!e

-or c&ange -ro!%86 to %>6?C

Percentage o- Xat @&ic&

co!"arison is!ade

;eplication ' 3>lend '4

$-$"


$-$"


-or c&ange -ro!%86 to %>6?C

Percentage o- Xat @&ic&

co!"arison is!ade


$-$"


$-$"Di=erence inreaction ti!e

-or c&ange -ro!5* to 59A

Te!"eratureB?C at @&ic&co!"arison is

!ade

;eplication ' 3>lend '4')')


')

')


-or c&ange -ro!5* to 59A

Te!"eratureB?C at @&ic&co!"arison is

!ade

;eplication ' 3>lend '4')')


')

')


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>ased on these comparisons, some importantobservations can be made about how temperature andthe percentage of G aect reaction time0

The replications produced very similar results,indicating that there were no important eects ofblends. 6ll four cases resulted in lower temperatureassociated with longer reaction time. The magnitude

of the dierence was large enough to be ofimportance in the process. The temperature comparisons when G was at $-

percent produced larger dierences than when G wasat $" percent. This observation along with the

consistency of the replications suggests aninteraction between temperature and percentage ofG.

Three of the four comparisons of the eect of high

versus low percentage of G on reaction time resulted


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6t this point, it is up to those familiar with the processto decide what action should be taken ne!t based on

their degree of belief. The ne!t cycle could be furtherlaboratory tests under dierent conditions or possiblycon9rmation of the results from this cycle in the plant.&nce the important eects have been determined,the relationship between the response and the

important factors should be graphically displayed. Ilotof this type were introduced earlier and are calledresponse plots. The response plots will provide theanalyst with further insight into the causeandeect

relationships and will simplify the presentation of theimportant 9ndings in the study.


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G 3K4 Temperature 3MD4

') ')$" .") /.((

$- -.)( '.")

' "

(

"

$

/

-

'(

' "

(

"

$

/

-

'(

Temperature3a4

G 3K43b4

$" $-')

')

; e a c t i o n

t i m

e

; e a c t i o n

t i m

e

G 2 $"K

G 2 $-K

T 2 ')

T 2 ')

Figure ). ;esponse plots for the plastici*er

e!periment


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The points on the plot are obtained by averaging thetwo replications for each of the four conditions in thee!perimental pattern 3i.e., averaging the two resultsat each corner of the square in Fig. ).-4.

The relationship between reaction time andpercentage of G for the two levels of temperature is

plotted in Fig. ).b. The plots in Fig. ).a and bcontain the same information displayed in twodierent ways, so only one is actually needed. Theplots in Fig. ). are especially useful in presenting

the results of the study to those not directly involvedin the factorial e!periments.

The ne!t e!ample e!tends these concepts to ane!periment with three factors.


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A The data in Table 5. resulted from a factorial

design on a de process. The aim of experiment wasto :uantif the eects of three factors on the shadeof ded material and to use the information todetermine settings of the factors. The three factors

studied were

'aterial :ualit '7xidation temperature T 7pen pressure (

9mportant bac-ground variables were identi2ed andheld constant. The response variable was a measureof shade using an optical instrument. 9t was desired

to choose settings of the factors to obtain a shade

:!ample)."


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&venpressure

&!idationtemperature


Jow High Jow High

Jow'-3$4

')3-4

""-34

"((3/4

High"'-354

"5-3"4

")3)4

"$'3'4 Table ).$ Tabular esign Eatri!

3with data4 for ye Irocess


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To begin the analysis, a run chart was madeL itappears in Fig. ).'(. For an unreplicated design,

only eight points are available to plot, so that onlygross trends or outlying values can be identi9ed fromthe run chart. No such values appear to be present inthe chart in Fig. ).'(.

8#65.*%

*#

*6

*5

*.

**

*%

*

%>

%9

S & a d e

Run C&art -or d2e "rocess e"eri!ent


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To begin the analysis, a run chart was madeL itappears in Fig. ).'(. For an unreplicated design, onlyeight points are available to plot, so that only gross

trends or outlying values can be identi9ed from therun chart. No such values appear to be present in thechart in Fig. ).'(.Figure ).'' contains the cube labeled with the three

factors studied in this e!periment. 6t each corner, thevalue of the response variable at that set ofcombinations of the factors is given.

>y appropriate analysis of the cube, the eects of thefactors can be estimated. To study the eect ofmaterial quality, the four values on the left side of thecube are compared to the four values on the rightside. The eect of o!idation temperature is obtained

by comparing the top and bottom of the cube.


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'-

""-

EaterialOuality

6

"$'

")

"5-

')

"((

"'

-

>

&venpressure

Jow

High

Jow

High


Figure ).'' 7eometric display 3cube4 for the dyeprocess e!periment.


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6 more detailed analysis of the data must beperformed if interactions between factors or specialcauses of variation in the data are to be found. The

factorial design can also be thought of as a series ofpaired comparisons. :ach edge of the cuberepresents one of these comparisons.

For e!ample, the bottom front edge connects twocorners between which the only dierence in thetest conditions is that material quality 6 is used atthe left corner and material of quality > is used at

the right corner. 6t both corners, o!idationtemperature is low and oven pressure is low. Thereare three other comparisons of the eect of materialquality during which the other factors were heldconstant. These comparisons are performed by

comparing the corners connected by the top front


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>ecause of symmetry of the design, there are alsofour pairs of tests that can be used to study the eectof o!idation temperature. These pairs found at thecorners connected by edges running from the top tothe bottom of the cube.

The four pairs used to study the eect of oven

pressure are found along the edges connecting thefront of the cube to the back. The computation ofthese eects from paired comparisons is continued in

Table ).).

The paired comparisons can also be displayedgraphically. These graphs are shown in Fig. ).'".

These pairedcomparison graphs shown here do notusually need to be developed unless it is dicult to

visuali*e the comparisons on the cube.

