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TABLE 28.7
4
Two Half-Fractions of the Full 2 Factorial Design
Factor
1 2 3 4
I - Run 1 − − − −
II + – − −
II − + − −
I - Run 4 + + − −
II − − + −
I - Run 6 + − + −
I - Run 7 − + + −
II + + + −
II − − − +
I - Run 2 + − − +
I - Run 3 − + − +
II + + − +
I - Run 5 − − + +
II + − + +
II − + + +
I - Run 8 + + + +
Note: The eight settings opposite I - Run 1, I - Run 2, etc. were used in the
4−1
2 fractional design of the case study. The other eight combinations, marked
4−1
II, are a second 2 fractional design that could be run as a follow-up.
Fractional factorial designs are often used in an iterative experimental strategy. Chapter 29 illustrates
7−4
5−1
a 2 design for evaluating five factors in 16 runs. Box et al. (1978) give two examples of 2 designs
3
for evaluating eight variables in 2 = 8 runs. They also show other fractional factorial designs, their
confounding pattern, and give a detailed explanation of how the confounding pattern is discovered.

References
k−p
Box, G. E. P. and J. S. Hunter (1961a). “The 2 Fractional Factorial Designs. Part I,” Technometrics, 3(3),
311–351.
k−p
Box, G. E. P. and J. S. Hunter (1961b). “The 2 Fractional Factorial Designs. Part II,” Technometrics, 3(4),
449–458.
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.
Tiao, George et al., Eds. (2000). Box on Quality and Discovery with Design, Control, and Robustness, New York,
John Wiley & Sons.

Exercises

28.1 Membrane Bioreactor. The ZeeWeed internal-membrane reactor was tested using a 2 4−1
fractional factorial design to evaluate four factors in eight runs, as shown in the table below.
The response is the permeate flow rate. Evaluate the main effects and interactions.

Backpulse Backpulse Membrane Permeate
Duration Frequency Airflow MLSS Flow Rate
3
(min) (min) (m /hr) (mg/L) (L/min)
−1 −1 −1 −1 83
+1 −1 −1 +1 72
−1 +1 −1 +1 64
+1 +1 −1 −1 72
© 2002 By CRC Press LLC