Transcript Design and Analysis of Multi-Factored Experiments Two-level Factorial Designs L. M. Lye

Design and Analysis of
Multi-Factored Experiments
Two-level Factorial Designs
L. M. Lye
DOE Course
1
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
L. M. Lye
DOE Course
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Chemical Process Example
A = reactant concentration, B = catalyst amount,
y = recovery
L. M. Lye
DOE Course
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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
L. M. Lye
DOE Course
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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
L. M. Lye
DOE Course
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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
a0b1 - a0b0
difference/2
estimate of effect of B at high A
estimate of effect of B at low A
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
L. M. Lye
DOE Course
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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.
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DOE Course
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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.
L. M. Lye
DOE Course
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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.
L. M. Lye
DOE Course
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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.
L. M. Lye
DOE Course
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Table of signs
The following table is useful:
a0b0 (1)
a0b1 (b)
a1b0 (a)
a1b1 (ab)
A
+
+
B
+
+
AB
+
+
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
L. M. Lye
DOE Course
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15
14
13
12
Y11
10
9
Example 1
A+
A-1
B
Example 1
0
-2
Low
1
A
B+
A
0
1
Example
Example 3
3
Low
B+
B-
A
High
-2
Example 2 Low B High
Low
A
High
10
(1)
15
a
15
b
15
ab
Example 2
B+
15
14
13
12
Y11
10
9
A=2.5
B-2
-1
0
A
1
B-1
15
14
13
12
Y11
10
9
High
10 12
(1) b
13 15
a ab
High
A=3
B
Low
B=2
-2
15
14
13
Y12
11
10
9
Example 1
-1
A
Example
1
2
3
4
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0
A
3*
2.5
0
0
B
Example 4
Low High
10
(1)
13
a
13
b
10
ab
Low
A
High
B
Low High
12
(1)
12
a
12
b
12
ab
Example 4
15
14
13
12
Y11
10
9
B-,
B+
-2
-1
A
0
1
B
2
2.5
0
0
AB
0
-2.5
-3
0
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
DOE Course
= 3
12
1
• 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.
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DOE Course
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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
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DOE Course
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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
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DOE Course
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Effects in The 23 Factorial Design
A  y A  y A
B  yB  yB
C  yC   yC 
etc, etc, ...
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DOE Course
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Estimating effects in three-factor two-level
designs (23)
Estimate of A
(1) a - 1
(2) ab - b
(3) ac - c
(4) abc - bc
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estimate of A, with B low and C low
estimate of A, with B high and C low
estimate of A, with B low and C high
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)
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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,
L. M. Lye
= [(abc+ab+c+1) - (a+b+ac+bc)]/4
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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,
L. M. Lye
=[abc+a+b+c - (1+ab+ac+bc)]/4
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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.
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DOE Course
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Sign Table for a 23 design
1
a
b
ab
c
ac
bc
abc
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A
+
+
+
+
B
+
+
+
+
AB
+
+
+
+
C
+
+
+
+
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AC
+
+
+
+
BC ABC
+
+
+
+
+
+
+
+
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Example
Yield of nitration process discussed earlier:
Y =
A
B
AB
C
AC
BC
ABC
=
=
=
=
=
=
=
1 a b ab c ac bc
7.2 8.4 2.0 3.0 6.7 9.2 3.4
main effect of nitric acid time
main effect of stirring time
interaction of A and B
main effect of heel
interaction of A and C
interaction of B and C
interaction of A, B, and C
abc
3.7
= 1.25
= -4.85
= -0.60
= 0.60
= 0.15
= 0.45
= -0.50
NOTE: ac = largest yield; AC = smallest effect
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DOE Course
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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
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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.
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DOE Course
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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.
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DOE Course
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Main effect
2 factor interaction
3 factor interaction
4 factor interaction
5 factor interaction
6 factor interaction
7 factor interaction
22
2
1
3
23
3
3
1
7
24
4
6
4
1
15
25
5
10
10
5
1
31
26
6
15
20
15
6
1
27
7
21
35
35
21
7
1
63
127
In a 2k design, the number of r-factor effects is Ckr = k!/[r!(k-r)!]
