l. m. lyedoe course1 design and analysis of multi-factored experiments two-level factorial designs

L. M. Lye DOE Course 1

Design and Analysis ofMulti-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


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 Bdifference/2 estimate of effect of B on the effect of A

called as the interaction of A and B


a0b1 - a0b0 estimate of effect of B at low Adifference/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


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.

Estimating effects, contd...


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 willreveal the effect, on the variable under study, of factorsacting jointly. This is what we have called interaction. Thesimple experimental design discussed here evidently provides a way of estimating such interaction, with the latterdefined in a way which corresponds to what many scientistshave in mind when they think of interaction.

It is useful to note that interaction was not invented bystatisticians. It is a joint effect existing, often prominently, inthe real world. Statisticians have merely provided ways andmeans 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

A B ABa0b0 (1) - - +a0b1 (b) - + -a1b0 (a) + - -

a1b1 (ab) + + +

12DOE CourseL. M. Lye

9101112131415

-2 -1 0 1

Y

Example 1

10(1)

13b

13a

10ab

Example 3 BLow High

Low

High

A

B

10(1)

15b

15a

15ab

Example 2 Low High

Low

High

A

10(1)

12b

13a

15ab

Example 1 BLow High

Low

HighA

12(1)

12b

12a

12ab

Example 4 BLow High

Low

High

A

A+

A-

B

B=2

9101112131415

-2 -1 0 1

Y

Example 1

A

B+

B-A=3

9101112131415

-2 -1 0 1

Y

Example 3

A

B+B-

Example A B AB1 3* 2 02 2.5 2.5 -2.53 0 0 -34 0 0 0

Discussion of examples:Notice that in examples 2 & 3 interaction is as large as or larger than main effects.

9101112131415

-2 -1 0 1

Y

B+

B-

A

A=2.5

Example 2

9101112131415

-2 -1 0 1

YB-, B+

Example 4

A

*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 abUsing modern notation and Yates’ order, the estimates of effectsbecome:

A = (-1 + a - b + ab)/2B = (-1 - a + b +ab)/2AB = (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

etc, etc, ...

A A

B B

C C

A y y

B y y

C y y


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)


Estimate of AB

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

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


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

Estimate of ABC


This is our first encounter with a three-factor interaction. Itmeasures the impact, on the yield of the nitration process, ofinteraction AB when C (heel) goes from C absent to Cpresent. Or it measures the impact on yield of interaction ACwhen B (stirring time) goes from 1/2 hour to 4 hours. Or finally, it measures the impact on yield of interaction BCwhen A (time of addition of nitric acid) goes from 2 hours to7 hours.

As with two-factor two-level factorial designs, the formationof estimates in three-factor two-level factorial designs can besummarized in a table.


Sign Table for a 23 design

A B AB C AC BC ABC1 - - + - + + -a + - - - - + +b - + - - + - +ab + + + - - - -c - - + + - - +

ac + - - + + - -bc - + - + - + -

abc + + + + + + +


Example

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

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

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.


22 23 24 25 26 27

2 3 4 5 6 71 3 6 10 15 21

1 4 10 20 351 5 15 35

1 6 211 7

1

Main effect2 factor interaction

3 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 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.


A B AB C AC

BC

AB

CD AD

BD

AB

DC

DA

CD

BC

DA

BC

DE AE

BE

AB

EC

EA

CE

BC

EA

BC

ED

EA

DE

BD

EA

BD

EC

DE

AC

DE

BC

DE

AB

CD

E

1 - - + - + + - - + + - + - - + - + + - + - - + + - - + - + + -a + - - - - + + - - + + + + - - - - + + + + - - + + - - - - + +b - + - - + - + - - + + - - + - - + - + + - + - + - + - - + - +ab + + + - - - - - - - - + + + + - - - - + + + + + + + + - - - -

c - - + + - - + - + + - - + + - - + + - - + + + + - - + + - - +ac + - - + + - - - - + + - - + + - - + + - - + + + + - - + + - -bc - + - + - + - - + - + - + - + - + - + - + - + + - + - + - + -abc + + + + + + + - - - - - - - - - - - - - - - - + + + + + + + +

