part iv process design and improvement with designed

Part IV

Process Design and Improvementwith Designed Experiments

163

Chapter 12

Factorial and Fractional FactorialExperiments for Process Designand Improvement

We look at using factorial and fractional factorial experiments in process design andimprovement.

12.1 What is Experimental Design?

The design of the experiment involves investigating many factors, to determine thefactors which both most influence and/or optimize a response variable and to reducevariability in the process. The particular two–level factorial and fractional factorialdesigns investigated in this chapter are useful in the early stages of an experimentto identify the important factors in the process. Whereas the previously discussedstatistical process control (SPC) techniques emphasize passively monitoring a process,the current experimental design technique actively seeks to determine the levels of themany factors that will reduce the variability in a process.

12.2 Examples of Designed Experiments in Pro-

cess Improvement

We look at an example of a two–level factorial designed experiment. The two mainobjectives related to this type of experiment are identifying the important factors andchoosing the levels of the chosen factors to optimize the response variable.

Exercise 12.1 (Examples of Designed Experiments in Process Improve-ment)

165

166Chapter 12. Factorial and Fractional Factorial Experiments . . . (ATTENDANCE 11)

Consider the effect of seven factors on the rate of oxygen consumption of mice. Eachfactor is studied at the following low and high levels.

factor low level, (−) high level, (+)temperature, A 0o F 30o Fnoise, B 10 dB 100 dBhumidity, C 50% 75%pressure, D 28.5 inches 30.5 incheslighting, E 50 watts 100 wattsliving space, F 25 in3 50 in3

water, G 10 fluid ounces 30 fluid ounces

1. A 27 full factorial design: exploratory design.There are (choose one) 7 / 14 / 128factors (treatments) in this two–level full factorial design.Most likely, this is an exploratory experiment where, to begin with, we aresimply interested, after checking the interactions, in identifying which of theseven factors are active (significant),

H0 : βq = 0 vs Ha : βq 6= 0, ; q = 1, . . . , 7

Having identified the (two, three or four, say) active factor effects, we wouldthen go on to analyze these active factor effects in greater detail, to determine,for example, the levels of the significant factors which maximize the rate ofoxygen consumption.

2. Choosing the optimal responseAssume we are able to reduce the seven–factor experiment to a two–factorexperiment involving the effect of air temperature and noise on the rate ofoxygen consumption of mice. The two factors are studied at the following low(−) and high (+) levels.

temperature ↓ noise → 10 dB (low, B = −) 100 dB (high, B = +)0o F (low, A = −) 10.3 11.430o F (high, A = +) 9.4 9.7

This information could also be written as (choose none, one or more)

(a) Data candidate A

response temperature, A noise, B10.3 0o F 10 dB11.4 0o F 100 dB9.4 30o F 10 dB9.7 30o F 100 dB

Section 2. Examples of Designed Experiments in Process Improvement (ATTENDANCE 11)167

(b) Data candidate B

response temperature, A noise, B10.3 − −

11.4 − +9.4 + −

9.7 + +

as long as, in this second case, we rememberA = − means 0o F, A = + means 30o F and thatB = − means 10 dB, B = + means 100 dB.

In this case, the rate of oxygen consumption (response) is maximized (11.4)when (choose one)

(a) factors A and B are both low (−)

(b) factor A is low (−) and factor B is high (+)

(c) factor A is low (+) and factor B is high (−)

(d) factors A and B are both low (−)

3. Notation and terminologyReconsider the two–factor experiment involving the effect of air temperatureand noise on the rate of oxygen consumption of mice.

run temperature, A = x1 noise, B = x2 response1 (1) − − 10.32 b − + 11.43 a + − 9.44 ab + + 9.7

(a) Each observation of the experiment is called a run. There are(choose one) 2 / 3 / 4 runs in this experiment.