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DOE Course
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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 231 = 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.
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DOE Course
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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
T
r
e
a
t
m
e
n
t
s
A
22
25
Effects
24
23
1
a
b
ab
+
+
+
+
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-
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abc
+
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e
ae
be
abe
ce
ace
bce
abce
de
ade
bde
abde
cde
acde
bcde
abcde
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28
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
L. M. Lye
DOE Course
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Yates’ Forward Algorithm (2)
Example:
Yield of a nitration process
Tr.
Yield
1stCol 2ndCol 3rdCol
1
a
b
ab
c
ac
bc
abc
7.2
8.4
2.0
3.0
6.7
9.2
3.4
3.7
15.6
5.0
15.9
7.1
1.2
1.0
2.5
0.3
20.6 43.6
23.0
5.0
2.2 -19.4
2.8 -2.4
-10.6
2.4
-8.8
0.6
-0.2
1.8
-2.2 -2.0
Contrast of µ
Contrast of A
Contrast of B
Contrast of AB
Contrast of C
Contrast of AC
Contrast of BC
Contrast of ABC
Again, note the line-by-line correspondence between treatments
and estimates; both are in Yates’ order.
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DOE Course
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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.
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DOE Course
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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
Y
L. M. Lye
750
700
650
600
611.0
-2
-1
P+
625.75 PDOE Course
0
1 N
-38
656
-121
Keep both
low is best
32
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.
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DOE Course
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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
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DOE Course
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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?
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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.
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DOE Course
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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
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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
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Set
DOE Course
Jump
38
Treatment Combinations and Results
Order
A
B
C
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
L. M. Lye
DOE Course
Combination
Y
39
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
L. M. Lye
DOE Course
40
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
L. M. Lye
DOE Course
41
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
L. M. Lye
DOE Course
42
Checking for goodness-of-fit
DESIGN-EXPERT Plot
Baskets
Predicted
Value
9.13
5.13
6.88
2.87
6.13
4.63
3.88
2.37
7.34
5.56
3.78
2.00
2.00
3.78
5.56
7.34
9.13
Actual
Amazing fit!!
L. M. Lye
9.13
Predicted
Actual
Value
9.00
5.00
7.00
3.00
6.00
5.00
4.00
2.00
Predicted vs. Actual
DOE Course
43
Interpreting Results
10
# out
of 10
8
6
4
2
Effect of B=4-6.25= -2.25
(9+5+6+5)/4=6.25
(7+3+4+2)/4=4
0
10
8
# out 6
of 10 4
2
90
C: Shot type
Interaction of A and C = 1.25
C(-1)
C (+1)
2m
L. M. Lye
B
5m
DOE Course
A
At 5m, Jump or set shot
about the same BUT at
2m, set shot gave higher
values compared to jump
shots
44
Design and Analysis of
Multi-Factored Experiments
Analysis of 2k Experiments
Statistical Details
L. M. Lye
DOE Course
45
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.
L. M. Lye
DOE Course
46
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.
Note: 2(kx) = k2 2(x)
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.
L. M. Lye
DOE Course
47
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.
L. M. Lye
1
a
b
abc
d
-
-
-
-
-
---
DOE Course
---
48
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.
L. M. Lye
DOE Course
49
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.
L. M. Lye
DOE Course
50
Effects, Sum of Squares and Regression Coefficients
Effect 
Contrast
n2k 1
2

Contrast 
SS 
n2k
b 0  grandmean
Effect i
b1 
2
L. M. Lye
DOE Course
51
Judging Significance of Effects
a) p- values from ANOVA
MSi
Fi 
MSE
Compute p-value of calculated F. IF p < , then effect is
significant.
b) Comparing std. error of effect to size of effect
1
 Contrast 
V(effect )  V 

V Contrast 
k 1 
k 1 2
 n2
 (n2 )
V(Contrast )  n2k  2
L. M. Lye
DOE Course
52
Hence
V(Effect ) 
se(Effect ) 
1
n2  
2
k
(n2k 1 )
1
(n2
k 2
)
2
1
(n2k 2 )
2
MSE
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)
L. M. Lye
DOE Course
53
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.