d - - + - + + - + - - + - + + - - + + - + - - + - + + - + - - +ad + - - - - + + + + - - - - + + - - + + + + - - - - + + + + - -bd - + - - + - + + - + - - + - + - + - + + - - - + - + + - + -abd + + + - - - - + + + + - - - - - - - - + + + + - - - - + + + +cd - - + + - - + + - - + + - - + - + + - - + + - - + + - - + + -acd + - - + + - - + + - - + + - - - - + + - - + + - - + + - - + +bcd - + - + - + - + - + - + - + - - + - + - + - + - + - + - + - +abcd + + + + + + + + + + + + + + + - - - - - - - - - - - - - - - -

e - - + - + + - - + + - + - - + + - - + - + + - - + + - + - - +ae + - - - - + + - - + + + + - - + + - - - - + + - - + + + + - -be - + - - + - + - + - + + - + - + - + - - + - + + + - + + - + -abe + + + - - - - - - - - + + + + + + + + - - - - - - - - + + + +ce - - + + - - + - + + - - + + - + - - + + - - + - + + - - + + -ace + - - + + - - - - + + - - + + + + - - + + - - - - + + - - + +bce - + - + - + - - + - + - - + + - + - + - + - + - + - + - +abce + + + + + + + - - - - - - - - + + + + + + + + - - - - - - - -de - - + - + + - + - - + - + + - + - - + - + + - + - - + - + + -ade + - - - - + + + + - - - - + + + + - - - - + + + + - - - - + +bde - + - - + - + + - + - - + - + + - + - - + - + + - + - - + - +abde + + + - - - - + + + + - - - - + + + + - - - - + + + + - - - -cde - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - +acde + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - -bcde - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + -abcde + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

22 23 24 25

Treatment s

E f f e c t s


Yates’ Forward Algorithm (1)

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

1. Applied to Complete Factorials (Yates, 1937)


Yates’ Forward Algorithm (2)Example:Yield of a nitration process

1 7.2 15.6 20.6 43.6 Contrast of µ a 8.4 5.0 23.0 5.0 Contrast of Ab 2.0 15.9 2.2 -19.4 Contrast of Bab 3.0 7.1 2.8 -2.4 Contrast of ABc 6.7 1.2 -10.6 2.4 Contrast of Cac 9.2 1.0 -8.8 0.6 Contrast of ACbc 3.4 2.5 -0.2 1.8 Contrast of BCabc 3.7 0.3 -2.2 -2.0 Contrast of ABC

Tr. Yield 1stCol 2ndCol 3rdCol

Again, note the line-by-line correspondence between treatmentsand 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.


low level (-1) high level (+1)N (A) blood sulphate of ammoniaP (B) superphosphate steamed bone flower;

Example (Adapted from Kempthorne)Yields are in bushels of potatoes per plot. The two factors are nitrate (N) and phosphate (P) fertilizers.

The yields are1 = 746.75 n = 625.75 p = 611.00 np = 656.00the estimates areN = -38.00 P = -52.75 NP = 83.00In the face of such high interaction we now specialize the maineffect of each factor to particular levels of the other factor.Effect of N at high level P = np-p = 656.00-611.00 = 45.0Effect of N at low level P = n-1 = 625.71-746.75 = -121.0, whichappear to be more valuable for fertilizer policy than the mean (-38.00)of such disparate numbers

600

650700

750

-2 -1 0 1

Y

N

P+

P-

-38-121

746.75

611.0656

625.75

Keep bothlow is best


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 AverageUsing 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 = 0 + 1 X1 + 2 X2 + 3 X3 + 13 X1X3

0 = overall average = 5.13

1 = Effect [A]/2 = -2.75/2 = -1.375

2 = Effect [B]/2 = -2.25/2 = -1.125

3 = Effect [C]/2 = -1.75/2 = - 0.875

13 = 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 PredictedValue Value9.00 9.135.00 5.137.00 6.883.00 2.876.00 6.135.00 4.634.00 3.882.00 2.37

Amazing fit!!