(b) True / FalseEach factor is designated as A, B, C, . . . if treated as part of an analysis ofvariance or as x1, x2, x3, . . . , if treated as part of a regression. Furthermore,A = −, then x1 = −1; if B = −, then x2 = −1.

(c) In a two–level for each of k factors, 2k, experiment, the levels are denotedeither as low (−) or high (+). Furthermore, the high level of a main effectis denoted by the presence of a small letter and the low level of a maineffect is denoted by the absence of a small letter. For example, “a” meansthat factor A is set at its high level and factor B is set at the low level. So“ab” means (choose one)


i. factor A and factor B are both set at the low level

ii. factor A and factor B are both set at the high level

12.3 Guidelines for Designing Experiments

Possible guidelines for designing experiments include

1. statement of problem

2. choice of factors and levels

3. selection of response variable

4. choice of experimental design

5. perform the experiment

6. data analysis

7. conclusions and recommendations

12.4 Factorial Experiments

This section looks into factorial experiments that involve more than two levels. Thisis not covered.

12.5 The 2k Factorial Design

SAS program: att11-12-5-mice-23fullfactorial

The 2k factorial design is analyzed in two steps. First, the factors which both mostinfluence the response variable are identified and then, second, the factors whichremain are each set at levels that optimize the response variable. Along the way, itis demonstrated that for unreplicated two–level studies, there is no estimated errorvariance. Consequently, it is not possible test for active factor effects because the Fstatistics necessary to do this require the estimated error variance (MSE ). There aretwo methods for obtaining estimated error variance

• pool higher order interactions and form the error variance from these interac-tions.

Section 5. The 2k Factorial Design (ATTENDANCE 11) 169

• add more observations (replications) called center points and form the errorvariance from these added replications.

Exercise 12.2 (The 2k Factorial Design)

1. The 22 full factorial design, replication.Consider the effect of air temperature and noiseon the rate of oxygen consumption of mice. We assumex1 = −1 means 0o F, x1 = 1 means 30

o F and thatx2 = −1 means 10 dB, x2 = 1 means 100 dB.

response temperature, x1 noise, x2

10.3 − −

11.4 − +9.4 + −

9.7 + +8.3 − −

10.4 − +9.5 + −

8.7 + +

This experiment has been replicated (choose one) once / twice.

2. The 22 full factorial design, interaction.Consider the effect of air temperature and noiseon the rate of oxygen consumption of mice. We assumex1 = −1 means 0o F, x1 = 1 means 30

o F and thatx2 = −1 means 10 dB, x2 = 1 means 100 dB.This 22 full factorial design, has one interaction term, x12, (choose one)

(a) temperature × noise

(b) temperature × temperature

(c) noise × noise

and is represented by the following coding scheme,

response temperature, x1 noise, x2 temp × noise, x12

10.3 − − +11.4 − + −

9.4 + − −

9.7 + + +


3. The 22 full factorial model, including intercept x0 and interaction x12.Consider the effect of air temperature and noiseon the rate of oxygen consumption of mice. We assumex1 = −1 means 0o F, x1 = 1 means 30

o F and thatx2 = −1 means 10 dB, x2 = 1 means 100 dB.Using the following regression version of the 22 full factorial model, match ap-propriately,

yi = β0xi0 + β1xi1 + β2xi2 + β12xi12 + εi,

model example

(a) yi (a) error(b) β0 (b) interaction temp × noise effect(c) β1 (c) main noise effect(d) β2 (d) main temperature effect(e) β12 (e) (grand) average of all responses(f) εi (f) individual roc response

model (a) (b) (c) (d) (e) (f)example

where the design matrix is given by

response intercept, x0 temperature, x1 noise, x2 temp × noise, x12

10.3 + − − +11.4 + − + −

9.4 + + − −

9.7 + + + +

4. Standard order for a 22 full factorial design.The standard order of the runs is a listing of the runs such that the level of thefirst factor1 changes most frequently and the level of the second factor changessecond most frequently and so on.Consider the effect of air temperature and noiseon the rate of oxygen consumption of mice. We assumex1 = −1 means 0o F, x1 = 1 means 30

o F and thatx2 = −1 means 10 dB, x2 = 1 means 100 dB.The standard order of the design matrix of the main factors in this study is(choose one)

(a) Factor Candidate List A

1The first factor is x1, not x0, which is associated with the overall average.