L. M. Lye
DOE Course
54
Design and Analysis of
Multi-Factored Experiments
Examples of Computer Analysis
L. M. Lye
DOE Course
55
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
L. M. Lye
DOE Course
56
Chemical Process Example
A = reactant concentration, B = catalyst amount,
y = recovery
L. M. Lye
DOE Course
57
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
The effect estimates are:
A
= 8.33, B = -5.00, AB = 1.67
C = (ab + 1 - a - b)/2n
= 1.67
L. M. Lye
Design-Expert analysis
DOE Course
58
Estimation of Factor Effects
Form Tentative Model
Model
Model
Model
Model
Error
Error
Term
Effect
SumSqr
% Contribution
Intercept
A
8.33333
208.333
64.4995
B
-5
75
23.2198
AB
1.66667
8.33333
2.57998
Lack Of Fit 0
0
P Error
31.3333
9.70072
Lenth's ME
Lenth's SME
L. M. Lye
6.15809
7.95671
DOE Course
59
Statistical Testing - ANOVA
Response:Conversion
ANOVA for Selected Factorial Model
Analysis of variance table [Partial sum of squares]
Sum of
Source
Squares
Model
291.67
A
208.33
B
75.00
AB
8.33
Pure Error 31.33
Cor Total 323.00
DF
3
1
1
1
8
11
Mean
Square
97.22
208.33
75.00
8.33
3.92
F
Value
24.82
53.19
19.15
2.13
Prob > F
0.0002
< 0.0001
0.0024
0.1828
Std. Dev.
Mean
C.V.
1.98
27.50
7.20
R-Squared
Adj R-Squared
Pred R-Squared
0.9030
0.8666
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?
L. M. Lye
DOE Course
60
Statistical Testing - ANOVA
Coefficient
Factor
Intercept
A-Concent
B-Catalyst
AB
Standard
Estimate DF Error
27.50
1 0.57
4.17
1 0.57
-2.50
1 0.57
0.83
1 0.57
95% CI
Low
26.18
2.85
-3.82
-0.48
95% CI
High
28.82
5.48
-1.18
2.15
VIF
1.00
1.00
1.00
General formulas for the standard errors of the model coefficients and
the confidence intervals are available. They will be given later.
L. M. Lye
DOE Course
61
Refine Model
Response:Conversion
ANOVA for Selected Factorial Model
Analysis of variance table [Partial sum of squares]
Sum of
Source
Squares
Model
283.33
A
208.33
B
75.00
Residual 39.67
Lack of Fit 8.33
Pure Error 31.33
Cor Total 323.00
DF
2
1
1
9
1
8
11
Mean
Square
141.67
208.33
75.00
4.41
8.33
3.92
F
Value
32.14
47.27
17.02
Prob > F
< 0.0001
< 0.0001
0.0026
2.13
0.1828
Std. Dev.
Mean
C.V.