E= t - C 'i ti - &i &


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E=ect o- o)en"ressure

Co!'ination -or @&ic&co!"arison is !ade


Eaterial

"'- % '- 2 ""5- % ') 2 $5") % ""- 2 5'"$' % "(( 2 $'

JowHighJowHigh

66>>

:ect of materialquality


&venpressure

""- % '- 2 5"(( % ') 2 )") % "'- 2 $'"$' % "5- 2 5

JowHighJowHigh

JowJowHighHigh

:ect of

o!idationtemperature

&ven pressure Eaterial

") % '- 2 /"5- % "'- 2 "(

"(( % ""- 2"-

"$' % ") 2

JowHighJowHigh

66>>

Table ).)

computation of

eectsfrom

IairedDomparis

ons for


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From Table ).) and Fig. ).'", it is apparent that asoven pressure increases, the measure of shadeincreases and that this increase is reasonably

consistent 3from " to $54 over the four sets ofconditions. is usedis "" units higher than that when material 6 is used.

That is, the average eect of material is "".

F E i l O li F t &


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' " 5 $')

'-)

')

"()

"')"")

"5)

"$)

"))

"/)

")

' " 5 $')

'-)

')

"()

"')

"")

"5)

"$)

"))

"/)

")Factor 2 Eaterial Ouality3E4

Factor 2 &ven pressure3I4

Factor 2 &!idationtemperature 3T4

=ndicates 8 level

=ndicates level

' " 5 $')

'-)

')

"()

"')"")

"5)

"$)

"))

"/)

")

s h a

d e

s h

a d e

s

h a d e


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6s shown in Table ).) and Figure ).'", the results arenot consistent over all the conditions. The eect ofmaterial is large when o!idation temperature is low.+hen o!idation temperature is high, material haslittle eect.

That is, material quality and o!idation temperatureinteract. The eect of material cannot be given

without 9rst specifying the o!idation temperature. The average eect of material when o!idationtemperature is low is $(. when o!idation temperatureis high, the average eect of material is $. The

magnitude of the interaction is % '- and is computedaccording to the convention outlined in Table )./.Ne!t the eect of the o!idation temperature isstudied. =n Table ).) and Fig. ).'", it can be seen thatshade increases as temperature increases in two

cases and shade decreases as temperature increases


45/135


The positive eects of temperature were observedwhen material quality 6 was used, and the negative

eects of temperature were seen when material ofquality > was used. This is another way of viewingthe materialtemperature interaction.

6verage eect of materialwhen o!idation temperature ishigh

$

6verage eect of material ""

6verage eect of materialwhen o!idation temperature islow

$(

=nteraction 2 $ ""2 '-

Table )./ Domputation of interaction :ect from IairedDomparisons


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Test

Factorial E=ectRes"

onseRun

order M P T MP MT PT0+C

' $ 8 8 8 '-

" 8 8 ""-

5 5 8 8 "'-

$ ) 8 ")

) - 8 8 ')

/ / 8 8 "((

" 8 8 "5-

- ' 8 8 8 8 8 "$'

ivisor :ect 2$

"" 5/ ) ( '- / '

Table ). esign Eatri! Domputation of :ects fromye Irocess


47/135


Esti!ating e=ects using t&e design!atri

The computation of the eects of the factors and the

interactions can be performed by using an algorithmbased on the e!tended design matri!. Table ).illustrates the standard form of a design matri! for adesign and the estimated eects. The columnscorresponding to the various interactions areobtained by multiplying the signs for the factorscontained in the interactions.

:ach of the eects is estimated by adding or

subtracting the value of the response variable,depending on whether the sign of the appropriatecolumn is plus or minus. For e!ample, the averageeect of oven pressure is


48/135


=t is easy to see that this computation is the same assubtracting the average shade value when ovenpressure is high. The estimate of the interactionbetween material and o!idation temperature is

Figure ).'5 ot diagram of eects in dye processe!periment.


49/135


This is same as the estimate obtained in Table )./.&nce the estimates are obtained, they can be plottedto help determine which are the most important

eects. The eects are plotted in Fig. ).'5. This plotis called a dot diagram by >o! et al. 3'-, p.")4.:ects clustered near *ero on the diagram cannot bedistinguished from variation due to nuisance

variables. =t is clear from the plot and the analysis ofthe cube that the most important eects are theaverage eect of oven pressure, the average eect ofmaterial, and the interaction between materialquality and o!idation temperature.uring the analysis of data from any e!periment, aclose watch must be kept for evidence of specialcauses of variation due to background or nuisancevariables.


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These estimates should be veri9ed by a moredetailed analysis of the data. The estimates ofaverage eects from the design matri! should

initially be used to provide some preliminary sense ofwhich factors or interaction might be important.Psually in studying two or three factors, this methodof estimating eects is not necessary since the

analysis can be done using a square or cube.However, the approach is very useful for four or morefactors.&nce the preliminary estimates are obtained, thedata in the cube should be studied for consistencybetween the individual comparisons contained in theaverage eects. Ilots of the individual comparisonscontained in the average eects. Ilots of theindividual comparisons such as those in Fig. ).'" are

also important for verifying the validity of the

Res"onse


51/135


&nce the important eects have been identi9ed andestimated, the relationship can be graphically

summari*ed using simple response plots such asthose in Fig. ).'$. This 9gure shows three interactionplots, one for each of the three combinations of twofactors.

Donstruction of the plots involves only some simplearithmetic. For e!ample, Fig. ).'$a shows the eectof material quality and o!idation temperature onshade. To construct this plot, the " " table shownunder the plot is needed. :ach entry in the table isthe average of the two values of shade correspondingto the same level of material quality and o!idationtemperature. For e!ample, "(5.) is the average ofthe two values 3'- and "'-4 obtained when

temperature was low and material quality 6.

