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Design and Analysis of Multi-Factored Experiments

- Two-level Factorial Designs

The 2k Factorial Design

- Special case of the general factorial design k

factors, all at two levels - The two levels are usually called low and high

(they could be either quantitative or

qualitative) - Very widely used in industrial experimentation
- Form a basic building block for other very

useful experimental designs (DNA) - Special (short-cut) methods for analysis
- We will make use of Design-Expert for analysis

Chemical Process Example

A reactant concentration, B catalyst amount,

y recovery

The Simplest Case The 22

- and denote the low and high levels of a

factor, respectively Low and high are arbitrary

terms Geometrically, the four runs form the

corners of a square Factors can be quantitative

or qualitative, although their treatment in the

final model will be different

Estimating effects in two-factor two-level

experiments

- Estimate of the effect of A
- a1b1 - a0b1 estimate of effect of A at high B
- a1b0 - a0b0 estimate of effect of A at low B
- sum/2 estimate of effect of A over all B
- Or average of high As average of low As.
- Estimate of the effect of B
- a1b1 - a1b0 estimate of effect of B at high A
- a0b1 - a0b0 estimate of effect of B at high A
- sum/2 estimate of effect of B over all A
- Or average of high Bs average of low Bs

Estimating effects in two-factor two-level

experiments

- Estimate the interaction of A and B
- a1b1 - a0b1 estimate of effect of A at high B
- a1b0 - a0b0 estimate of effect of A at low B
- difference/2 estimate of effect of B on the

effect of A - called as the interaction of A and B
- a1b1 - a1b0 estimate of effect of B at high A
- a0b1 - a0b0 estimate of effect of B at low A
- difference/2 estimate of the effect of A on the

effect of B - Called the interaction of B and A
- Or average of like signs average of unlike

signs

Estimating effects, contd...

- Note that the two differences in the interaction

estimate are - identical by definition, the interaction of A

and B is the - same as the interaction of B and A. In a given

experiment one - of the two literary statements of interaction may

be preferred - by the experimenter to the other but both have

the same - numerical value.

Remarks on effects and estimates

- Note the use of all four yields in the estimates

of the effect of - A, the effect of B, and the effect of the

interaction of A and - B all four yields are needed and are used in

each estimates. - Note also that the effect of each of the factors

and their - interaction can be and are assessed separately,

this in an - experiment in which both factors vary

simultaneously. - Note that with respect to the two factors

studied, the factors - themselves together with their interaction are,

logically, all - that can be studied. These are among the merits

of these - factorial designs.

Remarks on interaction

- Many scientists feel the need for experiments

which will - reveal the effect, on the variable under study,

of factors - acting jointly. This is what we have called

interaction. The - simple experimental design discussed here

evidently - provides a way of estimating such interaction,

with the latter - defined in a way which corresponds to what many

scientists - have in mind when they think of interaction.
- It is useful to note that interaction was not

invented by - statisticians. It is a joint effect existing,

often prominently, in - the real world. Statisticians have merely

provided ways and - means to measure it.

Symbolism and language

- A is called a main effect. Our estimate of A is

often simply written A. - B is called a main effect. Our estimate of B is

often simply written B. - AB is called an interaction effect. Our estimate

of AB is often simply written AB. - So the same letter is used, generally without

confusion, to describe the factor, to describe

its effect, and to describe our estimate of its

effect. Keep in mind that it is only for economy

in writing that we sometimes speak of an effect

rather than an estimate of the effect. We should

always remember that all quantities formed from

the yields are merely estimates.

Table of signs

- The following table is useful
- Notice that in estimating A, the two treatments

with A at high level are compared to the two

treatments with A at low level. Similarly B. This

is, of course, logical. Note that the signs of

treatments in the estimate of AB are the products

of the signs of A and B. Note that in each

estimate, plus and minus signs are equal in number

Example 2

B

Example 1

B

Example 2

Low

High

A

B

Low

High

A2.5

B2

B-

A-

Low

Low

A

A

B

A

High

High

B

A3

B-

B

Example 4

B

Example 4

Example 3

Low

High

Low

High

A

B-, B

Low

Example 3

Low

A

15

A

14

13

A

12

High

High

11

Y

10

9

-2

-1

0

1

Discussion of examples Notice that in examples 2

3 interaction is as large as or larger than

main effects.

A -(1) - b a ab/2

-10 - 12 13 15/2 3

- Change of scale, by multiplying each yield by a
- constant, multiplies each estimate by the

constant but does not affect the relationship of

estimates to each other. - Addition of a constant to each yield does not

affect the estimates. - The numerical magnitude of estimates is not

important here it is their relationship to each

other.