2.10
27.50
7.63
R-Squared 0.8772
Adj R-Squared
Pred R-Squared
0.8499
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
L. M. Lye
DOE Course
62
Regression Model for the Process
Coefficient
Factor
Estimate DF
Intercept
27.5
A-Concentration
4.166667
B-Catalyst
-2.5
Standard 95% CI
Error
Low
1 0.60604 26.12904
1 0.60604 2.79571
1 0.60604 -3.87096
95% CI
High
28.87096
5.537623
-1.12904
VIF
1
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 =
18.33333
0.833333 * Concentration
-5 * Catalyst
L. M. Lye
DOE Course
63
Residuals and Diagnostic Checking
DESIGN-EXPERT Plot
Conversion
Normal plot of residuals
DESIGN-EXPERT Plot
Conversion
Residuals vs. Predicted
2.16667
99
95
0.916667
80
70
Res iduals
Norm al % probability
90
50
-0.333333
30
20
10
-1.58333
5
2
1
-2.83333
-2.83333
-1.58333
-0.333333
0.916667
20.83
2.16667
27.50
30.83
34.17
Predicted
Residual
L. M. Lye
24.17
DOE Course
64
The Response Surface
DESIGN-EXPERT Plot
T Plot
Conversion
3
Conversion
X = A: Concentration
3
2.00
Y = B: Catalyst
ation
34.1667
23.0556
s
30.8333
1.75
24.1667
25.2778
Conversion
B: Catalys t
27.5
27.5
1.50
20.8333
29.7222
1.25
31.9444
2.00
25.00
1.75
3
22.50
3
1.50
1.00
15.00
17.50
20.00
22.50
25.00
B: Catalyst
20.00
1.25
17.50
A: Concentration
1.00
A: Concentration
L. M. Lye
DOE Course
15.00
65
An Example of a 23 Factorial Design
A = carbonation, B = pressure, C = speed, y = fill deviation
L. M. Lye
DOE Course
66
Estimation of Factor Effects
Model
Error
Error
Error
Error
Error
Error
Error
Error
Error
L. M. Lye
Term
Effect
Intercept
A
3
B
2.25
C
1.75
AB
0.75
AC
0.25
BC
0.5
ABC
0.5
LOF
0
P Error
SumSqr % Contribution
Lenth's ME
Lenth's SME
1.25382
1.88156
36
20.25
12.25
2.25
0.25
1
1
46.1538
25.9615
15.7051
2.88462
0.320513
1.28205
1.28205
5
6.41026
DOE Course
67
ANOVA Summary – Full Model
Response:Fill-deviation
ANOVA for Selected Factorial Model
Analysis of variance table [Partial sum of squares]
Sum of
Source
Squares
Model
73.00
A
36.00
B
20.25
C
12.25
AB
2.25
AC
0.25
BC
1.00
ABC
1.00
Pure Error 5.00
Cor Total 78.00
L. M. Lye
DF
7
1
1
1
1
1
1
1
8
15
Mean
Square
10.43
36.00
20.25
12.25
2.25
0.25
1.00
1.00
0.63
F
Value
16.69
57.60
32.40
19.60
3.60
0.40
1.60
1.60
Prob > F
0.0003
< 0.0001
0.0005
0.0022
0.0943
0.5447
0.2415
0.2415
Std. Dev.
Mean
C.V.
0.79
1.00
79.06
R-Squared 0.9359
Adj R-Squared
Pred R-Squared
0.8798
0.7436
PRESS
20.00
Adeq Precision
13.416
DOE Course
68
Model Coefficients – Full Model
Coefficient
Standard
Factor
L. M. Lye
Estimate
95% CI 95% CI
DF
Error
Low
High
Intercept
1.00
1
0.20
0.54
1.46
A-Carbonation
B-Pressure
C-Speed
AB
AC
BC
ABC
1.50
1.13
0.88
0.38
0.13
0.25
0.25
1
1
1
1
1
1
1
0.20
0.20
0.20
0.20
0.20
0.20
0.20
1.04
0.67
0.42
-0.081
-0.33
-0.21
-0.21
1.96
1.58
1.33
0.83
0.58
0.71
0.71
DOE Course
VIF
1.00
1.00
1.00
1.00
1.00
1.00
1.00
69
Refine Model – Remove Nonsignificant Factors
Response:
Fill-deviation
ANOVA for Selected Factorial Model
Analysis of variance table [Partial sum of squares]
Sum of
Source Squares
Model
70.75
A
36.00
B
20.25
C
12.25
AB
2.25
Residual 7.25
LOF
2.25
Pure E 5.00
C Total 78.00
L. M. Lye
DF
4
1
1
1
1
11
3
8
15
Mean
Square
17.69
36.00
20.25
12.25
2.25
0.66
0.75
0.63
F
Value
26.84
54.62
30.72
18.59
3.41
Prob > F
< 0.0001
< 0.0001
0.0002
0.0012
0.0917
1.20
0.3700
Std. Dev. 0.81
Mean
1.00
C.V.