10.3 − −

11.4 − +9.4 + −

9.7 + +

(b) Factor Candidate List B


10.3 − −

9.4 + −

11.4 − +9.7 + +

(c) Factor Candidate List C


9.4 + −

9.7 + +10.3 − −

11.4 − +

5. A 23 full factorial design.Consider the effect of air temperature, noise and humidity on the rate of oxygenconsumption of mice. The three factors are studied at the following low andhigh levels.

factor low level, − high level, +temperature, x1 0o F 30o Fnoise, x2 10 dB 100 dBhumidity, x3 50% 75%

The model is given by

yi = β0x0 + β1xi1 + β2xi2 + β3xi3

+ β12xi12 + β13xi13 + β23xi23

+ β123xi123 + εijk

where the complete design matrix X is


y x0 x1 x2 x3 x12 x13 x23 x123

9.8 + − − − + + + −

10.2 + + − − − − + +10.4 + − + − − + − +8.5 + + + − + − − −

11.1 + − − + + − − +10.7 + + − + − + − −

10.7 + − + + − − + −

9.9 + + + + + + + +

There are 23 = (choose one) 8 / 16 / 32 responsesand the design matrix (choose one) is / is not in standard order.There are four interaction terms(choose four) β1 / β2 / β12 / β13 / β23 / β123

If x1 = x2 = x3 = −1, y = (choose one) 9.8 / 10.7 / 11.1If x1 = x2 = −1 and x3 = 1, y = (choose one) 9.9 / 10.7 / 11.1If x1 = x2 = x3 = −1, x13 = x1 × x3 = −1×−1 = (choose one) −1 / 1If x1 = x2 = −1 and x3 = 1, x13 = x1 × x3 = −1× 1 = (choose one) −1 / 1If x1 = x2 = −1 and x3 = 1, x123 = −1×−1× 1 = (choose one) −1 / 1Furthermore, this experiment has been replicated (choose one) once / twiceand so will have no (zero) error variance and, as a consequence, it is(choose one) still / not possible to calculate the p–value for the various effects.

6. A 24 full factorial design.Consider the effect of air temperature, noise, humidity and air pressure on therate of oxygen consumption of mice. The four factors are studied at the followinglow and high levels,

factor low level, − high level, +temperature, x1 0o F 30o Fnoise, x2 10 dB 100 dBhumidity, x3 50% 75%pressure, x4 28.5 inches 30.5 inches

where the design matrix of the main factors and intercept are given by,


ROC response, y x0 x1 x2 x3 x4

9.8 + − − − −

10.2 + + − − −

10.4 + − + − −

8.5 + + + − −

11.1 + − − + −

10.7 + + − + −

10.7 + − + + −

9.9 + + + + −

9.9 + − − − +10.4 + + − − +10.2 + − + − +9.2 + + + − +11.8 + − − + +9.5 + + − + +10.7 + − + + +9.8 + + + + +

which (choose one) is / is not in standard orderAlso, there are (choose one) 4 / 16 / 32 factorsand 24 = (choose one) 4 / 16 / 32 responsesin this two–level full factorial designIf x1 = x2 = −1 and x3 = x4 = 1, y = (choose one) 9.9 / 10.7 / 11.8If x1 = x2 = x3 = x4 = −1, x14 = x1 × x4 = −1×−1 =(choose one) −1 / 1If x1 = x2 = −1 and x3 = x4 = 1, x13 = x1 × x3 = −1× 1 =(choose one) −1 / 1If x1 = x2 = −1 and x3 = x4 = 1, x134 = −1×−1× 1 =(choose one) −1 / 1and which (choose one) is / is not in replicatedand so it is not possible to calculate the p–values for the various effects.