Res"onsePlots


52/135


The response plot is then constructed by plottingshade versus o!idation temperature separately for

each material quality. The numbers "(5.) and "'/.)are connected by a straight line to display therelationship between shade and o!idationtemperature when material quality 6 is used.6 straight line between "$5.) and ""(.) displays the

same relationship when material > is used. Theusefulness of the plot depends on the assumption thatshade increases appro!imately linearly between thetwo e!tremes of o!idation temperature for both levels

of material quality.

The remaining two plots shown in Fig. ).'$ areconstructed in the same manner. :ach of the threeplots could be shown with the factors reversed, as inFig. )..

& id ti &


53/135


Eaterial

Oualit

y


Jow High

6 "(5.) "'/.)

> "$5.) ""(.)

Eaterial

Oualit

y

&ven Iressure

Jow High

6 '".( ""-.(

> "'$.( ")(.(

&!idationtemperat

ure

&venpressure

Jow High

6 "(-.) "5-.)

> '.) "5.)

' "'-(

'(

"((

"'(""(

"5(

"$(

")(

Eaterial 6

Eaterial >


3a4

s h a

d e

"/(


54/135


' "'(

"((

"'(

""(

"5(

"$(

")(

"/(

' "'(

"((

"'(

""(

"5(

"$(

")(

"/(

Eaterial 6

Eaterial >

&!idationtemperature high

&!idationtemperaturelow

&ven pressure 3b4

&ven pressure3c4

Figure ).'$ =nteraction plots for factors in dye processe!periment.


55/135


The line in the response plot shown in Fig. ).'$a arenot parallel. This indicates that the factors ofmaterial quality and o!idation temperature interact.


56/135


6s the numbers of factors included in thee!periment increases, the number of response plotsneeded to plot all combinations of two factors

becomes rather large. To reduce the numbers ofplots necessary, only those eects determined to beimportant from the design matri! are plotted.

Therefore, in this e!ample, one plot of average

shade versus oven temperature is needed sinceoven pressure is an important eect but does notinteract with either of the other factors. 6nother plotis needed to display the interaction betweenmaterial quality and o!idation temperature. These

plots are both shown in Fig. ).').

")( ")(


57/135


Eaterial

Ouality

&!idation

temperature

Jow High

6 "(5.) "'/.)

&ven pressure

Jow High"(5 "5

' "'-(

'(

"((

"'(

""(

"5(

"$(

")(

' "'-(

'(

"((

"'(

""(

"5(

"$(

")(

&!idationtemperature3a4

&ven pressure3b4

Eaterial >

Eaterial 6

Figure ).') ;esponse plots for theimportant eects in dye process

e!periment.

Conclusions -or Ea!"le


58/135


'. =ncreases in oven pressure increase shade at a rateindependent of material quality and o!idation

temperature 3within the bounds of the e!periment4.". &!idation temperature and material qualityinteract. =ncreases in o!idation temperatureincrease shade with material quality 6 but decreaseshade with material >.

5. ;unning the process at high o!idation temperaturewould make the process less sensitive to variationin material quality and would result in a moreuniform shade.

$. 6fter the o!idation temperature is set high, theoven pressure could be set near the low level usedin the e!periment to obtained the desired dyeshade level of "((.

). Followup to this study should be a veri9cation ofthe avera e shade level and of the reduced

Conclusions -or Ea!"le6$*

:!ample


59/135


$ manufacturer of solenoid valves designed a solenoidto be used with a pollution control device on anautomotive engine. The solenoid was used to turn the

pollution control device on and o. >ngineers

responsible for the design of the solenoid ran afactorial design to determine the eect of some of theimportant components in the solenoid valve on the!ow


60/135


Factor Jevel

8

6 2 Jength of the

armature, in

(.)) (./()

< 2


61/135


%$ O'(ecti)e


62/135


+$ Factor under stud24e)els

'. 6rmature length 3in4". obbin depth 3in$. Tube length 3in44C$ +ac,ground )aria'lesMet&od o- control

'.

".

5.

:nvironment

;esistance of wire

Flow tester

5$ Re"lication/Four solenoids were assembled for each of the '/ conditionsin the study.

(.)) (./()

( '(('.() '.'()(.)(( (.)'(

Iressure, humidity, andtemperature recorded at timeof testDontrol chart used at thewinders to monitor

resistance.Dalibration checked at thebeginning, middle, and end ofthe test


63/135


6$ Met&od o- rando!i7ation/ The order in which the '/ combinations were assembled wasrandomi*ed using a random permutation table. The four

solenoids for each combination were all assembled at thesame time. The order of test was the same as the order ofassembly to save time.

#$ Design !atri/ 3attach copy4

8$ Data collection -or!s/

9$ Planned !et&od o- statistical anal2sis/Domplete the average and standard deviation of the fourCow readings for each of the '/ combinations. 6naly*e theeects of the factors on both these statistics.

>$ Esti!ated cost sc&edule and ot&erresources/


64/135


Test

Runord

er

Factor

Flo@ c-!

0 S + T0

S

0

+

0

T

S

+ST +T

0S

+

0S

T

0+

T

S+

T

0S

+TS

' ' 8 8 8 8 8 8 8 (.$/ (.($" '" 8 8 8 8 8 8 8 (.$" (.'/5 '$ 8 8 8 8 8 8 8 (.) (.("

$ '( 8 8 8 8 8 8 8 (.$) (.'() - 8 8 8 8 8 8 8 (.5 (.("/ 8 8 8 8 8 8 8 (.' (.(' '5 8 8 8 8 8 8 8 (.( (.()- 5 8 8 8 8 8 8 8 (.( (.(' ') 8 8 8 8 8 8 (.$" (.($'( 8 8 8 8 8 8 8 8 (."- (.')'' ) 8 8 8 8 8 8 8 8 (./( (.('" $ 8 8 8 8 8 8 (." (.(/'5 / 8 8 8 8 8 8 (.( (.("'$ '/ 8 8 8 8 8 8 8 8 (.' (.("

') '' 8 8 8 8 8 8 8 8 (." (.("

Test

Runord

er

Factor

Flo@ c-!