Modern notation and Yates order

- Modern notation
- a0b0 1 a0b1 b a1b0 a a1b1 ab
- We also introduce Yates (standard) order of

treatments and yields - each letter in turn followed by all combinations

of that letter and - letters already introduced. This will be the

preferred order for the - purpose of analysis of the yields. It is not

necessarily the order in - which the experiment is conducted that will be

discussed later. - For a two-factor two-level factorial design,

Yates order is - 1 a b ab
- Using modern notation and Yates order, the

estimates of effects - become
- A (-1 a - b ab)/2
- B (-1 - a b ab)/2
- AB (1 -a - b ab)/2

Three factors each at two levels

- Example The variable is the yield of a nitration

process. The yield forms the base material for

certain dye stuffs and medicines. - Low high
- A time of addition of nitric acid 2 hours 7

hours - B stirring time 1/2 hour 4 hours
- C heel absent present
- Treatments (also yields) (i) old notation (ii)

new notation. - (i) a0b0c0 a0b0c1 a0b1c0 a0b1c1 a1b0c0

a1b0c1 a1b1c0 a1b1c1 - (ii) 1 c b bc

a ac ab abc - Yates order
- 1 a b ab c

ac bc abc

Effects in The 23 Factorial Design

Estimating effects in three-factor two-level

designs (23)

- Estimate of A
- (1) a - 1 estimate of A, with B low and C low
- (2) ab - b estimate of A, with B high and C low
- (3) ac - c estimate of A, with B low and C high
- (4) abc - bc estimate of A, with B high and C

high - (aabacabc - 1-b-c-bc)/4,
- (-1a-bab-cac-bcabc)/4
- (in Yates order)

Estimate of AB

Effect of A with B high - effect of A with B low,

all at C high plus effect of A with B high -

effect of A with B low, all at C low

- Note that interactions are averages. Just as our

estimate of A is an average of response to A over

all B and all C, so our estimate of AB is an

average response to AB over all C. - AB (4)-(3) (2) - (1)/4
- 1-a-babc-ac-bcabc)/4, in Yates order
- or, (abcabc1) - (abacbc)/4

Estimate of ABC

- interaction of A and B, at C high
- minus
- interaction of A and B at C low
- ABC (4) - (3) - (2) - (1)/4
- (-1ab-abc-ac-bcabc)/4, in Yates order
- or, abcabc - (1abacbc)/4

- This is our first encounter with a three-factor

interaction. It - measures the impact, on the yield of the

nitration process, of - interaction AB when C (heel) goes from C absent

to C - present. Or it measures the impact on yield of

interaction AC - when B (stirring time) goes from 1/2 hour to 4

hours. Or - finally, it measures the impact on yield of

interaction BC - when A (time of addition of nitric acid) goes

from 2 hours to - 7 hours.
- As with two-factor two-level factorial designs,

the formation - of estimates in three-factor two-level factorial

designs can be - summarized in a table.

- Sign Table for a 23 design

Example

Yield of nitration process discussed earlier

1 a b ab c ac

bc abc Y 7.2 8.4

2.0 3.0 6.7 9.2 3.4 3.7

- A main effect of nitric acid time 1.25
- B main effect of stirring time -4.85
- AB interaction of A and B -0.60
- C main effect of heel 0.60
- AC interaction of A and C 0.15
- BC interaction of B and C 0.45
- ABC interaction of A, B, and C -0.50

NOTE ac largest yield AC smallest effect

- We describe several of these estimates, though on

later - analysis of this example, taking into account the

unreliability - of estimates based on a small number (eight) of

yields, some - estimates may turn out to be so small in

magnitude as not to - contradict the conjecture that the corresponding

true effect is - zero. The largest estimate is -4.85, the estimate

of B an - increase in stirring time, from 1/2 to 4 hours,

is associated - with a decline in yield. The interaction AB

-0.6 an increase - in stirring time from 1/2 to 4 hours reduces the

effect of A, - whatever it is (A 1.25), on yield. Or

equivalently

- an increase in nitric acid time from 2 to 7 hours

reduces - (makes more negative) the already negative effect

(B -485) - of stirring time on yield. Finally, ABC -0.5.

