A - Temperature, B - Pressure, C - Concentration of Formaldehyde, D - Stirring Rate
4 quantitative variables:
Pilot Plant Filtration Experiment
Response - Filtration Rate
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Stat 402 (Spring 2016): slide set 11
Single Replicate of a 2 Factorial: Example
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Last update: March 28, 2016
Stat402 (Spring 2016)
Slide set 11
Stat 402 (Spring 2016): slide set 11
For example, a 26 factorial has 64 treatment combinations. So if you
have 3 replications, 64 × 3 = 192 runs will be needed.
Because of limitations of resources (experimental material, equipment,
labor, time or cost) only a single replicate may be possible, unless the
experiment is made smaller by eliminating some of the factors from the
study.
A major problem with running a single replicate is that there are no
degrees of freedom left to estimate the error (or noise) directly.
One approach to alleviate this is to assume that certain higher-order
interactions are negligible and combine their mean squares to estimate
the error.
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(1), a, b, ab, c, ac, ab, abc, d, ad, bd, abd, cd, acd, abd, abcd
The 16 treatment combinations are:
Contrasts for Effects of a 24
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Stat 402 (Spring 2016): slide set 11
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For even a small k (number of factors), the number of treatment
combinations, and therefore the number of runs needed is large.
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Single Replicate of the 2 Factorial
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Stat 402 (Spring 2016): slide set 11
The main effects A, C, and D are plotted in the the figure below:
All three effects are positive. If the aim is to maximize filtration rate, we
would run the three factors A, C, and D at the high levels.
However, we need to consider any important interactions. The AC and
AD interactions are shown in the figure below:
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From Fig. 6.1, it can be observed that the important effects are the
main effects of A, C, and D and the AC and the AD interactions.
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From the AD interaction plot: Stirring rate (D) effect is small at low
Temperature (A) but large and positive at highTemperature (A).
Conclusion: Best filtration rate is obtained when A and D at the high
levels and C is at the low level. This will also keep the formaldehyde
concentration C to a lower level (another objective).
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From the AC interaction plot: Temperature (A) effect is small when
Concentration (C) is at the high level but is very high when Concentration
(C) is at the low level.
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Stat 402 (Spring 2016): slide set 11
Stat 402 (Spring 2016): slide set 11
Interpretation of the important effects
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Fig. 6.11 shows the normal probabilty plot of the effects. The straight line drawn goes
through majority of points and the point (0, 50 percentile), and thus thus capture most of
the effects that might be considered non-significant (i.e., close to zero).
Inference from unreplicated factorials
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The 15 factorial effects estmates and sums of squares are:
Estimates and SS for Effects of a 2
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Stat 402 (Spring 2016): slide set 11
Stat 402 (Spring 2016): slide set 11
So the experiment may be regarded as a 23 factorial in A, C, and D with
two replicates (i.e., the two responses observed at each level of B). To
see this, examine Table 6.10 ignoring the column for factor B.
We say that an unreplicated 24 is projected into a replicated 23. The
implication of this is that we now get an estimate of error with 8 degrees
of freedom.
See Table 6.13 for the ANOVA table for the prejected experiment.
This method of analysis is also called pseudo-replication.
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Because Pressure (B) is not important and all interactions involving B
are negligible, we may act as if B was not a factor in the experiment.
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Another analysis: design projection
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Stat 402 (Spring 2016): slide set 11