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(+) 44.9 50.3 (+) 45. 9 50.6
Input Flow
Location Rate
(–) 44.25 45.1 (–) 43.9 44.1
(–) Stirring (+) (–) Dilution (+)
Ratio

FIGURE 28.3 Two-factor interactions of S and L, and of F and D.

averaging the positive and negative effects) and looking at it alone might lead us to wrongly conclude
that the factor is inactive.
In this case study, the SL + DF interaction effect appears large enough to be real. Interpreting the SL
interaction is done by viewing the experiment as one conducted in the interacting factors only. This is
shown in Figure 28.3, where the responses averaged over all runs with the same signs for L and S. The
same examination must be made of the DF interaction, which is also shown in Figure 28.3.
The sample flow rate is not very important at the low dilution ratio and dilution ratio is not very
important at the low sample flow rate. But when the sample flow rate is high and the dilution ratio is
large, the response increases dramatically. A similar interpretation can be made for the SL interaction.
Stirring in conjunction with injecting the sample at the bottom gives higher oxygen measurements while
the other three combinations of L and S show much the same DO levels.
In short, the significant two-factor interaction means that the effect of any factor depends on the level
of some other factor. It may be, however, that only one of the interactions SL or DF is real and the other
is an artifact caused by the confounding of the two interactions. Additional experiments could be done
4−1
to untangle this indefinite situation. One option would be to run the other half of the 2 design.
For the case study problem, a resolution of the two-factor interaction is not needed because all four factors
S, D, L, and F do influence the oxygen measurements and it is clear that all four should be set at their +
levels. That is, the best of the measurement techniques investigated would be to inject the sample at the
bottom of the bottle, stir, use the 4:1 dilution ratio, and use the high sample flow rate (8.2 mL/sec). Using
even higher dilution ratios, faster stirring, and a higher sample flow rate might yield even better results.

Comments
This example has shown how four variables can be evaluated with just eight runs. The eight runs were
a half-fraction of the sixteen runs that would be used to evaluate four factors in a full factorial design.
This design provided estimates of the main effects of the four factors just as would have been obtained
4
from the full 2 design, if we are willing to make the rather mild assumption that three-factor interactions
are negligible.
There was a price paid for forcing the extra factor into the eight-run design. Only a total of eight
effects and interactions can be estimated from the eight runs. These include the average and the four
main effects. The estimated two-factor interactions are confounded pairwise with each other and their
interpretation is not as clear as it would have been from the full factorial design.
Often we are interested primarily in the main effects, at least in the early iterations of the learning
cycle. If the main effects are significant and some of the two-factor interactions hold interest, additional
runs could be performed to untangle the interpretation. The most logical follow-up experiment would
4−1
4
be to run the other half of the 2 design. Table 28.7 shows how the full 2 design was divided into two
half-fractions, one of which was used in the case study experiment, and the other which could be used
4
as a follow-up. The two half-fractions combined are a saturated 2 design from which all main effects
and all interactions can be estimated independently (i.e., without confounding). A further advantage is
that the two half-fractions are blocks and any nuisance factors that enter the experiment between the
execution of the half-fractions will not distort the final results. (This is not obvious. For details, see Box
et al., 1978.)
© 2002 By CRC Press LLC