Going from no - heel to heel, the negative interaction effect AB

on yield - becomes even more negative. Or going from low to

high - stirring time, the positive interaction effect

AC is reduced. - Or going from low to high nitric acid time, the

positive - interaction effect BC is reduced. All three

descriptions of - ABC have the same numerical value but the

chemist would - select one of them, then say it better.

Number and kinds of effects

- We introduce the notation 2k. This means a

factor design with each factor at two levels. The

number of treatments in an unreplicated 2k design

is 2k. - The following table shows the number of each

kind of effect for each of the six two-level

designs shown across the top.

Main effect

2 factor interaction 3 factor interaction 4

factor interaction 5 factor interaction 6 factor

interaction 7 factor interaction

3

7

15

31

63

127

In a 2k design, the number of r-factor effects is

Ckr k!/r!(k-r)!

- Notice that the total number of effects

estimated in any design is always one less than

the number of treatments

In a 22 design, there are 224 treatments we

estimate 22-1 3 effects. In a 23 design, there

are 238 treatments we estimate 23-1 7 effects

One need not repeat the earlier logic to

determine the forms of estimates in 2k designs

for higher values of k. A table going up to 25

follows.

25

E f f e c t s

24

23

22

Treatment s

Yates Forward Algorithm (1)

1. Applied to Complete Factorials (Yates, 1937)

- A systematic method of calculating estimates of

effects. - For complete factorials first arrange the yields

in Yates - (standard) order. Addition, then subtraction of

adjacent - yields. The addition and subtraction operations

are - repeated until 2k terms appear in each line for

a 2k there - will be k columns of calculations

Yates Forward Algorithm (2)

- Example
- Yield of a nitration process

Tr.

Yield

1stCol

2ndCol

3rdCol

1 7.2 15.6 20.6 43.6 Contrast of µ a 8.4

5.0 23.0 5.0 Contrast of A b 2.0 15.9

2.2 -19.4 Contrast of B ab 3.0 7.1 2.8

-2.4 Contrast of AB c 6.7 1.2 -10.6 2.4

Contrast of C ac 9.2 1.0 -8.8 0.6

Contrast of AC bc 3.4 2.5 -0.2 1.8

Contrast of BC abc 3.7 0.3 -2.2 -2.0

Contrast of ABC

Again, note the line-by-line correspondence

between treatments and estimates both are in

Yates order.

Main effects in the face of large interactions

- Several writers have cautioned against making

statements - about main effects when the corresponding

interactions - are large interactions describe the dependence

of the - impact of one factor on the level of another in

the - presence of large interaction, main effects may

not be - meaningful.

Example (Adapted from Kempthorne) Yields are in

bushels of potatoes per plot. The two factors are

nitrate (N) and phosphate (P) fertilizers.

- low level (-1) high level (1)
- N (A) blood sulphate of ammonia
- P (B) superphosphate steamed bone flower

The yields are 1 746.75 n 625.75 p

611.00 np 656.00 the estimates are N -38.00 P

-52.75 NP 83.00 In the face of such high

interaction we now specialize the main effect of

each factor to particular levels of the other

factor. Effect of N at high level P np-p

656.00-611.00 45.0 Effect of N at low level P

n-1 625.71-746.75 -121.0, which appear to be

more valuable for fertilizer policy than the mean

(-38.00) of such disparate numbers

746.75

P

-121

Keep both low is best

Y

656

-38

611.0

P-

625.75

N

- Note that answers to these specialized questions

are based on fewer than 2k yields. In our

numerical example, with interaction NP prominent,

we have only two of the four yields in our

estimate of N at each level of P. - In general we accept high interactions wherever

found and seek to explain them in the process of

explanation, main effects (and lower-order

interactions) may have to be replaced in our

interest by more meaningful specialized or

conditional effects.

Specialized or Conditional Effects

- Whenever there is large interactions, check
- Effect of A at high level of B A A AB
- Effect of A at low level of B A- A AB
- Effect of B at high level of A B B AB
- Effect of B at low level of A B- B - AB

Factors not studied

- In any experiment, factors other than those

studied may be influential. Their presence is

sometimes acknowledged under the dubious title

experimental error. They may be neglected, but

the usual cost of neglect is high. For they often

have uneven impact, systematically affecting some

treatments more than others, and thereby

seriously confounding inferences on the studied

factors. It is important to deal explicitly with

them even more, it is important to measure their

impact. How?

- 1. Hold them constant.
- 2. Randomize their effects.
- 3. Estimate their magnitude by replicating the

experiment. - 4. Estimate their magnitude via side or earlier

experiments. - 5. Argue (convincingly) that the effects of some

of these non-studied factors are zero, either in

advance of the experiment or in the light of the

yields. - 6. Confound certain non-studied factors.

