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Design and Analysis of Multi-Factored Experiments. Two-level Factorial Designs. The 2 k 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)

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design and analysis of multi factored experiments

Design and Analysis ofMulti-Factored Experiments

Two-level Factorial Designs

DOE Course

the 2 k factorial design
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

DOE Course

chemical process example
Chemical Process Example

A = reactant concentration, B = catalyst amount, y = recovery

DOE Course

the simplest case the 2 2
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

DOE Course

estimating effects in two factor two level experiments
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

DOE Course

estimating effects in two factor two level experiments1
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

DOE Course

estimating effects contd
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.

DOE Course

remarks on effects and estimates
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.

DOE Course

remarks on interaction
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.

DOE Course

symbolism and language
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.

DOE Course

table of signs
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

DOE Course

slide12

B+

B-

A

Example 2

Example 1

B

B

Example 2

A+

Low

High

B+

Low

High

A=2.5

B=2

B-

A-

Low

Low

A

A

B

A

High

High

B+

A=3

B-

B

B

Example 4

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

DOE Course

slide13
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.

DOE Course

modern notation and yates order
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

DOE Course

three factors each at two levels
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.

Lowhigh

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

DOE Course

estimating effects in three factor two level designs 2 3
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

= (a+ab+ac+abc - 1-b-c-bc)/4,

= (-1+a-b+ab-c+ac-bc+abc)/4

(in Yates’ order)

DOE Course

estimate of ab
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-b+ab+c-ac-bc+abc)/4, in Yates’ order

or, = [(abc+ab+c+1) - (a+b+ac+bc)]/4

DOE Course

slide19

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

=(-1+a+b-ab+c-ac-bc+abc)/4, in Yates’ order

or, =[abc+a+b+c - (1+ab+ac+bc)]/4

DOE Course

slide20
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.

DOE Course

example
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

DOE Course

slide23
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

DOE Course

slide24
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.

DOE Course

number and kinds of effects
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.

DOE Course

slide26

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)!]

DOE Course

slide27
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 22=4 treatments; we estimate 22-1= 3 effects. In a 23 design, there are 23=8 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.

DOE Course

slide28

E f f e c t s

25

24

23

22

Treatment s

DOE Course

yates forward algorithm 1
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

DOE Course

yates forward algorithm 2
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.

DOE Course

main effects in the face of large interactions
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.

DOE Course

slide32

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+

Keep both

low is best

-121

Y

656

-38

611.0

P-

625.75

N

DOE Course

slide33
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.

DOE Course

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

DOE Course

factors not studied
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?

DOE Course

slide36
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.

DOE Course

simplified analysis procedure for 2 level factorial design
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

DOE Course

example shooting baskets
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

DOE Course

treatment combinations and results
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

DOE Course

estimating effects
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

DOE Course

effects and overall average
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

DOE Course

formulate model
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

DOE Course

checking for goodness of fit
Checking for goodness-of-fit

ActualPredictedValueValue

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!!

DOE Course

interpreting results
Interpreting Results

10

Effect of B=4-6.25= -2.25

(9+5+6+5)/4=6.25

# out

of 10

8

6

(7+3+4+2)/4=4

4

2

0

90

B

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)

2m

5m

A

DOE Course

design and analysis of multi factored experiments1

Design and Analysis ofMulti-Factored Experiments

Analysis of 2k Experiments

Statistical Details

DOE Course

errors of estimates in 2 k designs
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.

DOE Course

errors of estimates in 2 k designs contd
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-1[generalized (+ 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)

DOE Course

errors of estimates in 2 k designs contd1
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

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

-

---

---

DOE Course

errors of estimates in 2 k designs contd2
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.

DOE Course

slide50
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.

DOE Course

judging significance of effects
Judging Significance of Effects

a) p- values from ANOVA

Compute p-value of calculated F. IF p < , then effect is significant.

b) Comparing std. error of effect to size of effect

DOE Course

slide53
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)

DOE Course

slide54
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.

