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The final interpretation of the results is:

1. The average density over the eight experimental design conditions is 114.7.
3
2. Increasing water content from 4 to 10% increases the density by an average of 12.45 lb/ft .
3
3. Increasing compaction effort from 60 to 260 psi increases density by an average of 6.40 lb/ft .
3
4. Increasing reaction time from 5 to 20 min decreases density by an average of 7.50 lb/ft .
5. These main effects are additive because the interactions are zero. Therefore, increasing both
water content and compaction effort from their low to high values increases density by 12.45 +
3
6.40 = 18.85 lb/ft .

Comments
Two-level factorial experiments are a way of investigating a large number of variables with a minimum
k
number of experiments. In general, a k variable two-level factorial experiment will require 2 experimental
2
3
runs. A 2 experiment evaluates two variables in four runs, a 2 experiment evaluates three variables in
4
eight runs, a 2 design evaluates four variables in sixteen runs, etc. The designs are said to be full or
saturated. From this small number of runs it is possible to estimate the average level of the response, k
main effects, all two-factor interactions, and all higher-order interactions. Furthermore, these main effects
and interactions are estimated independently of each other. Each main effect independently estimates
the change associated with one experimental factor, and only one.
Why do so few experimental runs provide so much information? The strength and beauty of this design
arise from its economy and balance. Each data point does triple duty (at least) in estimating main effects.
Each observation is used in the computation of each factor main effect and each interaction. Main effects
are averaged over more than one setting of the companion variables. This is the result of varying all
experimental factors simultaneously. One-factor-at-a-time (OFAT) designs have none of this efficiency or
power. An OFAT design in eight runs would provide only estimates of the main effects (no interactions)
and the estimates of the main effects would be inferior to those of the two-level factorial design.
The statistical significance of the estimated effects can be evaluated by making the normal plot. If the
effects represent only random variation, they will plot as a straight line. If a factor has caused an effect
to be larger than expected due to random error alone, the effect will not fall on a straight line. Effects
of this kind are interpreted as being significant. Another way to evaluate significance is to compute a
confidence interval, or a reference distribution. This is shown in Chapter 28.
Factorial designs should be the backbone of an experimenter’s design strategy. Chapter 28 shows how
four factors can be evaluated with only eight runs. Experimental designs of this kind are called fractional
factorials. Chapter 29 extends this idea. In Chapter 30 we show how the effects are estimated by linear
algebra or regression, which is more convenient in larger designs and in experiments where the inde-
pendent variables have not been set exactly according to the orthogonal design. Chapter 43 explains
how factorial designs can be used sequentially to explore a process and optimize its performance.

References
Box, G. E. P., W. G. Hunter, and J. S. Hunter (1978). Statistics for Experimenters: An Introduction to Design,
Data Analysis, and Model Building, New York, Wiley Interscience.
Box, G. E. P. and N. R. Draper (1987). Empirical Model Building and Response Surfaces, New York, John
Wiley.
Davies, O. L. (1960). Design and Analysis of Industrial Experiments, New York, Hafner Co.
Edil, T. B., P. M. Berthouex, and K. Vesperman (1987). “Fly Ash as a Potential Waste Liner,” Proc. Conf.
Geotechnical Practice in Waste Disposal, Geotech. Spec. Pub. No. 13, ASCE, pp. 447–461.
Tiao, George et al., Eds. (2000). Box on Quality and Discovery with Design, Control, and Robustness, New York,
John Wiley & Sons.
© 2002 By CRC Press LLC