Page 264 - Statistics for Environmental Engineers

L1592_Frame_C29 Page 267 Tuesday, December 18, 2001 2:48 PM

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Fractional factorial experiments offer an efficient way of evaluating a large number of variables with a
reasonable number of experimental runs. In this example we have evaluated the importance of five
5
variables in 16 runs. This was a half-fraction of a full 2 = 32-run factorial experiment. Recalling the
adage that there is “no free lunch,” we wonder what was given up in order to study five variables with
just 16 runs. In the case study, the main effect of each factor was confounded with a four-factor interaction,
and each two-factor interaction was confounded with a three-factor interaction. If the higher-order inter-
actions are small, which is expected, the design produces excellent estimates of the main effects and
identifies the most important factors. In other words, we did have a free lunch.
In the case study, all interactions and three main effects were insignificant. This means that the
experiment can be interpreted by collapsing the design onto the two significant factors. Figure 29.3
shows permeability in terms of factors 1 and 2, which now appear to have been replicated four times.
You can confirm that the main effects calculated from this view of the experiment are exactly as obtained
from the previous analysis.
This gain in apparent replication is common in screening experiments. It is one reason they are so
efficient, despite the confounding that the inexperienced designer fears will weaken the experiment. To
appreciate this, suppose that three factors had been significant in the case study. Now the collapsed design
3
is equivalent to a 2 design that is replicated twice at each condition. Or suppose that we had been even
more ambitious with the fractional design and had investigated the five factors in just eight runs with a 2 5−2
2
design. If only two factors proved to be significant, the collapsed design would be equivalent to a 2
experiment replicated twice at each condition.
Finding an insignificant factor in a fractional factorial experiment always has the effect of creating
apparent replication. Screening experiments are designed with the expectation that some factors will be
inactive. Therefore, confounding usually produces a bonus instead of a penalty. This is not the case in
an experiment where all factors are known to be important, that is, in an experiment where the objective
is to model the effect of changing prescreened variables.
Table 29.6 summarizes some other fractional factorial designs. It shows five designs that use only
eight runs. In eight runs we can evaluate three or four factors and get independent estimates of the main
effects. If we try to evaluate five, six, or seven factors in just eight runs, the main effects will be
confounded with second-order interactions. This can often be an efficient design for a screening exper-
iment. The table also shows seven designs that use sixteen runs. These can handle four or five factors
without confounding the main effects. Lack of confounding with three-factor interactions (or higher) is
indicated by “OK” in the last two columns of the table, while “Confounded” indicates that the mentioned
effect is confounded with at least one second-order interaction. 4.654

Factor 2 - Percentage of Fly Ash Ave. = 6.546 Ave. = 4.039
7.307
5.858
4.007
100%
6.413
3.689
6.607
3.807

6.932
5.247
7.265
6.363
6.016
50%
7.258
7.948
5.273
Ave. = 7.351
A Ave. = 5.725
B
Factor 1 - Type of Fly Ash
FIGURE 29.3 The experimental results shown in terms of the two significant main effects. Because three main effects
2
are not significant, the fractional design is equivalent to a 2 design replicated four times.
© 2002 By CRC Press LLC