Simplified Analysis Procedure for 2-level

Factorial Design

- Estimate factor effects
- Formulate model using important effects
- Check for goodness-of-fit of the model.
- Interpret results
- Use model for Prediction

Example Shooting baskets

- Consider an experiment with 3 factors A, B, and

C. Let the response variable be Y. For example,

- Y number of baskets made out of 10
- Factor A distance from basket (2m or 5m)
- Factor B direction of shot (0 or 90 )
- Factor C type of shot (set or jumper)

Factor Name Units

Low Level (-1) High Level (1)

A Distance m 2 5 B

Direction Deg. 0 90

C Shot type Set

Jump

Treatment Combinations and Results

Order A B C Combination Y 1 -1 -1 -1

(1) 9 2 1 -1 -1 a 5 3 -1 1 -1

b 7 4 1 1 -1 ab 3 5 -1 -1 1

c 6 6 1 -1 1 ac 5 7 -1 1 1

bc 4 8 1 1 1 abc 2

Estimating Effects

Order A B AB C AC BC ABC Comb Y 1 -1 -1 1 -1

1 1 -1 (1) 9 2 1 -1 -1 -1 -1 1 1

a 5 3 -1 1 -1 -1 1 -1 1 b 7

4 1 1 1 -1 -1 -1 -1 ab 3 5 -1 -1 1 1

-1 -1 1 c 6 6 1 -1 -1 1 1 -1 -1 ac 5

7 -1 1 -1 1 -1 1 -1 bc 4 8 1 1 1 1

1 1 1 abc 2

Effect A (a ab ac abc)/4 - (1 b c

bc)/4 (5 3 5 2)/4 - (9 7

6 4)/4 -2.75

Effects and Overall Average

Using the sign table, all 7 effects can be

calculated Effect A -2.75 ? Effect B -2.25

? Effect C -1.75 ? Effect AC 1.25 ? Effect AB

-0.25 Effect BC -0.25 Effect ABC -0.25 The

overall average value (9 5 7 3 6 5

4 2)/8

5.13

Formulate Model

The most important effects are A, B, C, and AC

Model Y b0 b1 X1 b2 X2 b3 X3 b13

X1X3 b0 overall average 5.13 b1 Effect

A/2 -2.75/2 -1.375 b2 Effect B/2

-2.25/2 -1.125 b3 Effect C/2 -1.75/2 -

0.875 b13 Effect AC/2 1.25/2 0.625 Model

in coded units Y 5.13 -1.375 X1 - 1.125 X2 -

0.875 X3 0.625 X1 X3

Checking for goodness-of-fit

Actual Predicted Value

Value 9.00 9.13 5.00 5.13 7.00

6.88 3.00 2.87 6.00 6.13 5.00

4.63 4.00 3.88 2.00 2.37

Amazing fit!!

Interpreting Results

10

Effect of B4-6.25 -2.25

(9565)/46.25

out of 10

8

6

(7342)/44

4

2

B

0

90

C Shot type

10

Interaction of A and C 1.25

8

out of 10

C(-1)

6

At 5m, Jump or set shot about the same BUT at 2m,

set shot gave higher values compared to jump shots

4

2

C (1)

A

2m

5m

Design and Analysis of Multi-Factored Experiments

- Analysis of 2k Experiments
- Statistical Details

Errors of estimates in 2k designs

- 1. Meaning of ?2
- Assume that each treatment has variance ?2. This

has the following meaning consider any one

treatment and imagine many replicates of it. As

all factors under study are constant throughout

these repetitions, the only sources of any

variability in yield are the factors not under

study. Any variability in yield is due to them

and is measured by ?2.

Errors of estimates in 2k designs, Contd..

- 2. Effect of the number of factors on the error

of an estimate - What is the variance of an estimate of an

effect? In a 2k design, 2k treatments go into

each estimate the signs of the treatments are

or -, depending on the effect being estimated. - So, any estimate 1/2k-1generalized ( or -)

sum of 2k treatments - ?2(any estimate) 1/22k-2 2k ?2 ?2/2k-2
- The larger the number of factors, the smaller

the error of each estimate.

Note ?2(kx) k2 ?2(x)

Errors of estimates in 2k designs, Contd..

- 3. Effect of replication on the error of an

estimate - What is the effect of replication on the error

of an estimate? Consider a 2k design with each

treatment replicated n times.