DESIGN-EXPERT PlotBaskets

Actual

Pre

dic

ted

Predicted vs. Actual

2.00

3.78

5.56

7.34

9.13

2.00 3.78 5.56 7.34 9.13


Interpreting Results

# out

of 10

2

6

10

8

4

B0 90

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

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

Effect of B=4-6.25= -2.25

A2m 5m

# out

of 10

108642

C: Shot type

C(-1)

C (+1)

Interaction of A and C = 1.25

At 5m, Jump or set shot about the same BUT at 2m, set shot gave higher values compared to jump shots



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-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)


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


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

2

Effect

grandmean

2n

ContrastSS

2n

ContrastEffect

i1

0

k

2

1k


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

MSE

MSF i

i

2k

21k1k

2n)Contrast(V

ContrastV)2n(

1

2n

ContrastV)effect(V


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)

MSE)2n(

1)Effect(se

)2n(

12n

)2n(

1)Effect(V

2k

22k

2k21k


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

The effect estimates are: A = 8.33, B = -5.00, AB = 1.67

Design-Expert analysis

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


Estimation of Factor EffectsForm Tentative Model

Term Effect SumSqr % ContributionModel InterceptModel A 8.33333 208.333 64.4995Model B -5 75 23.2198Model AB 1.66667 8.33333 2.57998Error Lack Of Fit 0 0Error P Error 31.3333 9.70072

Lenth's ME 6.15809 Lenth's SME 7.95671


Statistical Testing - ANOVAResponse:Conversion ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]

Sum of Mean FSource Squares DF Square Value Prob > FModel 291.67 3 97.22 24.82 0.0002A 208.33 1 208.33 53.19 < 0.0001B 75.00 1 75.00 19.15 0.0024AB 8.33 1 8.33 2.13 0.1828Pure Error 31.33 8 3.92Cor Total 323.00 11

Std. Dev. 1.98 R-Squared 0.9030Mean 27.50 Adj R-Squared 0.8666C.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?


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

Sum of Mean FSource Squares DF Square Value Prob > FModel 283.33 2 141.67 32.14 < 0.0001A 208.33 1 208.33 47.27 < 0.0001B 75.00 1 75.00 17.02 0.0026Residual 39.67 9 4.41Lack of Fit 8.33 1 8.33 2.13 0.1828Pure Error 31.33 8 3.92Cor Total 323.00 11



There is now a residual sum of squares, partitioned into a “lack of fit” component (the AB interaction) and a “pure error” component


Coefficient Standard 95% CI 95% CIFactor Estimate DF Error Low High VIFIntercept 27.5 1 0.60604 26.12904 28.87096A-Concentration4.166667 1 0.60604 2.79571 5.537623 1B-Catalyst -2.5 1 0.60604 -3.87096 -1.12904 1

Final Equation in Terms of Coded Factors:

Conversion =27.5

4.166667 * A-2.5 * B

Final Equation in Terms of Actual Factors:

Conversion =18.333330.833333 * Concentration

-5 * Catalyst

Regression Model for the Process


Residuals and Diagnostic Checking

DESIGN-EXPERT PlotConversion

Residual

Nor

mal

% p

roba

bilit

yNormal plot of residuals

-2.83333 -1.58333 -0.333333 0.916667 2.16667

1

5

10

20

30

50

70

80

90

95

99

DESIGN-EXPERT PlotConversion

22

Predicted

Re

sid

ua

ls

Residuals vs. Predicted

-2.83333

-1.58333

-0.333333

0.916667

2.16667

20.83 24.17 27.50 30.83 34.17


The Response SurfaceDESIGN-EXPERT Plot

ConversionX = A: ConcentrationY = B: Catalyst

Design Points

Conversion

A: Concentration

B: C

ata

lys

t

15.00 17.50 20.00 22.50 25.00

1.00

1.25

1.50

1.75

2.00

23.0556

25.2778

27.5

29.7222

31.9444

3 3

3 3

DESIGN-EXPERT Plot

ConversionX = A: ConcentrationY = B: Catalyst

20.8333

24.1667

27.5

30.8333

34.1667

C

on

vers

ion

15.00

17.50

20.00

22.50

25.00

1.00

1.25

1.50

1.75

2.00

A: Concentration B: Catalyst


An Example of a 23 Factorial Design

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


Estimation of Factor Effects

Term Effect SumSqr % ContributionModel InterceptError A 3 36 46.1538Error B 2.25 20.25 25.9615Error C 1.75 12.25 15.7051Error AB 0.75 2.25 2.88462Error AC 0.25 0.25 0.320513Error BC 0.5 1 1.28205Error ABC 0.5 1 1.28205Error LOF 0Error P Error 5 6.41026