81.18
R-Squared
Adj R-Squared
Pred R-Squared
0.9071
0.8733
0.8033
PRESS
Adeq Precision
15.424
15.34
DOE Course
70
Model Coefficients – Reduced Model
Coefficient
Factor
Estimate
Intercept
1.00
A-Carbonation 1.50
B-Pressure
1.13
C-Speed
0.88
AB
0.38
L. M. Lye
Standard 95% CI 95% CI
DF
Error
Low
High
1
0.20
0.55
1.45
1
0.20
1.05
1.95
1
0.20
0.68
1.57
1
0.20
0.43
1.32
1
0.20
-0.072 0.82
DOE Course
71
Model Summary Statistics
• R2 and adjusted R2
SS Model 73.00
R 

 0.9359
SST
78.00
2
R
2
Adj
SS E / df E
5.00 / 8
 1
 1
 0.8798
SST / dfT
78.00 /15
• R2 for prediction (based on PRESS)
PRESS
20.00
2
RPred  1 
 1
 0.7436
SST
78.00
L. M. Lye
DOE Course
72
Model Summary Statistics
• Standard error of model coefficients
se( bˆ )  V ( bˆ ) 
2
MS E
0.625


 0.20
k
k
n2
n2
2(8)
• Confidence interval on model coefficients
bˆ  t / 2,df se( bˆ )  b  bˆ  t / 2,df se( bˆ )
E
L. M. Lye
E
DOE Course
73
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
L. M. Lye
DOE Course
74
Residual Plots are Satisfactory
DESIGN-EXPERT Plot
Fill-deviation
Normal plot of residuals
99
95
Norm al % probability
90
80
70
50
30
20
10
5
1
-1.67
-0.84
0.00
0.84
1.67
Studentized Res iduals
L. M. Lye
DOE Course
75
Model Interpretation
DESIGN-EXPERT Plot
Fill-deviation
Interaction Graph
B: Pres s ure
6
Moderate
interaction between
carbonation level
and pressure
X = A: Carbonation
Y = B: Pressure
3.75
Fill-deviation
B- 25.000
B+ 30.000
Actual Factor
C: Speed = 225.00
1.5
-0.75
-3
10.00
10.50
11.00
11.50
12.00
A: Carbonation
L. M. Lye
DOE Course
76
Model Interpretation
DESIGN-EXPERT Plot
Cube Graph
Fill-deviation
Fill-deviation
X = A: Carbonation
Y = B: Pressure
Z = C: Speed
1.13
B+
B: Pres s ure
-0.63
Cube plots are
often useful visual
displays of
experimental
results
4.88
3.13
-0.37
1.88
C+
C: Speed
BA-
-2.13
0.12
CA+
A: Carbonation
L. M. Lye
DOE Course
77
Contour & Response Surface Plots –
Speed at the High Level
DESIGN-EXPERT Plot
DESIGN-EXPERT Plot
Fill-deviationFill-deviation2
2
30.00
X = A: Carbonation
Y = B: Pressure
Fill-deviation
X = A: Carbonation
Y = B: Pressure
Actual Factor
C: Speed = 250.00
Design Points
28.75
3.5625
3.125
Actual Factor
C: Speed = 250.00
4.875
0.9375
Fill-deviation
B: Pres s ure
2.25
2.25
27.50
-0.375
1.375
0.5
26.25
30.00
12.00
28.75
2
2
25.00
10.00
10.50
11.00
11.50
12.00
11.50
27.50
11.00
B: Pressure 26.25
10.50
A: Carbonation
25.00
A: Carbonation
L. M. Lye
DOE Course
10.00
78
Design and Analysis of
Multi-Factored Experiments
Unreplicated Factorials
L. M. Lye
DOE Course
79
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”?