1

a

b

abc

d

- - - -

- - - -

- - - -

- - - -

- - - -

---

---

Errors of estimates in 2k designs, Contd..

- Any estimate 1/2k-1 sums of 2k terms, all of

them means based on samples of size n - ?2(any estimate) 1/22k-2 2k ?2/n

?2/(n2k-2) - The larger the replication per treatment, the

smaller the error of each estimate.

- So, the error of an estimate depends on k (the

number of factors studied) and n (the replication

per factor). It also (obviously) depends on ?2.

The variance ?2 can be reduced holding some of

the non-studied factors constant. But, as has

been noted, this gain is offset by reduced

generality of any conclusions.

Effects, Sum of Squares and Regression

Coefficients

Judging Significance of Effects

- a) p- values from ANOVA
- Compute p-value of calculated F. IF p lt ?, then

effect is significant. - b) Comparing std. error of effect to size of

effect

- Hence
- If effect 2 (se), contains zero, then that

effect is not significant. These intervals are

approximately the 95 CI. - e.g. 3.375 1.56 (significant)
- 1.125 1.56 (not significant)

- c) Normal probability plot of effects
- Significant effects are those that do not fit on

normal probability plot. i. e. non-significant

effects will lie along the line of a normal

probability plot of the effects. - Good visual tool - available in Design-Expert

software.

Design and Analysis of Multi-Factored Experiments

- Examples of Computer Analysis

Analysis Procedure for a Factorial Design

- Estimate factor effects
- Formulate model
- With replication, use full model
- With an unreplicated design, use normal

probability plots - Statistical testing (ANOVA)
- Refine the model
- Analyze residuals (graphical)
- Interpret results

Chemical Process Example

A reactant concentration, B catalyst amount,

y recovery

Estimation of Factor Effects

A (a ab - 1 - b)/2n (100 90 - 60 -

80)/(2 x 3) 8.33 B (b ab - 1 - a)/2n

-5.00 C (ab 1 - a - b)/2n 1.67

The effect estimates are A 8.33, B

-5.00, AB 1.67 Design-Expert analysis

Estimation of Factor Effects Form Tentative Model

Term Effect SumSqr

Contribution Model Intercept Model A

8.33333 208.333 64.4995 Model B

-5 75 23.2198 Model

AB 1.66667 8.33333

2.57998 Error Lack Of Fit 0

0 Error P Error 31.3333

9.70072 Lenth's ME 6.15809 Lenth's

SME 7.95671

Statistical Testing - ANOVA

Response Conversion ANOVA for Selected

Factorial Model Analysis of variance table

Partial sum of squares Sum of Mean F Source

Squares DF Square Value Prob gt

F Model 291.67 3 97.22 24.82 0.0002 A 208.33 1 2

08.33 53.19 lt 0.0001 B 75.00 1 75.00 19.15 0.0024

AB 8.33 1 8.33 2.13 0.1828 Pure

Error 31.33 8 3.92 Cor Total 323.00 11 Std.

Dev. 1.98 R-Squared

0.9030 Mean 27.50 Adj R-Squared 0.8666 C.V. 7.2

0 Pred R-Squared 0.7817 PRESS 70.50 Adeq

Precision 11.669 The F-test for the model

source is testing the significance of the overall

model that is, is either A, B, or AB or some

combination of these effects important?

Statistical Testing - ANOVA

Coefficient Standard 95 CI 95

CI Factor Estimate DF Error Low High VIF

Intercept 27.50

1 0.57 26.18 28.82 A-Concent 4.17

1 0.57 2.85 5.48 1.00 B-Catalyst -2.50

1 0.57 -3.82 -1.18 1.00 AB

0.83 1 0.57

-0.48 2.15 1.00 General formulas for the

standard errors of the model coefficients and the

confidence intervals are available. They will be

given later.

Refine Model

Response Conversion ANOVA for Selected

Factorial Model Analysis of variance table

Partial sum of squares Sum of Mean F Source

Squares DF Square Value Prob gt

F Model 283.33 2 141.67 32.14 lt

0.0001 A 208.33 1 208.33 47.27 lt

0.0001 B 75.00 1 75.00 17.02 0.0026 Residual 39.

67 9 4.41 Lack of Fit 8.33 1 8.33 2.13 0.1828 Pu

re Error 31.33 8 3.92 Cor Total 323.00 11 Std.