ANOVA Summary – Full ModelResponse:Fill-deviation ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]

Sum of Mean FSource Squares DF Square Value Prob > FModel 73.00 7 10.43 16.69 0.0003A 36.00 1 36.00 57.60 < 0.0001B 20.25 1 20.25 32.40 0.0005C 12.25 1 12.25 19.60 0.0022AB 2.25 1 2.25 3.60 0.0943AC 0.25 1 0.25 0.40 0.5447BC 1.00 1 1.00 1.60 0.2415ABC 1.00 1 1.00 1.60 0.2415Pure Error 5.00 8 0.63Cor Total 78.00 15




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 ModelAnalysis of variance table [Partial sum of squares]

Sum of Mean FSource Squares DF Square Value Prob > FModel 70.75 4 17.69 26.84 < 0.0001A 36.00 1 36.00 54.62 < 0.0001B 20.25 1 20.25 30.72 0.0002C 12.25 1 12.25 18.59 0.0012AB 2.25 1 2.25 3.41 0.0917Residual 7.25 11 0.66LOF 2.25 3 0.75 1.20 0.3700Pure E 5.00 8 0.63C Total 78.00 15




Model Coefficients – Reduced Model

Coefficient Standard 95% CI 95% CI

Factor Estimate DF Error Low 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)

2

2

73.000.9359

78.00

/ 5.00 / 81 1 0.8798

/ 78.00 /15

Model

T

E EAdj

T T

SSR

SS

SS dfR

SS df

2Pred

20.001 1 0.7436

78.00T

PRESSR

SS


Model Summary Statistics

• Standard error of model coefficients

• Confidence interval on model coefficients

2 0.625ˆ ˆ( ) ( ) 0.202 2 2(8)

Ek k

MSse V

n n

/ 2, / 2,ˆ ˆ ˆ ˆ( ) ( )

E Edf dft se t se


The Regression Model


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 SatisfactoryDESIGN-EXPERT PlotFil l-deviation

Studentized Residuals

No

rma

l % p

rob

ab

ility

Normal plot of residuals

-1.67 -0.84 0.00 0.84 1.67

1

5

10

20

30

50

70

80

90

95

99


Model InterpretationDESIGN-EXPERT Plot

Fil l-deviation

X = A: CarbonationY = B: Pressure

B- 25.000B+ 30.000

Actual FactorC: Speed = 225.00

B: PressureInteraction Graph

Fill

-de

via

tion

A: Carbonation

10.00 10.50 11.00 11.50 12.00

-3

-0.75

1.5

3.75

6

Moderate interaction between carbonation level and pressure


Model InterpretationDESIGN-EXPERT Plot

Fill-deviationX = A: CarbonationY = B: PressureZ = C: Speed

Cube GraphFill-deviation

A: Carbonation

B: P

res

su

re

C: Speed

A- A+B-

B+

C-

C+

-2.13

-0.37

-0.63

1.13

0.12

1.88

3.13

4.88 Cube plots are often useful visual displays of experimental results


Contour & Response Surface Plots – Speed at the High Level

DESIGN-EXPERT Plot

Fill-deviationX = A: CarbonationY = B: Pressure

Design Points


Fill-deviation

A: Carbonation

B: P

res

su

re

10.00 10.50 11.00 11.50 12.00

25.00

26.25

27.50

28.75

30.00

0.5

1.375

2.25

3.125

2 2

2 2

DESIGN-EXPERT Plot

Fill-deviationX = A: CarbonationY = B: Pressure


-0.375

0.9375

2.25

3.5625

4.875

F

ill-

de

via

tio

n

10.00

10.50

11.00

11.50

12.00

25.00

26.25

27.50

28.75

30.00

A: Carbonation B: Pressure



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• Risks…if 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