DOE Course

design and analysis of multi factored experiments2

Design and Analysis of Multi-Factored Experiments

Examples of Computer Analysis

DOE Course

analysis procedure for a factorial design
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

DOE Course

chemical process example1
Chemical Process Example

A = reactant concentration, B = catalyst amount, y = recovery

DOE Course

estimation of factor effects
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-Expertanalysis

DOE Course

estimation of factor effects form tentative model
Estimation of Factor EffectsForm 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

DOE Course

statistical testing anova
Statistical Testing - ANOVA

Response:Conversion ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]

Sum ofMeanFSourceSquaresDFSquareValueProb > F Model 291.67 3 97.22 24.82 0.0002A208.331208.3353.19< 0.0001B75.00175.0019.150.0024AB8.3318.332.130.1828 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. 7.20 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?

DOE Course

statistical testing anova1
Statistical Testing - ANOVA

CoefficientStandard95% CI95% CI

FactorEstimateDFErrorLowHighVIF 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.

DOE Course

refine model
Refine Model

Response:Conversion ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]

Sum ofMeanFSourceSquaresDFSquareValueProb > F Model 283.33 2 141.67 32.14 < 0.0001A208.331208.3347.27< 0.0001B75.00175.0017.020.0026 Residual 39.67 9 4.41Lack of Fit8.3318.332.130.1828Pure Error31.3383.92 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

DOE Course

an example of a 2 3 factorial design
An Example of a 23 Factorial Design

A = carbonation, B = pressure, C = speed, y = fill deviation

DOE Course

estimation of factor effects1
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 Error 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

DOE Course

anova summary full model
ANOVA Summary – Full Model

Response:Fill-deviation ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]

Sum ofMeanFSourceSquaresDFSquareValueProb > F Model 73.00 7 10.43 16.69 0.0003A36.00136.0057.60< 0.0001B20.25120.2532.400.0005C12.25112.2519.600.0022AB2.2512.253.600.0943AC0.2510.250.400.5447BC1.0011.001.600.2415ABC1.0011.001.600.2415 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

DOE Course

model coefficients full model
Model Coefficients – Full Model

CoefficientStandard95% CI 95% CI

FactorEstimateDFErrorLowHighVIF

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

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refine model remove nonsignificant factors
Refine Model – Remove Nonsignificant Factors

Response:Fill-deviation ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]

Sum ofMeanFSourceSquaresDFSquareValueProb > F Model 70.75 4 17.69 26.84 < 0.0001A36.00136.0054.62< 0.0001B20.25120.2530.720.0002C12.25112.2518.590.0012AB2.2512.253.410.0917 Residual 7.25 11 0.66LOF2.2530.751.200.3700Pure E5.0080.63 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

DOE Course

model coefficients reduced model
Model Coefficients – Reduced Model

CoefficientStandard95% CI 95% CIFactorEstimateDFErrorLowHigh 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

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model summary statistics
Model Summary Statistics
  • R2 and adjusted R2
  • R2 for prediction (based on PRESS)

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model summary statistics1
Model Summary Statistics
  • Standarderror of model coefficients
  • Confidenceinterval on model coefficients

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the regression model
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

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model interpretation
Model Interpretation

Moderate interaction between carbonation level and pressure

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model interpretation1
Model Interpretation

Cubeplots are often useful visual displays of experimental results

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design and analysis of multi factored experiments3

Design and Analysis ofMulti-Factored Experiments

Unreplicated Factorials

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unreplicated 2 k factorial designs
Unreplicated 2kFactorial 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
  • Risks…if there is only one observation at each corner, is there a chance of unusual response observations spoiling the results?
  • Modeling “noise”?