L. M. Lye
DOE Course
80
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
L. M. Lye
DOE Course
81
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)
L. M. Lye
DOE Course
82
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
L. M. Lye
DOE Course
83
The Resin Plant Experiment
L. M. Lye
DOE Course
84
The Resin Plant Experiment
L. M. Lye
DOE Course
85
Estimates of the Effects
Model
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Term
Intercept
A
B
C
D
AB
AC
AD
BC
BD
CD
ABC
ABD
ACD
BCD
ABCD
Effect
SumSqr % Contribution
21.625
3.125
9.875
14.625
0.125
-18.125
16.625
2.375
-0.375
-1.125
1.875
4.125
-1.625
-2.625
1.375
1870.56
39.0625
390.062
855.563
0.0625
1314.06
1105.56
22.5625
0.5625
5.0625
14.0625
68.0625
10.5625
27.5625
7.5625
Lenth's ME
Lenth's SME
L. M. Lye
DOE Course
32.6397
0.681608
6.80626
14.9288
0.00109057
22.9293
19.2911
0.393696
0.00981515
0.0883363
0.245379
1.18763
0.184307
0.480942
0.131959
6.74778
13.699
86
The Normal Probability Plot of Effects
DESIGN-EXPERT Plot
Filtration Rate
A:
B:
C:
D:
T emperature
Pressure
Concentration
Stirring Rate
Normal plot
99
A
95
Norm al % probability
90
AD
80
C
70
D
50
30
20
10
5
AC
1
-18.12
-8.19
1.75
11.69
21.62
Effect
L. M. Lye
DOE Course
87
The Half-Normal Probability Plot
DESIGN-EXPERT Plot
Filtration Rate
A:
B:
C:
D:
T emperature
Pressure
Concentration
Stirring Rate
Half Normal plot
99
97
A
Half Norm al % probability
95
90
AC
85
AD
80
D
70
C
60
40
20
0
0.00
5.41
10.81
16.22
21.63
|Effect|
L. M. Lye
DOE Course
88
ANOVA Summary for the Model
Response:Filtration Rate
ANOVA for Selected Factorial Model
Analysis of variance table [Partial sum of squares]
L. M. Lye
Source
Model
A
C
D
AC
AD
Residual
Cor Total
Sum of
Squares
5535.81
1870.56
390.06
855.56
1314.06
1105.56
195.12
5730.94
Std. Dev.
Mean
C.V.
4.42
70.06
6.30
R-Squared 0.9660
Adj R-Squared
Pred R-Squared
0.9489
0.9128
PRESS
499.52
Adeq Precision
20.841
DF
5
1
1
1
1
1
10
15
Mean
Square
1107.16
1870.56
390.06
855.56
1314.06
1105.56
19.51
DOE Course
F
Value
56.74
95.86
19.99
43.85
67.34
56.66
Prob >F
< 0.0001
< 0.0001
0.0012
< 0.0001
< 0.0001
< 0.0001
89
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
L. M. Lye
DOE Course
90
Model Residuals are Satisfactory
DESIGN-EXPERT Plot
Filtration Rate
Normal plot of residuals
99
95
Norm al % probability
90
80
70
50
30
20
10
5
1
-1.83
-0.96
-0.09
0.78
1.65
Studentized Res iduals
L. M. Lye
DOE Course
91
Model Interpretation – Interactions
DESIGN-EXPERT Plot
Interaction Graph
Filtration Rate
DESIGN-EXPERT Plot
104
Filtration Rate
X = A: Temperature
Y = C: Concentration
D: Stirring Rate
104
X = A: Temperature
Y = D: Stirring Rate
88.4426
D- -1.000
D+ 1.000
Actual Factors
B: Pressure = 0.00
C: Concentration = 0.00
Filtration Rate
Filtration Rate
C- -1.000
C+ 1.000
Actual Factors
B: Pressure = 0.00
D: Stirring Rate = 0.00
72.8851
88.75
73.5
57.3277
58.25
41.7702
43
-1.00
-0.50
0.00
0.50
1.00
A: Tem perature
L. M. Lye
Interaction Graph
C: Concentration
-1.00
-0.50
0.00
0.50
1.00
A: Tem perature
DOE Course
92
Model Interpretation – Cube Plot
DESIGN-EXPERT Plot
If one factor is
dropped, the
unreplicated 24
design will project
into two replicates
of a 23
Cube Graph
Filtration Rate
Filtration Rate
X = A: Temperature
Y = C: Concentration
Z = D: Stirring Rate
72.25
92.38
Actual Factor
B: Pressure = 0.00