Estimates of the Effects

Term Effect SumSqr % ContributionModel InterceptError A 21.625 1870.56 32.6397Error B 3.125 39.0625 0.681608Error C 9.875 390.062 6.80626Error D 14.625 855.563 14.9288Error AB 0.125 0.0625 0.00109057Error AC -18.125 1314.06 22.9293Error AD 16.625 1105.56 19.2911Error BC 2.375 22.5625 0.393696Error BD -0.375 0.5625 0.00981515Error CD -1.125 5.0625 0.0883363Error ABC 1.875 14.0625 0.245379Error ABD 4.125 68.0625 1.18763Error ACD -1.625 10.5625 0.184307Error BCD -2.625 27.5625 0.480942Error ABCD 1.375 7.5625 0.131959



The Normal Probability Plot of EffectsDESIGN-EXPERT PlotFiltration Rate

A: TemperatureB: PressureC: ConcentrationD: Stirring Rate

Normal plot

No

rma

l %

pro

ba

bility

Effect

-18.12 -8.19 1.75 11.69 21.62

1

5

10

20

30

50

70

80

90

95

99

A

CD

AC

AD


The Half-Normal Probability Plot

DESIGN-EXPERT PlotFiltration Rate

A: TemperatureB: PressureC: ConcentrationD: Stirring Rate

Half Normal plot

Ha

lf N

orm

al %

pro

ba

bility

|Effect|

0.00 5.41 10.81 16.22 21.63

0

20

40

60

70

80

85

90

95

97

99

A

CD

AC

AD


ANOVA Summary for the Model

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

Sum of Mean FSource Squares DF Square Value Prob >FModel 5535.81 5 1107.16 56.74 < 0.0001A 1870.56 1 1870.56 95.86 < 0.0001C 390.06 1 390.06 19.99 0.0012D 855.56 1 855.56 43.85 < 0.0001AC 1314.06 1 1314.06 67.34 < 0.0001AD 1105.56 1 1105.56 56.66 < 0.0001Residual 195.12 10 19.51Cor Total 5730.94 15