Dev. 2.10 R-Squared 0.8772 Mean 27.50 Adj

R-Squared 0.8499 C.V. 7.63 Pred

R-Squared 0.7817 PRESS 70.52 Adeq

Precision 12.702 There is now a residual sum of

squares, partitioned into a lack of fit

component (the AB interaction) and a pure error

component

Regression Model for the Process

Residuals and Diagnostic Checking

The Response Surface

An Example of a 23 Factorial Design

A carbonation, B pressure, C speed, y

fill deviation

Estimation of Factor Effects

Term Effect SumSqr Contribution Model

Intercept Error A 3 36 46.1538 Error

B 2.25 20.25 25.9615 Error C 1.75 12.25 15.7051

Error AB 0.75 2.25 2.88462 Error

AC 0.25 0.25 0.320513 Error BC 0.5 1 1.28205 Er

ror ABC 0.5 1 1.28205 Error LOF 0 Error P

Error 5 6.41026 Lenth's

ME 1.25382 Lenth's SME 1.88156

ANOVA Summary Full Model

Response Fill-deviation ANOVA for

Selected Factorial Model Analysis of variance

table Partial sum of squares Sum

of Mean F Source Squares DF Square Value Prob

gt F Model 73.00 7 10.43 16.69 0.0003 A 36.00 1 3

6.00 57.60 lt 0.0001 B 20.25 1 20.25 32.40 0.0005

C 12.25 1 12.25 19.60 0.0022 AB 2.25 1 2.25 3.60

0.0943 AC 0.25 1 0.25 0.40 0.5447 BC 1.00 1 1.0

0 1.60 0.2415 ABC 1.00 1 1.00 1.60 0.2415 Pure

Error 5.00 8 0.63 Cor Total 78.00 15 Std.

Dev. 0.79 R-Squared 0.9359 Mean 1.00 Adj

R-Squared 0.8798 C.V. 79.06 Pred

R-Squared 0.7436 PRESS 20.00 Adeq

Precision 13.416

Model Coefficients Full Model

Coefficient

Standard 95 CI 95 CI Factor

Estimate DF Error Low High VIF Intercept

1.00 1 0.20 0.54 1.46 A-Carbonation

1.50 1 0.20 1.04 1.96 1.00

B-Pressure 1.13 1 0.20 0.67 1.58

1.00 C-Speed

0.88 1 0.20 0.42 1.33 1.00 AB

0.38 1 0.20 -0.081 0.83

1.00 AC 0.13 1 0.20

-0.33 0.58 1.00 BC

0.25 1 0.20 -0.21 0.71 1.00 ABC

0.25 1 0.20 -0.21 0.71 1.00

Refine Model Remove Nonsignificant Factors

Response Fill-deviation ANOVA for

Selected Factorial Model Analysis of variance

table Partial sum of squares Sum

of Mean F Source Squares DF Square Value Prob

gt F Model 70.75 4 17.69 26.84 lt

0.0001 A 36.00 1 36.00 54.62 lt

0.0001 B 20.25 1 20.25 30.72 0.0002 C 12.25 1 12

.25 18.59 0.0012 AB 2.25 1 2.25 3.41 0.0917 Resi

dual 7.25 11 0.66 LOF 2.25 3 0.75 1.20 0.3700 Pu

re E 5.00 8 0.63 C Total 78.00 15 Std.

Dev. 0.81 R-Squared 0.9071 Mean 1.00 Adj

R-Squared 0.8733 C.V. 81.18 Pred

R-Squared 0.8033 PRESS 15.34 Adeq

Precision 15.424

Model Coefficients Reduced Model

Coefficient Standard

95 CI 95 CI Factor Estimate DF Error L

ow High Intercept 1.00 1 0.20 0.55 1.45

A-Carbonation 1.50 1 0.20 1.05 1.95

B-Pressure 1.13 1 0.20 0.68 1.57

C-Speed 0.88 1 0.20 0.43 1.32 AB

0.38 1 0.20 -0.072 0.82

Model Summary Statistics

- R2 and adjusted R2
- R2 for prediction (based on PRESS)

Model Summary Statistics

- Standard error of model coefficients
- Confidence interval on model coefficients

The Regression Model

Final Equation in Terms of Coded Factors

Fill-deviation 1.00 1.50 A 1.13

B 0.88 C 0.38 A B Final

Equation in Terms of Actual Factors

Fill-deviation 9.62500 -2.62500

Carbonation -1.20000 Pressure 0.035000

Speed 0.15000 Carbonation Pressure

Residual Plots are Satisfactory

Model Interpretation

Moderate interaction between carbonation level

and pressure

Model Interpretation

Cube plots are often useful visual displays of

experimental results

Contour Response Surface Plots Speed at the

High Level

Design and Analysis of Multi-Factored Experiments

- Unreplicated Factorials

Unreplicated 2k Factorial Designs

- These are 2k factorial designs with one

observation at each corner of the cube - An unreplicated 2k factorial design is also

sometimes called a single replicate of the 2k - These designs are very widely used
- Risksif there is only one observation at each

corner, is there a chance of unusual response

observations spoiling the results? - Modeling noise?

