Slide set 10
Stat402 (Spring 2016)
Last update: March 22, 2016
Stat 402 (Spring 2016): Slide set 10
The 23 Factorial
The Treatment combinations are:
(1), a, b, ab, c, ac, bc, and abc.
The effects A, B, C, AB, BC, AC, ABC are difined as in the 22 case.
Example
Simple Effects of A are: a-(1), ab-b, ac-c, abc-bc
Main Effects of Factor A is the average of these:
A = 1/4[(a − (1)) + (ab − b) + (ac − c) + (abc − bc)]
= 1/4[abc + ab + ac + a − b − c − bc − (1)]
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Stat 402 (Spring 2016): Slide set 10
Algebraic Identities for the 23 case
•
Contrasts for main effects and interactions:
A
B
C
AB
•
:
:
:
:
(a-1)(b+1)(c+1)
(a+1)(b-1)(c+1)
(a+1)(b+1)(c-1)
(a-1)(b-1)(c+1)
=
=
=
=
abc+ab+ac+a-bc-b-c-(1)
abc+ab+bc+b-ac-a-c-(1)
abc+bc+ac+c-ab-a-b-(1)
abc+ab+c+(1)-ac-bc-a-b
The interaction effect AB is the average of the two interaction effects at
each level of C
Effect AB at level 0 of C: 1/2[(ab-b)-(a-(1))]
Effect AB at level 1 of C: 1/2[(abc-bc)-(ac-c)]
AB
=
1/2{1/2[ab − b − a + (1)] + 1/2[abc − bc − ac + c]}
=
1/4[abc + ab + c + (1) − ac − bc − a − b]
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Stat 402 (Spring 2016): Slide set 10
Algebraic Identities for the 23 case (Cont’d)
The same procedure is used to define the AC and BC effects. These
effects are called 2-factor or 1st order interactions (usually shortened to
2-fi’s).
•
Three-factor interactions (3-fi’s): It is defined as the average difference
between the interaction effects of AB at each level of C:
ABC
=
1/2{1/2[abc − bc − ac + c] − 1/2[ab − b − a + (1)]}
=
1/4[abc + a + b + c − ab − ac − bc − (1)]
The signs for the above contrast can be obtained from the identity
(a-1)(b-1)(c-1).
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Stat 402 (Spring 2016): Slide set 10
Contrasts for all effects in the 23 factorial
In the 23 factorial, the set of defining contrasts are:
Treatment
Combinations
(1)
a
b
ab
c
ac
bc
abc
I
+
+
+
+
+
+
+
+
A
+
+
+
+
B
+
+
+
+
Factorial Effect
AB C AC BC
+
+
+
+
+
+
+ +
+ +
+
+
+ + +
+
ABC
+
+
+
+
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Stat 402 (Spring 2016): Slide set 10
Notes
1. Each effect is a contrast of the observations.
2. These are orthogonal contrasts implying that the corresponding effects
are independent.
3. In a 23 experiment, there are 8 different treatment combinations and
hence there are 7 d.f. for treatment. What we have done so far is to
partition these 7 d.f. to 7 orthogonal single d.f. contrasts.
4. In general, in a 2k experiment we can find (2k -1) orthogonal contrasts
to partition the treatment sum of squares.
5. See Example 6.1 in Section 6.3 of Montgomery.
6. The general 2k factorial is discussed in Section 6.4
7. The unreplicated 2k factorial is discussed in Section 6.5.
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Stat 402 (Spring 2016): Slide set 10
An Example of a 23 Factorial: Plasma Etch Experiment
A 23 factorial design was used to develop a nitride etch process on a
single-wafer plasma etching tool. There are three quantitative factors-gap
between electrodes, gas flow and the RF power applied to the cathode. The
response is the etch rate for silicon nitride. Each treatment combination
was replicated twice and the 16 experimental runs were done in completely
random order.
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Stat 402 (Spring 2016): Slide set 10
Example 6.1 (Cont’d)
•
•
•
Since this is a completely randomized design with 2 replications per
treatment combination, the standard analysis is carried out as follows:
The model is yijkl = µijk + ijkl, i = 1, 2; j = 1, 2; k = 1, 2; l = 1, 2
iid
where ijkl ∼ N (0, σ 2).