0 S + T0

S

0

+

0

T

S

+ST +T

0S

+

0S

T

0+

T

S+

T

0S

+TS

' ' 8 8 8 8 8 8 8 (.$/ (.($" '" 8 8 8 8 8 8 8 (.$" (.'/5 '$ 8 8 8 8 8 8 8 (.) (.("

$ '( 8 8 8 8 8 8 8 (.$) (.'() - 8 8 8 8 8 8 8 (.5 (.("/ 8 8 8 8 8 8 8 (.' (.(' '5 8 8 8 8 8 8 8 (.( (.()- 5 8 8 8 8 8 8 8 (.( (.(' ') 8 8 8 8 8 8 (.$" (.($'( 8 8 8 8 8 8 8 8 (."- (.')'' ) 8 8 8 8 8 8 8 8 (./( (.('" $ 8 8 8 8 8 8 (." (.(/'5 / 8 8 8 8 8 8 (.( (.("'$ '/ 8 8 8 8 8 8 8 8 (.' (.("

') '' 8 8 8 8 8 8 8 8 (." (.("


65/135


gstudy.

Figure ).' ;un chart for the solenoid


66/135


gstudy.

Th h t f th d t d d


67/135


The run chart for the averages and standarddeviations of Cow from the four tests of each of the '/combinations appear in Fig. ).'. >oth charts have

some patterns worth noting. The chart for averagesshows eight points all near (.". 6ll eight points areassociated with a bobbin depth of Cow. The relativelysmall variation of the eight points led the engineers to

believe that some other component in the solenoidwas preventing the Cow from e!ceeding (.5. Furthertests were planned to investigate this possibility.

Three large standard deviations 3(.' cfm or greater4appear in the run chart. 6 check of data on the

background variables did not identify any specialcauses of variation. The three were all associated withlong armature length and short bobbin depth, so theengineers attributed them to an interaction between

the factors. The preliminary conclusion would be

Factor ori t ti

Esti!ate Bc-!

Factor ori t ti

Esti!ate Bc-!


68/135


interaction

s

0 (.(- (.(5S (.($ (.("

+ (."- (.(/

T (.($ (.((

0S (.(5 (.("

0+ (.( (.()

0T (.(5 (.('

S+ (.($ (.("

ST (.(' (.((

+T (.($ (.((0S+ (.(5 (.("

0ST (.(' (.('

0+T (.($ (.("

S+T (.(( (.('

0S+T (.(' (.('

interaction

s

0 (.(- (.(5S (.($ (.("

+ (."- (.(/

T (.($ (.((

0S (.(5 (.("

0+ (.( (.()

0T (.(5 (.('

S+ (.($ (.("

ST (.(' (.((

+T (.($ (.((0S+ (.(5 (.("

0ST (.(' (.('

0+T (.($ (.("

S+T (.(( (.('

0S+T (.(' (.('


69/135


$*$:$*:$5:$#

Dot"lot o- Standard de)iation Bs

D$*6D$*DD$%6D$%DD$D6D$DD:D$D6

Dot"lot o- 0)erage

Figure ).'- :ects of factors on Cow.

Design !atri and dot diagra!s


70/135


Table ). contains the design matri! for the solenoide!periment. Figure ).'- shows the eects of thefactors and the dot diagrams. :ects of the factors onboth the average Cow and the standard deviation ofCow were estimated from the design matri!. From Fig.).'-, it is seen that bobbin depth had a large eect

on average Cow. >obbin depth, armature length, andtheir interaction were the most important factorsaecting the standard deviation of Cow. Thisinteraction was observed in the run chart.

For a design, the dot diagrams can be modi9ed toprovide a display of the variation due to nuisancevariables. The magnitudes of the four threefactorinteractions and the single fourfactor interaction canusually be assumed to be primarily the result ofnuisance variables. The smaller the eect of nuisance

Design !atri and dot diagra!s

Fi ) ' t i di9 d d t di f th


71/135


Figure ).' contains a modi9ed dot diagram of the

eects on the standard deviation of Cow. =n the

9gure, the eects are spread out hori*ontally,

beginning with the eects of the individual factors

and ending with the fourfactor interaction. The

vertical line alerts the analyst to the range of

variation in the eects that could be e!pected due to

nuisance variables.

0nal2sis o- cu'es


72/135


0nal2sis o- cu'es

Figure )."( shows the two cubes representing the '/combinations in the e!periment. The paired

comparisons do not indicate the presence of anyspecial causes of variation and therefore con9rm theeects that were found using the design matri!. Theeight paired comparisons for each factor are obtained

in the same manner as for the design e!cept for thefactor of tube length.

The paired comparisons for tube length are found bycomparing the corresponding corners of the twocubesL e.g., compare the results at the bottom frontleft corner of the top cube, (.$/, to the result at thecorresponding corner of the bottom cube, (.$". Theconsistency of the eects can be evaluated by using

these plots.


73/135


(.$/3(.($4

(.$"3(.'/4

6rmature length3in4

(.)

)

(.(3(.('

4

(.$)3(.'(

4

(./(

)

obbindepth

3in4

(.(3(.()

4 (.'

3(.('4

(.)3(.("

4

(.5

3(.("4

Tu'elengt& $6 in


74/135


(.$"3(.($4

(."-3(.')4

6rmature length3in4

(.)

)

(."3(.('

4

(."3(.(/

4

(./(

)

obbindepth

3in4

(."3(.("

4 (.'

3(.("4

(.)3(.(

4

(.'

3(.("4

Tu'elengt& $6% in

Figure )."( Dubes for the solenoide!periment.