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spacing of factor levels in the unreplicated 2 k factorial designs
Spacing of Factor Levels in the Unreplicated 2kFactorial 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

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unreplicated 2 k factorial designs1
Unreplicated 2kFactorial 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)

DOE Course

example of an unreplicated 2 k design
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

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estimates of the effects
Estimates of the Effects

Term Effect SumSqr % Contribution Model Intercept Error A 21.625 1870.56 32.6397 Error 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

DOE Course

anova summary for the model
ANOVA Summary for the Model

Response:Filtration Rate ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]

Sum ofMeanFSourceSquaresDFSquareValueProb >F Model 5535.81 5 1107.16 56.74 < 0.0001A1870.5611870.5695.86< 0.0001C390.061390.0619.990.0012D855.561855.5643.85< 0.0001AC1314.0611314.0667.34< 0.0001AD1105.5611105.5656.66< 0.0001 Residual 195.12 10 19.51 Cor Total 5730.94 15

Std. Dev. 4.42 R-Squared 0.9660 Mean 70.06 Adj R-Squared 0.9489 C.V. 6.30 Pred R-Squared 0.9128

PRESS 499.52 Adeq Precision 20.841

DOE Course

the regression model1
The Regression Model

Final Equation in Terms of Coded Factors:

Filtration Rate = +70.06250 +10.81250 * Temperature +4.93750 * Concentration +7.31250 * Stirring Rate -9.06250 * Temperature * Concentration +8.31250 * Temperature * Stirring Rate

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model interpretation cube plot
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

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model interpretation response surface plots
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

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the drilling experiment
The Drilling Experiment

A = drill load, B = flow, C = speed, D = type of mud, y = advance rate of the drill

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effect estimates the drilling experiment
Effect Estimates - The Drilling Experiment

Term Effect SumSqr % Contribution Model Intercept Error A 0.9175 3.36722 1.28072 Error 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 0.529599 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

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residual plots
Residual Plots

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residual plots1
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

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selecting a transformation
Selecting a Transformation
  • Empirical selection of lambda
  • Prior (theoretical) knowledge or experience can often suggest the form of a transformation
  • Analytical selection of lambda…the Box-Cox (1964) method (simultaneously estimates the model parameters and the transformation parameter lambda)
  • Box-Cox method implemented in Design-Expert

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the box cox method
The Box-Cox Method

A log transformation is recommended

The procedure provides a confidenceinterval on the transformation parameter lambda

If unity is included in the confidence interval, no transformation would be needed

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effect estimates following the log transformation
Effect Estimates Following the Log Transformation

Three main effects are large

No indication of large interaction effects

What happened to the interactions?

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anova following the log transformation
ANOVA Following the Log Transformation

Response:adv._rateTransform:Natural logConstant: 0.000 ANOVA for Selected Factorial Model

Analysis of variance table [Partial sum of squares]Sum ofMeanFSourceSquaresDFSquareValueProb > F Model 7.11 3 2.37 164.82 < 0.0001B5.3515.35371.49< 0.0001C1.3411.3493.05< 0.0001D0.4310.4329.920.0001 Residual 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

DOE Course

following the log transformation
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

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the log advance rate model
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

DOE Course

other analysis methods for unreplicated 2 k designs
Other Analysis Methods for Unreplicated 2k Designs
  • Lenth’s 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

DOE Course

addition of center points to a 2 k designs
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:

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slide110

The hypotheses are:

This sum of squares has a single degree of freedom

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example1
Example

Usually between 3 and 6 center points will work well

Design-Expert provides the analysis, including the F-test for pure quadratic curvature

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anova for example
ANOVA for Example

Response:yield ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]

Sum ofMeanFSourceSquaresDFSquareValueProb > F Model 2.83 3 0.94 21.92 0.0060A2.4012.4055.870.0017B0.4210.429.830.0350AB2.500E-00312.500E-0030.0580.8213 Curvature 2.722E-003 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. 0.51 Pred R-Squared N/A

PRESS N/A Adeq Precision 14.234

DOE Course

slide113

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

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practical use of center points
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

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