The expected response from the treatment combination ijk is µijk which
may be expressed using factorial effects as
µijk = µ + τi + βj + (τ β)ij + γk + (τ γ)ik + (βγ)jk + (τ βγ)ijk
The ANOVA table (from JMP) is
SV
Trt
Error
Total
DF
7
8
15
SS
513,400.4375
18,020.5000
531,420.9347
MS
73342.9196
2
sE =2252.5625
F
32.56
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Stat 402 (Spring 2016): Slide set 10
Example 6.1 (Cont’d)
Now the 7 d.f. for Treatment Sum of Squares is partitioned into 7 single
degree of d.f. sums of squares corresponding to the 7 factorial effects:
Treatment
Combination
(1)
a
b
ab
c
ac
bc
abc
Contrast
Divisor for Estimate
Estimate of Effect
Divisor for SS
SSA =
SSC =
Observed
Total
1154
1319
1234
1277
2089
1617
2138
1589
I
+
+
+
+
+
+
+
+
12,417
16
776.0625
16
(−813)2
= 41, 310.5625
16
(2449)2
SSBC =
16
A
+
+
+
+
-813
8
-101.625
16
SSB =
(59)2
16 = 217.5625
(−1229)2
= 374, 850.0625 SSAC =
(−17)2
= 18.0625
16
SSABC =
B
+
+
+
+
59
8
7.375
16
Factorial Effect
AB
C
+
+
+
+
+
+
+
+
-199
2449
8
8
-24.875
306.125
16
16
16
SSAB =
AC
+
+
+
+
-1229
8
-153.625
16
BC
+
+
+
+
-17
8
-2.125
16
ABC
+
+
+
+
45
8
5.625
16
(−199)2
= 2475.0625
16
= 94, 402.5625
(45)2
16 = 126.5625
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Stat 402 (Spring 2016): Slide set 10
Example 6.1 (Cont’d)
This results in the summary and ANOVA tables:
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Stat 402 (Spring 2016): Slide set 10
Discussion
1. Since the above experiment was a replicated experiment it was possible to construct
the above analysis of variance table for testing hypotheses of interest.
2. Many factorial experiments in industry are not replicated experiments, since even for
a moderate number of factors the total number of treatment combinations in a 2k
factorial is large. Thus limited availability of resources or high costs may inhibit
experimenters from replicating factorial experiments.
3. The result of not replicating an experiment is that there will be no d.f. available for
estimating the error variance on; hence the anova table cannot be used for testing
hypotheses concerning main effects and interactions.
4. Other methods have to be devised to determine the effects that are significant from
such experiments. See Section 6.5 for a discussion of the analysis of a single replicate
of a 2k factorial experiment.
5. When an estimate of the error variance is available, as part of the analysis the estimates
of effects are usually reported (instead of an analysis of variance table) as follows:
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Stat 402 (Spring 2016): Slide set 10
Estimating Effects
Estimating of Effects
from the Plasma-Etch Example
√
2×sE
s.e.(E)= √
n×2k
= 2 √2252.56
= 23.73; t0.025,8 = 2.31 & 2.31 × 23.73 = 54.82
3
2(2 )
Thus approximate 95% CI’s are:
Effects
Main Effects
Gap(A)
C2F6 Flow (B)
Power(C)
Two-factor Interactions
AB
AC
BC
Three-factor Interactions
ABC
Estimate(E) ± t-value × s.e.(E)
-101.625±54.82
7.375±54.82
306.125±54.82
-24.875 ±54.82
-153.625±54.82
-2.125±54.82
5.625 ±54.82
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Stat 402 (Spring 2016): Slide set 10
Discussion
•
•
•
•
Comparison of the estimates with their standard errors suggests that the bold items
A, C and the two-factor interaction AC require interpretation, while the remaining
effects could be due to noise or error.
The main effect of a factor should be individually interpreted only if there is no evidence
that the variable interacts with other variables. When there is evidence of one or more
such interaction effects, the interacting variables should be considered jointly.
In the above table there are large Gap (A) and Power (C ) effects, but since a large
AC effect indicates interaction between the two, we make no statement of Gap (A)
and Power (C ) effects alone.
The effects of Gap (A) and Power (C) can best be considered using the two-way table
of means shown below and the corresponding graph shown next page:
Power (C)
275
325
Gap (A)
0.8
1.2
597.00
649.00
1059.75 801.50
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Stat 402 (Spring 2016): Slide set 10
Discussion (contd.)
Power
6
375
275
1056.75
801.5
597.0
649.0
-
0.8
•
Gap
1.2
The AC interaction evidently arises due to a difference in effect of the change in power
at the two different gaps. With Gap=0.8 the effect of temperature is to increase mean
etch rate by almost 460 but with Gap=1.2, the mean increases by only 152.
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