The follo ing are e amples of obser ations based on


75/135


The following are e!amples of observations based onevaluating the consistency of the eects.'. 6s bobbin depth increases, the average Cow

increases for each of the eight combinations of thefactors. This indicates an important positive eectof bobbin depth on the average Cow.

". 6s bobbin depth increases, there is no eect on

the standard deviation of Cow for low levels ofarmature length, and a consistent positive eectfor high levels of armature length. This indicatesan important armature length bobbin depthinteraction on the standard deviation of Cow.

5. 6s spring load increases, the standard deviation ofCow increases for two conditions, decreases for9ve conditions, and is appro!imately *ero for theremaining condition. This indicates that there is

probably not an important eect of spring load on

Res"onse "lots


76/135


Figure )."' shows the response plots summari*ing

the important results of the e!periment. The response

plots are constructed Aust as they were for the

design.


77/135


+o''in de"t&

'.() '.'()

(.$$ (.'

6rmaturelength

>obbin depth

'.() '.'()

(.)) (.($/ (.(5"

(./() (.'") (.(')

' "(

(.'

(."

(.5

(.$

(.)

(./

(.

' "(

(.("

(.($

(.(/

(.(-

(.'

(.'"

>obbin depth >obbin depth

6rmature length 2 ./()

6rmature length 2 .))

6 v e r

a g e C o w

3 D F E 4

< t d .

d e v .

C o w

3

D F E 4

Figure )."' ;esponse plots for the solenoide!periment.

The four standard deviations corresponding to a given


78/135


The four standard deviations corresponding to a givenlevel of bobbin depth and armature length are pooledtogether rather than averaged.

Donclusions for :!ample ).5, design of a solenoidvalve0'. There was no special causes of variation detected

in the run chart or the analysis of the paired

comparisons in the cube.". >obbin depth is the only factor in the e!perimentthat had a substantial eect on the average Cow ofthe solenoid. The longer the bobbin depth, thegreater the Cow of the valve.

5. The standard deviation of Cow was aected bybobbin depth and armature length. These twofactors interacted with each other. Thecombinations of a long armature and a short

bobbin produced large variations in Cow.

Design o- Factorial


79/135


=n chapter 5, the principles of e!perimental design

were discussed. =n this section, the discussion centerson how the following relate to factorial designs0

Design o- FactorialE"eri!ents


80/135


The numbers of factors to be included in the

e!periment will obviously depend on the obAectiveand the available resources. =n the early stages ofe!perimentation, when little is known about theprocess or the product, the obAective of thee!periment may be to screen out the unimportant

variables. =f it is reasonable to assume that only a fewof the variables will be important, a fractional factorialdesign should be considered. These designs will bediscussed in chapter /. there are some particularly

useful fractional factorial designs for screening up to'/ factors.=f it is desired to study in depth the relationshipbetween the factors and the response variables,including interactions between the factors, two to 9ve

=f th bA ti f th i t i l d th t d


81/135


=f the obAective of the e!periment includes the studyof the important factors under a range of conditionsto increase degree of belief, a chunk variable could

be included as one of the factors. 6 chunk factor wasde9ned in chapter 5 as a combination of backgroundvariables.

The diculty of changing levels is another

consideration for how factors may require little morethan a turn of a dial, and their eect on the responseis seen very quickly. &ther factors are hard tochange. 6 physical change to the equipment maytake hours or days to accomplish. For someprocesses, it may take hours to reach equilibriumafter changes in operating conditions have beenmade. =t is usually unwise to run a study in whichmore than half of the factors are hard to change. =n

that case, it would be advisable to run a series of

Figure ).'( Dhoosing the Number of Factors


82/135


O'(ecti)e Nu!'er o--actors

Design


83/135


Dhoice of the speci9c levels used for each factor willbe based on knowledge of the process or product andthe conditions of the study. =t is desirable to set thelevels of the factors far enough apart that the eectsof the factors will be large relative to the variationcaused by the nuisance variables. However, the levels

should not be so far apart that there is a good chancefor trouble to develop.:!amples of such trouble are the following0 Donditions that make the e!periment run unsafely

Donditions that cause substantial disruption of amanufacturing facility =mportant nonlinearities or discontinuities hidden

between the levels


84/135


p pfactor constant at the desired level because the factorcannot be controlled that precisely. 6lthough it is

desirable, it is not necessary that a factor be heldconstant at the planned level. However, the variationin the factor should be small relative to the distancebetween the planned levels. =f this cannot beachieved, the factor should be made a background

variable and measured during the e!periment.+hen continuous variables such as temperature,pressure, and line speed are used as factors, then arun at the center of the design can be used to check

for discontinuities or nonlinear eects.

Factor Jevel

8

Temperature34 ""( "$(

Iressure 3lb4 )( -(

Doncentration3K4

'( '"

Factor Jevel

8

""( "$(

)( -(

Doncentration

3K4

'( '"

For e!ample, supposethe following factorsand levels were used ina factorial design0

The center point of this design is Denter point


85/135


The center point of this design is

Temperature "5(

Iressure /)

Doncentration

''

The e!perimenter will also need to take into accountthe amount of unusable material generated during thestudy and the safety of those conducting the

e!periment. >o! and raper 3'/4 discuss ways ofdesigning e!periments for processes already inproduction.

=n their approach, levels are set close together tominimi*e unacceptable product. The small eect of afactor when the levels are close together is overcomeby running many e!perimental units per factorcombination. This usually can be done when therocess is in full roduction. This a roach is

T&is "oint @ould 'e in

t&e !iddle o- t&e cu'e$


86/135


6s indicated in chapter 5, there are two importantdecisions to be made concerning background variables0'. How to control the background variables so that the

eects of the factors are not distorted by them". How to use the background variables to establish a

wide range of conditions for the study to increase

degree of belief

+hen a sequential approach to a study is taken, it isoften useful to hold background variables constant and

study the eects of the factors. To design the ne!tstudy, the background variables should be used toestablish conditions for the study that diersubstantially from those of previous study. This willallow the e!perimenter to determine of the eects of

the factors are consistent over a wide range of


87/135


;eplication is the primary means of studying thestability of the eects of the factors and increasing thedegree of belief in the eects. ierent types ofreplication that can be used in an e!periment werediscussed in chapter 5.