7. More 24 full factorial design.Consider the effect of air temperature, noise, humidity and air pressure on therate of oxygen consumption of mice. The four factors are studied at the followinglow and high levels.



The model is given by

yi = β0x0 + β1xi1 + β2xi2 + β3xi3 + β4xi4

+ β12xi12 + β13xi13 + β14xi14

+ β23xi23 + β24xi24 + β34xi34

+ β123xi123 + β124xi124 + β134xi134 + β234xi234

+ β1234xi1234 + εijk

where

x1 =

{

1, if case from first level of factor 1−1, if case from second level of factor 1,

x2 =

{

1, if case from first level of factor 2−1, if case from second level of factor 2,

and so on and where, for example, x12 = x1 × x2 = x1x2.So, x123 = (choose one) x1x2 / x1x2x3 / x2x3

We would first check if any of the interaction effects are significant.If not, we are then free to test the main effects.If so, we would then investigate the interaction effects further.

8. Looking at a 23 design in detailConsider the effect of air temperature, noise and humidity on the rate of oxygenconsumption of mice. The three factors are studied at the following low andhigh levels.


where

ROC response, y x0 x1 x2 x3

9.8 + − − −

10.2 + + − −

10.4 + − + −

8.5 + + + −

11.1 + − − +10.7 + + − +10.7 + − + +9.9 + + + +


and where the regression equation is

yi = β0 + β1xi1 + β2xi2 + β3xi3

+ β12xi12 + β13xi14 + β23xi12

+ β123xi123 + εijk

(a) A 23 full factorial design, one replication.From the SAS output, although there is no (zero) error variance (and sono p–values), the coefficients of the various effects in the model are (chooseone)

i. Candidate model Acoefficient bq p–value

b0 10.1625b1 -0.3375 nab2 -0.2875 nab3 0.4375 nab12 -0.3375 nab13 0.0375 nab23 -0.0125 nab123 0.2375 na

ii. Candidate model Bcoefficient bq p–value

b0 13.1625b1 -0.3375 nab2 -0.2875 nab3 0.4375 nab12 -0.6375 nab13 0.0375 nab23 -0.0125 nab123 0.2375 na

It is not possible to tell, from this information, which of the various effectsare significant are not. In other words, it is not possible to determine ifthis model can be simplified or not.

(b) A 23 full factorial design, pool higher order interactions.From the SAS output, match the following two different pooling methods(I or II) with the two estimated models (A or B).

Pooling Method I IICoefficients (A)/(B) (A)/(B)

(I) Pooling Method I:x123 is used alone


(II) Pooling Method II:x12, x13, x23 and x123 are pooled together

(A) Candidate model A

coefficient bq p–valueb0 10.1625 0.0149b1 -0.3375 0.3904b2 -0.2875 0.4396b3 0.4375 0.3166b12 -0.3375 0.3904b13 0.0375 0.9003b23 -0.0125 0.9665

(B) Candidate model B

coefficient bq p–valueb0 10.1625 0.0001b1 -0.3375 0.1788b2 -0.2875 0.2377b3 0.4375 0.1024

In both pooling methods, none of the main or interaction effects are sig-nificant (have a p–value less than 0.05) except the intercept (the grandaverage); in other words, the mice roc is not influenced by any of thesefactors.