The Regression Model


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


Model Residuals are SatisfactoryDESIGN-EXPERT PlotFiltration Rate

Studentized Residuals

No

rma

l % p

rob

ab

ility


-1.83 -0.96 -0.09 0.78 1.65

1

5

10

20

30

50

70

80

90

95

99


Model Interpretation – InteractionsDESIGN-EXPERT Plot

Filtration Rate

X = A: TemperatureY = C: Concentration

C- -1.000C+ 1.000

Actual FactorsB: Pressure = 0.00D: Stirring Rate = 0.00

C: ConcentrationInteraction Graph

Filt

ratio

n R

ate

A: Temperature

-1.00 -0.50 0.00 0.50 1.00

41.7702

57.3277

72.8851

88.4426

104

DESIGN-EXPERT Plot

Filtration Rate

X = A: TemperatureY = D: Stirring Rate

D- -1.000D+ 1.000

Actual FactorsB: Pressure = 0.00C: Concentration = 0.00

D: Stirring RateInteraction Graph

Filt

ratio

n R

ate

A: Temperature

-1.00 -0.50 0.00 0.50 1.00

43

58.25

73.5

88.75

104


Model Interpretation – Cube PlotDESIGN-EXPERT Plot

Filtration RateX = A: TemperatureY = C: ConcentrationZ = D: Stirring Rate

Actual FactorB: Pressure = 0.00

Cube GraphFiltration Rate

A: Temperature

C: C

on

cen

tra

tion

D: Stirring Rate

A- A+C-

C+

D-

D+

46.25

44.25

74.25

72.25

69.38

100.63

61.13

92.38

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

DESIGN-EXPERT Plot

Filtration RateX = A: TemperatureY = D: Stirring Rate

Actual FactorsB: Pressure = 0.00C: Concentration = -1.00

44.25

58.3438

72.4375

86.5313

100.625

Fi

ltra

tio

n R

ate

-1.00

-0.50

0.00

0.50

1.00

-1.00

-0.50

0.00

0.50

1.00

A: Temperature D: Stirring Rate

DESIGN-EXPERT Plot

Filtration RateX = A: TemperatureY = D: Stirring Rate

Actual FactorsB: Pressure = 0.00C: Concentration = -1.00

Filtration Rate

A: Temperature

D: S

tirri

ng R

ate

-1.00 -0.50 0.00 0.50 1.00

-1.00

-0.50

0.00

0.50

1.00

64.62571

77.375

83.75

90.125

56.93551.9395

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 % ContributionModel InterceptError A 0.9175 3.36722 1.28072Error B 6.4375 165.766 63.0489Error C 3.2925 43.3622 16.4928Error D 2.29 20.9764 7.97837Error AB 0.59 1.3924 0.529599Error AC 0.155 0.0961 0.0365516Error AD 0.8375 2.80563 1.06712Error BC 1.51 9.1204 3.46894Error BD 1.5925 10.1442 3.85835Error CD 0.4475 0.801025 0.30467Error ABC 0.1625 0.105625 0.0401744Error ABD 0.76 2.3104 0.87876Error ACD 0.585 1.3689 0.520661Error BCD 0.175 0.1225 0.0465928Error ABCD 0.5425 1.17722 0.447757



Half-Normal Probability Plot of Effects

DESIGN-EXPERT Plotadv._rate

A: loadB: flowC: speedD: mud

Half Normal plot

Ha

lf N

orm

al %

pro

ba

bili

ty

|Effect|

0.00 1.61 3.22 4.83 6.44

0

20

40

60

70

80

85

90

95

97

99

B

C

D

BCBD


Residual PlotsDESIGN-EXPERT Plotadv._rate

Residual

No

rma

l % p

rob

ab

ility


-1.96375 -0.82625 0.31125 1.44875 2.58625

1

5

10

20

30

50

70

80

90

95

99

DESIGN-EXPERT Plotadv._rate

Predicted

Re

sid

ua

ls


-1.96375

-0.82625

0.31125

1.44875

2.58625

1.69 4.70 7.70 10.71 13.71


• 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

Residual Plots

*y y


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


The Box-Cox MethodDESIGN-EXPERT Plotadv._rate

LambdaCurrent = 1Best = -0.23Low C.I. = -0.79High C.I. = 0.32

Recommend transform:Log (Lambda = 0)

Lambda

Ln

(Re

sid

ua

lSS

)

Box-Cox Plot for Power Transforms

1.05

2.50

3.95

5.40

6.85

-3 -2 -1 0 1 2 3

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

DESIGN-EXPERT PlotLn(adv._rate)

A: loadB: flowC: speedD: mud

Half Normal plotH

alf

No

rma

l % p

rob

ab

ility

|Effect|

0.00 0.29 0.58 0.87 1.16

0

20

40

60

70

80

85

90

95

97

99

B

C

D

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 > FModel 7.11 3 2.37 164.82 < 0.0001B 5.35 1 5.35 371.49 < 0.0001C 1.34 1 1.34 93.05 < 0.0001D 0.43 1 0.43 29.92 0.0001Residual 0.17 12 0.014Cor Total 7.29 15




Following the Log Transformation


Ln(adv._rate) =+1.60+0.58 * B+0.29 * C+0.16 * D


Following the Log Transformation


Residual

No

rma

l % p

rob

ab

ility


-0.166184 -0.0760939 0.0139965 0.104087 0.194177

1

5

10

20

30

50

70

80

90

95

99


PredictedR

es

idu

als


-0.166184

-0.0760939

0.0139965

0.104087

0.194177

0.57 1.08 1.60 2.11 2.63


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

• 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



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:

01 1

20

1 1 1

First-order model (interaction)

Second-order model

k k k

i i ij i ji i j i

k k k k

i i ij i j ii ii i j i i

y x x x

y x x x x


no "curvature"F Cy y

The hypotheses are:

01

11

: 0

: 0

k

iii

k

iii

H

H

2

Pure Quad

( )F C F C

F C

n n y ySS

n n

This sum of squares has a single degree of freedom


Example

5Cn

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 ModelAnalysis of variance table [Partial sum of squares]

Sum of Mean FSource Squares DF Square Value Prob > FModel 2.83 3 0.94 21.92 0.0060A 2.40 1 2.40 55.87 0.0017B 0.42 1 0.42 9.83 0.0350AB 2.500E-003 1 2.500E-003 0.058 0.8213Curvature 2.722E-003 1 2.722E-003 0.063 0.8137Pure Error 0.17 4 0.043Cor Total 3.00 8

Std. Dev. 0.21 R-Squared 0.9427Mean 40.44 Adj R-Squared 0.8996

C.V. 0.51 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

l. m. lyedoe course1 design and analysis of multi-factored experiments two-level factorial designs

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