Spacing of Factor Levels in the Unreplicated 2k

Factorial Designs

If the factors are spaced too closely, it

increases the chances that the noise will

overwhelm the signal in the data More aggressive

spacing is usually best

Unreplicated 2k Factorial Designs

- Lack of replication causes potential problems in

statistical testing - Replication admits an estimate of pure error (a

better phrase is an internal estimate of error) - With no replication, fitting the full model

results in zero degrees of freedom for error - Potential solutions to this problem
- Pooling high-order interactions to estimate error
- Normal probability plotting of effects (Daniels,

1959)

Example of an Unreplicated 2k Design

- A 24 factorial was used to investigate the

effects of four factors on the filtration rate of

a resin - The factors are A temperature, B pressure, C

mole ratio, D stirring rate - Experiment was performed in a pilot plant

The Resin Plant Experiment

The Resin Plant Experiment

Estimates of the Effects

Term Effect SumSqr Contribution Model

Intercept Error A 21.625 1870.56 32.6397 Er

ror B 3.125 39.0625 0.681608 Error

C 9.875 390.062 6.80626 Error

D 14.625 855.563 14.9288 Error

AB 0.125 0.0625 0.00109057 Error

AC -18.125 1314.06 22.9293 Error

AD 16.625 1105.56 19.2911 Error

BC 2.375 22.5625 0.393696 Error

BD -0.375 0.5625 0.00981515 Error

CD -1.125 5.0625 0.0883363 Error

ABC 1.875 14.0625 0.245379 Error

ABD 4.125 68.0625 1.18763 Error

ACD -1.625 10.5625 0.184307 Error

BCD -2.625 27.5625 0.480942 Error

ABCD 1.375 7.5625 0.131959 Lenth's

ME 6.74778 Lenth's SME 13.699

The Normal Probability Plot of Effects

The Half-Normal Probability Plot

ANOVA Summary for the Model

Response Filtration Rate ANOVA for

Selected Factorial Model Analysis of variance

table Partial sum of squares Sum

of Mean F Source Squares DF Square Value Prob

gtF Model 5535.81 5 1107.16 56.74 lt

0.0001 A 1870.56 1 1870.56 95.86 lt

0.0001 C 390.06 1 390.06 19.99 0.0012 D 855.56 1

855.56 43.85 lt 0.0001 AC 1314.06 1 1314.06 67.34

lt 0.0001 AD 1105.56 1 1105.56 56.66 lt

0.0001 Residual 195.12 10 19.51 Cor

Total 5730.94 15 Std. Dev. 4.42 R-Squared 0.966

0 Mean 70.06 Adj R-Squared 0.9489 C.V. 6.30 Pr

ed R-Squared 0.9128 PRESS 499.52 Adeq

Precision 20.841

The Regression Model

Final Equation in Terms of Coded Factors

Filtration Rate 70.06250 10.81250

Temperature 4.93750 Concentration 7.3125

0 Stirring Rate -9.06250 Temperature

Concentration 8.31250 Temperature

Stirring Rate

Model Residuals are Satisfactory

Model Interpretation Interactions

Model Interpretation Cube Plot

If one factor is dropped, the unreplicated 24

design will project into two replicates of a

23 Design projection is an extremely useful

property, carrying over into fractional factorials

Model Interpretation Response Surface Plots

With concentration at either the low or high

level, high temperature and high stirring rate

results in high filtration rates

The Drilling Experiment

A drill load, B flow, C speed, D type of

mud, y advance rate of the drill

Effect Estimates - The Drilling Experiment

Term Effect SumSqr Contribution Model

Intercept Error A 0.9175 3.36722 1.28072 Er

ror B 6.4375 165.766 63.0489 Error

C 3.2925 43.3622 16.4928 Error

D 2.29 20.9764 7.97837 Error AB 0.59 1.3924 0.52

9599 Error AC 0.155 0.0961 0.0365516 Error

AD 0.8375 2.80563 1.06712 Error

BC 1.51 9.1204 3.46894 Error BD 1.5925 10.1442 3

.85835 Error CD 0.4475 0.801025 0.30467 Error

ABC 0.1625 0.105625 0.0401744 Error

ABD 0.76 2.3104 0.87876 Error

ACD 0.585 1.3689 0.520661 Error

BCD 0.175 0.1225 0.0465928 Error

ABCD 0.5425 1.17722 0.447757 Lenth's

ME 2.27496 Lenth's SME 4.61851

Half-Normal Probability Plot of Effects

Residual Plots

Residual Plots

- The residual plots indicate that there are

problems with the equality of variance assumption - The usual approach to this problem is to employ a