=n factorial e!periments, replication is built into thee!perimental pattern. For e!ample, in a design thereare eight comparisons of the high and low levels ofeach factor. These comparisons are made under the

eight dierent combinations of the other three factors.

The amount of replication that is feasible dependspartly on how dicult it is to change the levels of thefactors. +hen multiple factors are included in an

e!periment, between - and '/ runs are desirable.

=f the aim of the study is to improve a process and


88/135


=f the aim of the study is to improve a process, andthe eects of the factors cannot be distinguishedfrom the nuisance variables in - to '/ runs, then the

wrong factors or levels have probably been chosen.Eore important than the number of runs is the rangeof conditions included in the study. ;eplication oversimilar conditions provides little increase in the

degree of belief.


89/135


;andomi*ation is the primary means of controlling theeect of nuisance variables. =f blocks are constructed tocontrol background variables, randomi*ation should beconsidered for choosing the order of the tests, assigningthe combinations of the factors to e!perimental units,and choosing the order in which the measurements are

made.=n factorial e!periments, sometimes randomi*ation isnot practical because of the diculty of changing thelevels of some of the factors. =n such a case, it is

desirable to repeat one or more of the earlier runs atthe end of the e!periment. These replications can beused to check for special causes of variation that mayhave occurred during the e!periment.

There is less need to randomi*e when e!periments are

conducted on stable rocesses. For rocesses

:!ample


90/135


Consider an experiment in a welding operation forassembling an automotive part. (revious improvementccles using fractional factorial designs indicated thatthe two most important factors were pressure andvacuum. 9mprovement ccles using control chartsindicated that da*toda variation in the process+ the

environmental conditions+ and operator techni:uewere important bac-ground factors. The responsevariable of interest was a particular dimension of theassembled part.

The primar aim of the experiment was to stud theeects of pressure and vacuum over a wide range ofconditions to increase degree of belief in their eects.Table 5.11 identi2es the design used to accomplish

this aim.

p).$

%a*to*da variation and operator techni:ue were


91/135


%a to da variation and operator techni:ue wereincorporated into a chun- factor. This chun- factor isused along with pressure and vacuum to form a

factorial design. ?ecause of the nature of the chun-factor+ randomi)ation is restricted to each chun-. Thefour tests in chun- 1 are done in random order+followed b the four tests in chun- in random order.

9f neither pressure nor vacuum is found to interactwith the chun- factor+ then an increased degree ofbelief will result that the estimated eects of

pressure and vacuum can be used in the future. That

is+ the eects can be used to set speci2cations for pressure and vacuum and also be used as a guide toad@ust the process


92/135


9f either pressure or vacuum is found to interact withthe chun- factor+ then the appropriate setting ofeither pressure or vacuum can be used to mitigate

the eect of the chun- factor+ i.e.+ da*to*davariation and operator techni:ue. This will lead toimproved consistenc of the welding process.

The analysis of this e!periment is discussed in


93/135


Chun- factor 0Jevel '0 Eonday, operator'

Jevel "0 +ednesday,operator "

%esign0 factorial

Factors Jevels

Iressure 5( )$Qacuum - '(

Dhunk ' "

Table ).'' esign for +elding:!periment

0nal2sis o- Factorial


94/135


=n sec. ).' analysis of data from factorial designs withtwo, three, and four factors were illustrated. This

section provides more information on the analysis offactorial designs. The emphasis is on the analysis offactorial designs when things do not go as planned.6lso, some additional details about the various

graphical displays used in the earlier e!amples willbe given.

The approach to the analysis of factorial designs inthe previous e!amples followed four steps0'. Ilot the data in run order to look for trends and

obvious special causes. =f an uncontrolledbackground variable has been measured, then thedata should also be plotted in increasing order of

measurement of the background variable.

E"eri!ents

The purpose of the remaining three steps is to


95/135


The purpose of the remaining three steps is topartition the variation seen in the run chart betweenthe factors, background factors, and nuisance

variables.". :stimate the eects of the factors using the

design matri!, and plot the estimates on a dotdiagram. This step provides a preliminary

assessment of the factors that contribute most tothe variation in the run chart.5. 6naly*e the data using a square, a cube, or sets of

cubes to study the variation in the pairedcomparisons that make up the eects estimated in

step ". a study of the paired comparison will helpidentify the impact of nuisance variables on thevariation and will indicate the presence of specialcauses.

$.


96/135


e a o ese ou s eps s o pa o evariation in the measurements of the responsevariable among the factors, the background

variables, the nuisance variables, and anyinteractions between them. This partitioning, alongwith other knowledge of the product or process,allows the e!perimenter to determine the most

advantageous approach to improvement. This analysis can be carried out on the individualmeasurements of the response variable or on astatistic chosen based on knowledge of the processor product. =f multiple e!perimental units are

measured for each combination of the factors,statistics such as the average or range may becomputed. The eects of the factors on thesestatistics can then be estimated.

For e!ample, in a machining operation it may be

=n this case, several e!perimental units would be


97/135


measured for each condition, the range of the

measurements computed, and the fourstep analysiscarried out using the range as the response of

interest.

Taguchi 3'-4 suggests combining various statistics

into what he calls a signal-to-noise ratio and using

the signaltonoise ratio as the response of interest.

Rackar 3'-)4 provides an e!cellent summary of

these ideas. >o! 3'--4 discusses the use of signal

tonoise ratios and transformation.