(c) A 23 full factorial design, replications at center point.Consider the effect of air temperature, noise and humidity on the rate ofoxygen consumption of mice. The three factors are studied at the followinglow and high levels.


where


ROC response, y x0 x1 x2 x3

9.8 + − − −

10.2 + + − −

10.4 + − + −

8.5 + + + −

11.1 + − − +10.7 + + − +10.7 + − + +9.9 + + + +10.3 + 0 0 09.7 + 0 0 010.0 + 0 0 0

In this case, three additional center points are used to calculate error vari-ance and are given by (choose one)

i. Candidate Center Points A10.3, 9.7, 10.0

ii. Candidate Center Points B9.9, 10.3, 9.7

(d) More on a 23 full factorial design, replications at center point.The resulting coefficients and corresponding p–values are given below.

coefficient bq p–valueb0 10.1625 0.000b1 -0.3375 0.0427b2 -0.2875 0.0630b3 0.4375 0.0218b12 -0.3375 0.0427b13 0.0375 0.7317b23 -0.0125 0.9080b123 0.2375 0.0970

In this case, the significant effects at α = 0.05 are (choose one)

(i) β1

(ii) β2

(iii) β1 and β3

(iv) β3 and β12

(v) β1, β3, and β12

(e) More investigation of 23 full factorial design, replications at center point.

i. Residual plot versus predictedThe residual plot, from the SAS output, indicates


(choose one) constant / nonconstant varianceIn fact, there appears to be one bad outlier.

ii. Normal probability plot of the residualsThe normal probability plot, also obtained from the SAS output, in-dicates(choose one) normal / non–normal residuals.

(f) Investigation of reduced 23 full factorial design, no center point replicationsbut including only main factors 1 and 3.We will include onlymain factors 1 and 3 in a second analysis because thesemain factors were the most significant quantities in the initial analysis. Wewill then check to see if this change improves our results.

i. Residual plot versus predictedThe residual plot, from the SAS output, indicates(choose one) constant / nonconstant variance

ii. Normal probability plot of the residualsThe normal probability plot, also obtained from the SAS output, in-dicates(choose one) normal / non–normal residuals

iii. Which coefficients are significant?H0 : βq = 0 versus Ha : βq 6= 0The significant effects are (choose one)

(i) none

(ii) β3

(iii) β1, β3

(iv) β1 and β34

(v) β1, β3 and β34

iv. More graphical findingsFrom SAS main effect plot graph output,If temperature x1 is high, (1), roc is (choose one) low / highIf humidity x3 high, (1), roc is (choose one) low / high

v. Estimated modelThe estimated model chosen by this analysis is (choose one)

A. y = 9.1625− 0.3375x1 + 0.4375x3

B. y = 10.1625− 0.3375x1 + 0.4375x3

C. y = 11.1625− 0.3375x1 + 0.4375x3

and so, for example, if temperature and humidity were both set athigh levels, x1 = x3 = 1, then the roc of mice would be

y = 10.1625− 0.3375(1) + 0.4375(1) = 10.2625

Section 6. Fractional Replication of the 2k Design (ATTENDANCE 11) 179

vi. Run which maximizes responseFrom SAS, the roc response is maximized if (choose one)

A. temperature is low and humidity is low

B. temperature is low and humidity is high

C. temperature is high and humidity is low

D. temperature is high and humidity is high

12.6 Fractional Replication of the 2k Design

SAS program: att11-12-6-fracfactorial

Factional factorial designs allow the analysis of factorial designs with less (a fractionof the) data.

Exercise 12.3 (Fractional Replication of the 2k Design)

1. Two–Level Fractional Factorial Designs.Consider the effect of air temperature, noise, humidity and air pressure on therate of oxygen consumption of mice. The four factors are studied at the followinglow and high levels.


where, for a full 24 factorial design,



9.8 + − − − −

10.2 + + − − −

10.4 + − + − −

8.5 + + + − −

11.1 + − − + −

10.7 + + − + −

10.7 + − + + −

9.9 + + + + −

9.9 + − − − +10.4 + + − − +10.2 + − + − +9.2 + + + − +11.8 + − − + +9.5 + + − + +10.7 + − + + +9.8 + + + + +

where there are (choose one) 4 / 16 / 32 factorsand 24 = (choose one) 4 / 16 / 32 responses.

