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Design and Analysis of Multi-Factored Experiments

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Title: Design and Analysis of Multi-Factored Experiments


1
Design and Analysis of Multi-Factored Experiments
  • Two-level Factorial Designs

2
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

3
Chemical Process Example
A reactant concentration, B catalyst amount,
y recovery
4
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
5
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

6
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

7
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.

8
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.

9
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.

10
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.

11
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

12
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
13
  • 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.

14
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

15
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

16
Effects in The 23 Factorial Design
17
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)

18
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

19
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

20
  • 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.

21
  • Sign Table for a 23 design

22
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
23
  • 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

24
  • 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.

25
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.

26
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)!
27
  • 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.
28
25
E f f e c t s
24
23
22
Treatment s
29
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

30
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.
31
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.

32
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
33
  • 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.

34
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

35
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?

36
  • 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.

37
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

38
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
39
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
40
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
41
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
42
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
43
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!!
44
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
45
Design and Analysis of Multi-Factored Experiments
  • Analysis of 2k Experiments
  • Statistical Details

46
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.

47
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)
48
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
- - - -
- - - -
- - - -
- - - -
- - - -
---
---
49
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.

50
  • 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.

51
Effects, Sum of Squares and Regression
Coefficients
52
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

53
  • 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)

54
  • 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.

55
Design and Analysis of Multi-Factored Experiments
  • Examples of Computer Analysis

56
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

57
Chemical Process Example
A reactant concentration, B catalyst amount,
y recovery
58
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
59
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
60
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?
61
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.
62
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
63
Regression Model for the Process
64
Residuals and Diagnostic Checking
65
The Response Surface
66
An Example of a 23 Factorial Design
A carbonation, B pressure, C speed, y
fill deviation
67
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
68
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
69
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
70
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
71
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
72
Model Summary Statistics
  • R2 and adjusted R2
  • R2 for prediction (based on PRESS)

73
Model Summary Statistics
  • Standard error of model coefficients
  • Confidence interval on model coefficients

74
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
75
Residual Plots are Satisfactory
76
Model Interpretation
Moderate interaction between carbonation level
and pressure
77
Model Interpretation
Cube plots are often useful visual displays of
experimental results
78
Contour Response Surface Plots Speed at the
High Level
79
Design and Analysis of Multi-Factored Experiments
  • Unreplicated Factorials

80
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?

81
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
82
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)

83
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

84
The Resin Plant Experiment
85
The Resin Plant Experiment
86
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
87
The Normal Probability Plot of Effects
88
The Half-Normal Probability Plot
89
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
90
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
91
Model Residuals are Satisfactory
92
Model Interpretation Interactions
93
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
94
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
95
The Drilling Experiment
A drill load, B flow, C speed, D type of
mud, y advance rate of the drill
96
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
97
Half-Normal Probability Plot of Effects
98
Residual Plots
99
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

100
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

101
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
102
Effect Estimates Following the Log Transformation
Three main effects are large No indication of
large interaction effects What happened to the
interactions?
103
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
104
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
105
Following the Log Transformation
106
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

107
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

108
Design and Analysis of Multi-Factored Experiments
  • Center points

109
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

110
The hypotheses are
This sum of squares has a single degree of freedom
111
Example
Usually between 3 and 6 center points will work
well Design-Expert provides the analysis,
including the F-test for pure quadratic curvature
112
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
113
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
114
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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