Gra"&ical dis"la2s -or t&e anal2sis o-


98/135


" " 2 2-actorial designs

The following graphical displays have proven useful

in analy*ing data from factorial designs0 ;un charts ot diagrams 7eometric 9gures or other displays of paired

comparisons ;esponse plots

:ach of these displays has been illustrated in thee!amples. =n this section, some additional insight into

the use of these displays will be given.

Run C&art


99/135


The run chart is usually a display of themeasurements of the response variable in the order

that the tests were made. &ther ordering, such asorder of measurement or increasing order of ameasured background variable, are also useful. Therun chart displays the total variation in the responsevariable that is to be partitioned among the factors,background variables, and nuisance variables.6nalysis of run chart begins with partitioning.


100/135


The variation in the measurement of the response

variable that is due to the background variables

comprising the blocks can be assessed.


101/135


The second step in the analysis of data from afactorial e!periment is to estimate the eects of the

factors from the design matri! and to plot theestimates on a dot diagram. The dot diagram is usedto obtain a 9ner partitioning of the variation than ispossible in the run chart. The dot diagram identi9esfactors whose eects are clearly separated from thevariation due to nuisance variables.

ata from a factorial design will vary because of thefactors, background variables, and nuisance

variables, :ects clustered near *ero cannot bedistinguished from variation due to nuisancevariables. Nuisance variables and backgroundvariables can impact a study as either common or

special causes of variation.

=f a threeor fourfactor interaction is one of the


102/135


largest 3in absolute value4 eects in the dot diagram,then

=t may not be possible to separate any of theeects of the factors from variation due tonuisance variables.

6 special cause may be dominating the estimate ofthe eects 3the run chart and the cubes should bechecked4.

The interaction may be important 3the least likelyalternative4.

For a design, a modi9ed dot diagram can be used tobetter distinguish the eects of nuisance variables.>o! et al. 3'-4 use reference distributions incombination with dot diagrams to separate variation

due to nuisance variables from variation due to

;egardless of this method of analysis, caution should


103/135


g y ,

be e!ercised in placing a high degree of belief in

small, although discernible eects. =f an eect is

small compared to the variation caused by other

factors, background variables, or nuisance variables,

then it is liable to change substantially in the future

because of interactions with the conditions only

through a synthesis of knowledge of the subAect

matter and replication of the eect over a wide

variety of conditions.

>y following these simple guidelines one can make


104/135


the dot diagram more useful for separating theimportant eects from the variation due to nuisance

variables0'4 Donstruct the scale on the diagram such that *ero

is at the center point and endpoints of the scaleare symmetric 3for e!ample, )(, )(4. The

endpoints should be chosen such that all eectscan be plotted on a linear scale."4 :ects too close together to be plotted side by side

should be plotted one above the other, as in ahistogram.

54 6ll eects that are separated from the clusteraround *ero should be labeled.

$4 =f a complete factorial design is replicated in two ormore blocks, the eects should be computed

separately for each block. The eects from each of

Geo!etric gures and "lots o-i d i


105/135


Factorial designs can be depicted using a square orone or more cubes. From these geometric 9gures, the

paired comparisons that make up the factorialdesigns can be identi9ed and studied. 6nalysis of thepaired comparisons determines the factors with thegreatest eect on the response variable. This is thesame information obtained from the dot diagram.However, study of the paired comparisons providessome important additional information.>y studying the individual paired comparisons forconsistency, the e!perimenter can ascertain the

presence of special causes in the data. The e!istenceof special causes in the data may not be evident inthe run chart because of the variation attributable tochanges in the factors. =f special causes of variation

e!ist, they may also not be evident in the dot

"aired co!"arisons

:!ample ). 3to follow4 illustrates the e!istence of a


106/135


special cause that was identi9ed by analysis of the

paired comparisons.

=n e!ample ).", the paired comparisons were

analy*ed by identifying them on the cube, listing

them 3Table ).)4, and plotting them 3Fig.).'"4.

+hether the eects are listed or plotted, the

e!perimenter can analy*e them for consistencyL it is

not necessary to do both. The aim of the analysis is

the same in each case0 to con9rm that special causes

of variation are not distorting the eects estimated

from the desi n matri!.

Res"onsel t


107/135


;esponse plots are used to help e!perts in thesubAect matter visuali*e the impact of the eects of

the factors found to be important in steps " and 5.the number of response plots constructed dependson the factors and interactions found to be importantin steps ', ", and 5. The following items should beconsidered when response plots are constructed0 =f a factor is found to be important but does not

interact with any other factor, the response plotconsists of one line appro!imating the relationshipbetween the factor and the response.

=f two factors 6 and > interact, their relationship tothe response variable is displayed on one responseplot by using two lines. The relationship betweenthe response variable and factor 6 when > is at the

minus level is displayed by one line.

"lots

The relationship between response variable and


108/135


factor 6 when > is at the plus level is displayed byanother line. &nce this response plot has been

constructed, there is no need to construct individualresponse plots of the type in Fig. ).')a for each offactors 6 and >.

=f two factors 6 and > interact and, in addition, one

or both of them interact with a third factor D, theresponse plot displaying the interaction of 6 and >must be separately for the two levels of D.

The response plots constructed for a particulare periment pro ide a isual model of the relationships


109/135


e!periment provide a visual model of the relationshipsbetween the important factors and the response

variable. The response plots are interpreted as follows0'. The response plots are only a linear appro!imationof the relationshipL therefore, their usefulness islimited based on the linear relationship of the data.

The use of center points to check for lack of

linearity has been mentioned in this chapter andwill be discussed in greater details in chapter -.

". The slope of the response plot is proportional to theeect estimated from the design matri!.