However, an example of a half fractional factorial design is


9.8 + − − − −

10.2 + + − − −

10.4 + − + − −

8.5 + + + − −

11.1 + − − + −

10.7 + + − + −

10.7 + − + + −

9.9 + + + + −

where there are (choose one) 4 / 16 / 32 factorsbut only 24−1 = (choose one) 4 / 8 / 32 responses

An example of a another fractional factorial design is


9.8 + − − − −

10.2 + + − − −

10.4 + − + − −

8.5 + + + − −


where there are (choose one) 4 / 16 / 32 factorsbut only 24−2 = (choose one) 4 / 8 / 32 responsesand so this is an example of a 2−2 = half / quarter / eighth fractional design.

An example of a replicated quarter fractional factorial design is


9.8 + − − − −

10.2 + + − − −

10.4 + − + − −

8.5 + + + − −

8.8 + − − − −

9.2 + + − − −

10.6 + − + − −

8.8 + + + − −

where there are (choose one) 4 / 16 / 32 factorsbut only 24−2 = (choose one) 4 / 8 / 32 responsesand (choose one) + / 2 / 3 replications.

2. Confounding (Aliasing).Consider the following 24−2 fractional factorial design.


9.8 + − − − −

10.2 + + − − −

10.4 + − + − −

8.5 + + + − −

Notice that column vectors x3 and x4 are identical, x3 = x4. This means thatin the model of this design is given by

y = · · ·+ β3x3 + β4x4 + · · ·

= · · ·+ β3x3 + β4x3 + · · ·

= · · ·+ (β3 + β4)x3 + · · ·

and so we are not able to estimated factors 3 and 4 separately, but only theircombined effects. Factors 3 and 4 are said to be confounded (or aliased). Forthe following 24−1 fractional factorial design,



9.8 + − − − −

10.2 + + − − −

10.4 + − + − −

8.5 + + + − −

11.1 + − − + −

10.7 + + − + −

10.7 + − + + −

9.9 + + + + −

For this design, (choose one)

(a) factor 1 and factor 4 are confounded, x1 = x4.

(b) factor 1 and factor 2 are confounded, x1 = x2.

(c) factor 2 and factor 3 are aliased, x2 = x3.

(d) factor 3 and factor 4 are confounded, 3 = 4 .

(e) none of the main factors, 1, 2, 3 or 4 are confounded with one another(although this does not mean some of the main factors are confoundedwith some of the interaction factors).

3. Notation of confounding.Consider the following design,


9.8 + − − − −

10.2 + + − − −

10.4 + − + − −

8.5 + + + − −

11.1 + − − + −

10.7 + + − + −

10.7 + − + + −

9.9 + + + + −


(a) True / FalseSince x0 = 1 and so multiplying each term in the x0 vector by itself

2,

x0 × x0 = x20 =

11111111

×

11111111

=

11111111

= x0

we can state this using the following more succinct notation, as

I × I = I 2 = I

(b) True / FalseSince multiplying x0 by any other column x2 (say) gives x2, in other words,

x0 × x2 =

11111111

×

−1−111−1−111

=

−1−111−1−111

= x2

we can state this using the following more succinct notation, asI × B = B, or, in general, for any factor q

I × q = q

(c) True / FalseSince multiplying a column x2 (say) by itself, gives a column of 1s, in otherwords,

x2 × x2 =

−1−111−1−111

×

−1−111−1−111

=

11111111

= x0

2Notice that the multiplication does not involve transposing one of the vectors.