C+
C: Concentration
74.25
61.13
44.25
100.63
D: Stirring Rate
CA-
46.25
69.38
DA+
A: Tem perature
L. M. Lye
D+
DOE Course
Design projection is
an extremely useful
property, carrying
over into fractional
factorials
93
Model Interpretation – Response Surface Plots
DESIGN-EXPERT Plot
1.00
DESIGN-EXPERT Plot
Filtration Rate
Filtration Rate
X = A: Temperature
Y = D: Stirring Rate
Filtration Rate
X = A: Temperature
Y = D: Stirring Rate
90.125
Actual Factors
B: Pressure = 0.00
C: Concentration = -1.00
Actual Factors
100.625
B: Pressure = 0.00
C: Concentration = 86.5313
-1.00
0.50
83.75
0.00
51.9395
56.935
72.4375
58.3438
71
Filtration Rate
D: Stirring Rate
77.375
64.625
44.25
-0.50
1.00
1.00
0.50
0.50
-1.00
-1.00
-0.50
0.00
0.50
0.00
1.00
0.00
D: Stirring Rate-0.50
-0.50
A: Temperature
A: Tem perature
-1.00
-1.00
With concentration at either the low or high level, high temperature and
high stirring rate results in high filtration rates
L. M. Lye
DOE Course
94
The Drilling Experiment
A = drill load, B = flow, C = speed, D = type of mud,
y = advance rate of the drill
L. M. Lye
DOE Course
95
Effect Estimates - The Drilling Experiment
Model
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Term
Intercept
A
B
C
D
AB
AC
AD
BC
BD
CD
ABC
ABD
ACD
BCD
ABCD
Effect
SumSqr % Contribution
0.9175
6.4375
3.2925
2.29
0.59
0.155
0.8375
1.51
1.5925
0.4475
0.1625
0.76
0.585
0.175
0.5425
3.36722
165.766
43.3622
20.9764
1.3924
0.0961
2.80563
9.1204
10.1442
0.801025
0.105625
2.3104
1.3689
0.1225
1.17722
Lenth's ME
Lenth's SME
L. M. Lye
DOE Course
1.28072
63.0489
16.4928
7.97837
0.529599
0.0365516
1.06712
3.46894
3.85835
0.30467
0.0401744
0.87876
0.520661
0.0465928
0.447757
2.27496
4.61851
96
Half-Normal Probability Plot of Effects
DESIGN-EXPERT Plot
adv._rate
A:
B:
C:
D:
load
flow
speed
mud
Half Normal plot
99
97
B
Half Norm al % probability
95
90
C
85
D
80
BD
BC
70
60
40
20
0
0.00
1.61
3.22
4.83
6.44
|Effect|
L. M. Lye
DOE Course
97
Residual Plots
DESIGN-EXPERT Plot
Normal plot of residuals
adv._rate
PERT Plot
Residuals vs. Predicted
2.58625
99
95
1.44875
80
70
Res iduals
Norm al % probability
90
50
0.31125
30
20
10
-0.82625
5
1
-1.96375
-1.96375
-0.82625
0.31125
1.44875
2.58625
1.69
Res idual
L. M. Lye
4.70
7.70
10.71
13.71
Predicted
DOE Course
98
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
y y
*

• Transformations are typically performed to
– Stabilize variance
– Induce normality
– Simplify the model
L. M. Lye
DOE Course
99
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
L. M. Lye
DOE Course
100
The Box-Cox Method
DESIGN-EXPERT Plot
adv._rate
Box-Cox Plot for Power Transforms
A log transformation is
recommended
6.85
Lambda
Current = 1
Best = -0.23
Low C.I. = -0.79
High C.I. = 0.32
The procedure provides a
confidence interval on
the transformation
parameter lambda
5.40
Ln(Res idualSS)
Recommend transform:
Log
(Lambda = 0)
3.95
If unity is included in the
confidence interval, no
transformation would be
needed
2.50
1.05
-3
-2
-1
0
1
2
3
Lam bda
L. M. Lye
DOE Course
101
Effect Estimates Following the
Log Transformation
DESIGN-EXPERT Plot
Ln(adv._rate)
load
flow
speed
mud
99
97
B
Three main effects are
large
95
Half Norm al % probability
A:
B:
C:
D:
Half Normal plot
90
No indication of large
interaction effects
C
85
D
80
70
What happened to the
interactions?
