•
•
Stat 402 (Spring 2016): Slide set 10
(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
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]
=
=
Effect AB at level 1 of C: 1/2[(abc-bc)-(ac-c)]
Effect AB at level 0 of C: 1/2[(ab-b)-(a-(1))]
2
The interaction effect AB is the average of the two interaction effects at
each level of C
:
:
:
:
Contrasts for main effects and interactions:
A
B
C
AB
3
Algebraic Identities for the 2 case
Last update: March 22, 2016
Stat402 (Spring 2016)
Slide set 10
•
Stat 402 (Spring 2016): Slide set 10
1/2{1/2[abc − bc − ac + c] − 1/2[ab − b − a + (1)]}
1/4[abc + a + b + c − ab − ac − bc − (1)]
=
=
3
The signs for the above contrast can be obtained from the identity
(a-1)(b-1)(c-1).
ABC
Three-factor interactions (3-fi’s): It is defined as the average difference
between the interaction effects of AB at each level of C:
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).
Algebraic Identities for the 2 case (Cont’d)
3
A = 1/4[(a − (1)) + (ab − b) + (ac − c) + (abc − bc)]
= 1/4[abc + ab + ac + a − b − c − bc − (1)]
Main Effects of Factor A is the average of these:
Example
Simple Effects of A are: a-(1), ab-b, ac-c, abc-bc
1
Stat 402 (Spring 2016): Slide set 10
The effects A, B, C, AB, BC, AC, ABC are difined as in the 22 case.
(1), a, b, ab, c, ac, bc, and abc.
The Treatment combinations are:
The 2 Factorial
3
Stat 402 (Spring 2016): Slide set 10
3
I
+
+
+
+
+
+
+
+
A
+
+
+
+
B
+
+
+
+
6
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.
•
•
•
SV
Trt
Error
Total
DF
7
8
15
SS
513,400.4375
18,020.5000
531,420.9347
MS
73342.9196
s2E =2252.5625
F
32.56
μijk = μ + τi + βj + (τ β)ij + γk + (τ γ)ik + (βγ)jk + (τ βγ)ijk
The ANOVA table (from JMP) is
7
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
Example 6.1 (Cont’d)
Stat 402 (Spring 2016): Slide set 10
Stat 402 (Spring 2016): Slide set 10
7. The unreplicated 2k factorial is discussed in Section 6.5.
6. The general 2k factorial is discussed in Section 6.4
5. See Example 6.1 in Section 6.3 of Montgomery.
4. In general, in a 2k experiment we can find (2k -1) orthogonal contrasts
to partition the treatment sum of squares.
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.
5
ABC
+
+
+
+
Stat 402 (Spring 2016): Slide set 10
2. These are orthogonal contrasts implying that the corresponding effects
are independent.
1. Each effect is a contrast of the observations.
Notes
4
Factorial Effect
AB C AC BC
+
+
+
+
+
+
+ +
+ +
+
+
+ + +
+
An Example of a 2 Factorial: Plasma Etch Experiment
Treatment
Combinations
(1)
a
b
ab
c
ac
bc
abc
In the 23 factorial, the set of defining contrasts are:
Contrasts for all effects in the 2 factorial
3
Stat 402 (Spring 2016): Slide set 10
16
16
B
+
+
+
+
59
8
7.375
16
s.e.(E)= √
2×sE
Estimating Effects
Estimating of Effects
from the Plasma-Etch Example
√
10
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:
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.
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.
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.
2(2 )
Effects
Main Effects
Gap(A)
C2F6 Flow (B)
Power(C)
Two-factor Interactions
AB
AC
BC
Three-factor Interactions
ABC
5.625 ±54.82
-24.875 ±54.82
-153.625±54.82
-2.125±54.82
-101.625±54.82
7.375±54.82
306.125±54.82
Estimate(E) ± t-value × s.e.(E)
= 2 √2252.56
= 23.73; t0.025,8 = 2.31 & 2.31 × 23.73 = 54.82
3
Thus approximate 95% CI’s are:
n×2k
11
Stat 402 (Spring 2016): Slide set 10
Stat 402 (Spring 2016): Slide set 10
Stat 402 (Spring 2016): Slide set 10
9
ABC
+
+
+
+
45
8
5.625
16
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.
Discussion
BC
+
+
+
+
-17
8
-2.125
16
(−199)2
= 2475.0625
16
AC
+
+
+
+
-1229
8
-153.625
16
This results in the summary and ANOVA tables:
Example 6.1 (Cont’d)
8
= 94, 402.5625
(45)2
SSABC = 16 = 126.5625
= 374, 850.0625 SSAC =
SSAB =
Factorial Effect
AB
C
+
+
+
+
+
+
+
+
-199
2449
8
8
-24.875
306.125
16
16
(59)2
16 = 217.5625
(−1229)2
A
+
+
+
+
-813
8
-101.625
16
SSB =
I
+
+
+
+
+
+
+
+
12,417
16
776.0625
16
(−813)2
= 41, 310.5625
16
(2449)2
Observed
Total
1154
1319
1234
1277
2089
1617
2138
1589
(−17)2
SSBC = 16 = 18.0625
SSC =
SSA =
Treatment
Combination
(1)
a
b
ab
c
ac
bc
abc
Contrast
Divisor for Estimate
Estimate of Effect
Divisor for SS
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:
Example 6.1 (Cont’d)
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Stat 402 (Spring 2016): Slide set 10
275
325
Power (C)
Gap (A)
0.8
1.2
597.00
649.00
1059.75 801.50
12
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:
Discussion
•
6
0.8
597.0
1056.75
1.2
649.0
801.5
-
Gap
Stat 402 (Spring 2016): Slide set 10
13
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.
275
375
Power
Discussion (contd.)