we can state this using the following more succinct notation, asB × B = I, or, in general, for any factor q

q × q = I

(d) True / FalseSince multiplying one column x1 (say) by another column (not x0 and notitself) x2 (say), gives a new column of numbers, which is neither x0, x1 orx2

x1 × x2 =

−11−11−11−11

×

−1−111−1−111

=

1−1−111−1−11

6= x1, or x2

we can state this using the following more succinct notation, asA × B = AB, or, in general, for any two factors q and r

q × r = qr

4. More confoundingI × I = (choose one) I / A / BI × C = (choose one) I / A / CC × C = (choose one) I / A / CC × D = (choose one) I / C / CDC × C × C = (choose one) I / A / CC × C × D = (choose two) I / ID / DC × D × C = (choose two) I / ID / DCDC = (choose two) I / ID / DC2D = (choose two) I / ID / DC3D = (choose two) I / CD / DABC3D = (choose one) ABC / ABD / ABCD

5. More confounding.Match the confounding in the two columns.

Column I Column II

(a) B × ABCD (a) BC(b) AB × AC (b) ACD(c) I × C (c) ABC(d) CD × ABD (d) C


Column I (a) (b) (c) (d)Column II

6. Confounding scheme.Consider the following 24−2 fractional factorial design.

x0 x1 x2 x3 x4 x12 x13 x14 x23 x24 x34 x123 x124 x134 x234 x1234

+ − − − − + + + + + + − − − − +

+ + − − − − − − + + + + + + − −

+ − + − − − + + − − + − − − + −

+ + + − − + − − − − + − − + + +

For this design, (choose none, one or more)

(a) I = CD

(b) A = ACD

(c) B = BCD

(d) C = D

(e) AB = ABCD

(f) AC = AD

(g) BC = BD

(h) ABC = ABD

In fact, this is the entire confounding scheme and relation I = CD is said to bethe defining relation.

7. Defining relation generates confounding scheme.True / False The entire confounding scheme can be found from the definingrelation. For example, for the 24−2 fractional factorial design with definingrelation I = CD ,

I × A = CD × A = ACD

and all of the rest of the confounding scheme can be found in a similar way.

8. More defining relation generates confounding scheme.The entire confounding scheme for the 24−2 fractional factorial design with defin-ing relation I = AB , is given by (choose none, one or more)

(a) I = AB

(b) A = B

(c) C = ABC

(d) D = ABD


(e) AC = BC

(f) AD = BD

(g) CD = ABCD

(h) ACD = BCD

9. More defining relation generates confounding scheme.The entire confounding scheme for the 24−1 fractional factorial design with defin-ing relation I = ABC , is given by (choose none, one or more)

(a) I = ABC

(b) A = BC

(c) B = AC

(d) C = AB

(e) D = ABCD

(f) AD = BCD

(g) BD = ACD

(h) CD = ABD

10. Good confounding schemesIt is better that the main effects are confounded with high order interactioneffects rather than either other main effects or low order interaction effects be-cause it is often the case that the interaction effects, particularly high orderinteractions, are insignificant. So, although a main effect and an (assumedinsignificant) interaction effect are confounded, we would be inclined to be-lieve that the main effect, rather than the interaction effect, in the confoundingrelationship is the contributing significant factor. Consequently, the best con-founding scheme of the three given above, is (choose one)

(a) I = AB

(b) I = ABC

(c) I = CD

because in this case the main effects are confounded with, at worst, secondorder confounding interactions, whereas in the other two confounding schemes,at least one main effect is confounded with another main effect.

11. Defining Relation and Resolution.The number of factors in the lowest order effect in a defining relation is thecalled the resolution. For example, defining relation I = AB has resolution II.In a similar way, defining relation I = ABC has resolution


(choose one) I / II / III / IVIn general, the higher the resolution, the better the design; higher resolutiondesigns have confounding schemes where the main effects are confounded withhigher order interactions.

12. More resolution.From SAS, the resolution of a

25−1 design, since I = ABCD , for example, is(choose one) I / II / III / IV / V

25−2 design, since lowest of I = ADE and I = ABCD , for example, is(choose one) I / II / III / IV / V

26−3 design is, from table 12-23, page 631,(choose one) I / II / III / IV / V

29−3 design is, from table 12-23, page 631,(choose one) I / II / III / IV / V

part iv process design and improvement with designed

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