60
40
20
0
0.00
0.29
0.58
0.87
1.16
|Effect|
L. M. Lye
DOE Course
102
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 > F
Model
7.11
3
2.37
164.82 < 0.0001
B
5.35
1
5.35
371.49 < 0.0001
C
1.34
1
1.34
93.05
< 0.0001
D
0.43
1
0.43
29.92
0.0001
Residual 0.17
12
0.014
Cor Total 7.29
15
L. M. Lye
Std. Dev. 0.12
Mean
1.60
C.V.
7.51
R-Squared
Adj R-Squared
Pred R-Squared
0.9763
0.9704
0.9579
PRESS
Adeq Precision
34.391
0.31
DOE Course
103
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
L. M. Lye
DOE Course
104
Following the Log Transformation
DESIGN-EXPERT Plot
Ln(adv._rate)
Normal plot of residuals
DESIGN-EXPERT Plot
Ln(adv._rate)
Residuals vs. Predicted
0.194177
99
95
0.104087
80
70
Res iduals
Norm al % probability
90
50
0.0139965
30
20
10
-0.0760939
5
1
-0.166184
-0.166184
-0.0760939
0.0139965
0.104087
0.194177
Res idual
L. M. Lye
0.57
1.08
1.60
2.11
Predicted
DOE Course
105
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
L. M. Lye
DOE Course
106
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
L. M. Lye
DOE Course
107
Design and Analysis of
Multi-Factored Experiments
Center points
L. M. Lye
DOE Course
108
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:
k
k
k
First-order model (interaction) y  b 0   b i xi   b ij xi x j  
i 1
k
k
i 1 j i
k
k
Second-order model y  b 0   b i xi   b ij xi x j   b ii xi2  
i 1
L. M. Lye
DOE Course
i 1 j i
i 1
109
yF  yC  no "curvature"
The hypotheses are:
k
H 0 :  b ii  0
i 1
k
H1 :  b ii  0
i 1
SSPure Quad
nF nC ( yF  yC )2

nF  nC
This sum of squares has a
single degree of freedom
L. M. Lye
DOE Course
110
Example
nC  5
Usually between 3
and 6 center points
will work well
Design-Expert
provides the analysis,
including the F-test
for pure quadratic
curvature
L. M. Lye
DOE Course
111
ANOVA for Example
Response: yield
ANOVA for Selected Factorial Model
Analysis of variance table [Partial sum of squares]
L. M. Lye
Source
Model
A
B
AB
Curvature
Pure Error
Cor Total
Sum of
Squares
2.83
2.40
0.42
2.500E-003
2.722E-003
0.17
3.00
Std. Dev.
Mean
0.21
40.44
R-Squared
Adj R-Squared
C.V.
0.51
Pred R-Squared
N/A
PRESS
N/A
Adeq Precision
14.234
DF
3
1
1
1
1
4
8
Mean
Square
0.94
2.40
0.42
2.500E-003
2.722E-003
0.043
DOE Course
F
Value
21.92
55.87
9.83
0.058
0.063
Prob > F
0.0060
0.0017
0.0350
0.8213
0.8137
0.9427
0.8996
112
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
L. M. Lye
DOE Course
113
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. Lye
DOE Course
114