5. =f two factors 6 and > interact the interaction isdisplayed using two lines on the response plot 3asin Fig. ).')b4, then the dierence in the slopes isproportional to the interaction between 6 and >. if

6 and > do not interact, then the two lines will be

$. The slopes of the lines in a response plot are their


110/135


most important aspect. The intercepts of the linesdepend on the levels of other factors in the study

that are not in the response plot. 6s constructed inthis chapter, the intercepts correspond to the otherfactors being set midway between the minus andplus levels.

=n some cases, the analysis of data from a factorialdesign can proceed in a straightforward mannerthrough the four steps listed at the beginning of thissection. &ften, the analysis of the data is not so

straightforward because of problems such as thepresence of special causes of variation, a largeamount of variation caused by nuisance variables, orone or more data points missing. The primary purpose

of the remainder of this section is to provide some

0nal2sis o- -actorial designs @&en t&ingsgo @rong


111/135


S"ecial causes in t&e run c&art. There are times when trends or other obvious special

causes are seen in the run charts. =t is not advisableto proceed further with the analysis in such casesuntil the special cause is determined. The potentialinteractions of the special cause with the factors

should also be assessed. =f the cause is identi9ed, thedata can possibly be adAusted to remove its eect.;egardless of the method of analysis, e!treme careshould be taken in e!trapolating the results of thee!periment. Qeri9cation of a conclusion by futuree!periments is almost always necessary.

go @rong

:!ample) )


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Study of Automotive Emissions $ factorial designwas run to stud the eects of four factors onemissions from an automotive engine. The run chartsfor hdrocarbon


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Design of a Throttle Return Mechanism 9n astud to determine the durabilit of a throttle returnmechanism+ the following three factors+ which relatedto the pin in the mechanism+ were included4

)./

Factor Jevel

Iin coated No Ues

Iin length 8Iin diameter 8

$ design was used+ in which the response variable

was the number of ccles to failure on a stress test.,igure 5.3 contains the run chart. ,igure 5. showsthe data displaed on a cube.

) '$(,)


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$(,((( 3)4

5,-((

3$4

Iin coated

),'")3'4

'5,)(

354

8

Iinlength

8

8

Iindiamete

r

$$3-4 '$,(

$3"4

"",$$$3/4

5$,'

34

Figure )."$ Dube for the factorial design in:!ample )./.


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Figure )."5 ;un chart for :!ample )./.

From the run chart it is seen that the number ofl bt i d th 9 t 3 di t th


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cycles obtained on the 9rst run 3corresponding to theconditions of pin coated, plus length, and plus

diameter4 dier substantially from the rest of thedata.

These conditions represent an e!treme case becauseeach of the factors is set at the level that theengineer believed would provide increased durability.=t is not unusual that e!treme points produce resultsof greater magnitude than the sum of the individualeects.=n this e!periment a special 9!ture had to be set up

to run the tests.


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6 special cause of variation in the data from a

factorial e!periment may not be evident in a run

chart.


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Pilot line for a Tile Process 7n a pilot line built tostud a new process of ma-ing !ooring tile+ a designwas used to determine the eect of three factors onan important :ualit characteristics of the tile. Thefactors are related to the setup of the production line.The pilot line was run for 1 h at each of the eight

conditions. %uring each hour+ 1; tiles were selected+and the :ualit characteristic was measured. ?asedon an analsis of the run chart of the readings foreach condition+ it was deemed appropriate to

summari)e the data for each condition b an averageand a standard deviation.

).

'.''/.(3'4


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"(.-34

'/.'3)4

6Jow

'.'3-4

')."354

High

>

Jow

High

Jow

High

D

3'4

".(3/4

'".'3$4

"$.'

3"4

Figure )."/ Dube for :!ample )..

;un order

inparentheses


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Figure ).") ;un chart for :!ample )..

Figure ).") is the run chart of the averages. Noobvious special causes are seen in the run chart The


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obvious special causes are seen in the run chart. Thedata are displayed in a cube in Fig. )."/. The paired

comparisons, plotted in Fig. ).", show that of thefour comparisons for each factor, three are consistentand one is quite dierent. 6s they are plotted in9gure ).", the 9rst, second, and third comparisonsfor factors 6, >, and D respectively, are quitedierent from the other three in each plot.

These comparisons have one point in commonL it isthe 9fth run. This run was performed at the

conditions 6 2 8, > 2 , D 2 and resulted in anaverage response of '/.'. based on an e!aminationof the other comparisons, an average response of "5to "$ would have been consistent with the rest of the

data. 6fter a check of the notes kept during the

The analysis of the cube uncovered the possiblee!istence of a special cause although none was


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e!istence of a special cause, although none wasevident from the run chart. This provides an increased

opportunity to learn about the process. This e!ampleillustrates the importance of detailed analysis of thecube prior to estimating the factor eects. =f only theaverage eects estimated from the design matri! areused, some important information concerning specialcauses may be lost.

Missing Data$ =t occasionally happens that the


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data from one of the e!perimental conditions aremissing. +hen this happens, it is important to

determine whether the fact that the data are missingis related to the e!perimental conditions. Fore!ample, the data may be missing because theproduct not be made under those conditions.

=f the data are missing because of the e!perimentalconditions, the analysis should proceed, using thecube to determine better conditions under which torun the process.


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from :!ample ).", the design for a dye process,which is taken from Table )..

The threefactor interaction from Table ).'"a is % '. Table ).'"b shows a procedure for estimating amissing value. 6ssume, e.g., that the data for test )are missing, for reasons that have nothing to do withthe e!perimental conditions of that test.


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when the variation in the data from a factoriale!periment is caused primarily by nuisance variables

rather than by changes in the factors under study. The nuisance variables may be either common orspecial causes of variation.

The e!perimenter can be alerted to the presence ofnuisance variables dominating the variation in the

data by the presence of one or more of the followingconditions0 Qariation in the run chart that is of similar

magnitude to the variation seen in an e!isting

control chart for the process 6 measurement system

06 reducing the size of experiments

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