transformation on the response - Power family transformations are widely used
- Transformations are typically performed to
- Stabilize variance
- Induce normality
- Simplify the model

Selecting a Transformation

- Empirical selection of lambda
- Prior (theoretical) knowledge or experience can

often suggest the form of a transformation - Analytical selection of lambdathe Box-Cox (1964)

method (simultaneously estimates the model

parameters and the transformation parameter

lambda) - Box-Cox method implemented in Design-Expert

The Box-Cox Method

A log transformation is recommended The procedure

provides a confidence interval on the

transformation parameter lambda If unity is

included in the confidence interval, no

transformation would be needed

Effect Estimates Following the Log Transformation

Three main effects are large No indication of

large interaction effects What happened to the

interactions?

ANOVA Following the Log Transformation

Response adv._rate Transform Natural

log Constant 0.000 ANOVA

for Selected Factorial Model Analysis of

variance table Partial sum of squares Sum

of Mean F Source Squares DF Square Value Prob

gt F Model 7.11 3 2.37 164.82 lt

0.0001 B 5.35 1 5.35 371.49 lt 0.0001 C 1.34 1 1.

34 93.05 lt 0.0001 D 0.43 1 0.43 29.92 0.0001 Res

idual 0.17 12 0.014 Cor Total 7.29 15 Std.

Dev. 0.12 R-Squared

0.9763 Mean 1.60 Adj R-Squared 0.9704 C.V. 7.51

Pred R-Squared 0.9579 PRESS 0.31 Adeq

Precision 34.391

Following the Log Transformation

Final Equation in Terms of Coded Factors

Ln(adv._rate) 1.60 0.58 B 0.29

C 0.16 D

Following the Log Transformation

The Log Advance Rate Model

- Is the log model better?
- We would generally prefer a simpler model in a

transformed scale to a more complicated model in

the original metric - What happened to the interactions?
- Sometimes transformations provide insight into

the underlying mechanism

Other Analysis Methods for Unreplicated 2k

Designs

- Lenths method
- Analytical method for testing effects, uses an

estimate of error formed by pooling small

contrasts - Some adjustment to the critical values in the

original method can be helpful - Probably most useful as a supplement to the

normal probability plot

Design and Analysis of Multi-Factored Experiments

- Center points

Addition of Center Points to a 2k Designs

- Based on the idea of replicating some of the runs

in a factorial design - Runs at the center provide an estimate of error

and allow the experimenter to distinguish

between two possible models

The hypotheses are

This sum of squares has a single degree of freedom

Example

Usually between 3 and 6 center points will work

well Design-Expert provides the analysis,

including the F-test for pure quadratic curvature

ANOVA for Example

Response yield ANOVA for Selected

Factorial Model Analysis of variance table

Partial sum of squares Sum of Mean F Source

Squares DF Square Value Prob gt

F Model 2.83 3 0.94 21.92 0.0060 A 2.40 1 2.40 5

5.87 0.0017 B 0.42 1 0.42 9.83 0.0350 AB 2.500E-

003 1 2.500E-003 0.058 0.8213 Curvature 2.722E-00

3 1 2.722E-003 0.063 0.8137 Pure

Error 0.17 4 0.043 Cor Total 3.00 8 Std.

Dev. 0.21 R-Squared

0.9427 Mean 40.44 Adj R-Squared 0.8996 C.V. 0.5

1 Pred R-Squared N/A PRESS N/A Adeq

Precision 14.234

If curvature is significant, augment the design

with axial runs to create a central composite

design. The CCD is a very effective design for

fitting a second-order response surface model

Practical Use of Center Points

- Use current operating conditions as the center

point - Check for abnormal conditions during the time

the experiment was conducted - Check for time trends
- Use center points as the first few runs when

there is little or no information available about

the magnitude of error - Can have only 1 center point for computer

experiments hence requires a different type of

design