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426 Chapter Twelve

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factorial experiment. There will be abc n total number of trials if
there are n replicates. Clearly, the number of trials needed to run
the experiment will increase very rapidly with increase in the num-
ber of factors and the number of levels. In practical applications, we
rarely use general full factorial experiments for more than two fac-
tors; two-level factorial experiments are the most popular experi-
mental methods.

12.3 Two-Level Full Factorial Designs
The most popular experimental designs are two-level factorial designs,
factorial designs in which all factors have exactly two levels. These
designs are the most popular designs because

1. Two levels for each factor will lead to factorial designs with the
least number of runs, so they will lead to the most economical
experiments.
2. Two-level factorial designs will be ideal designs for screening
experiments.
3. Two-level factorial designs are the basis of the fractional two-level
factorial designs. They are the most flexible, efficient, and economi-
cal experimental designs. In practical applications of DOE, frac-
tional factorial designs are the most frequently used designs.

A two-level full factorial design is also called a “2 design,” where k
k
is the number of experimental factors and 2 means two levels. This
is because the number of treatment combinations in a two-level full
...
factorial of k factors is 2
2 2 2 . If there are n replicates at
k
each treatment combination, then the total number of experimental
k
trials is 2 n. Because there are only two levels for each factor, we
call the level with low setting the low level, and the level with the
high setting, the high level. For example, if a factor is temperature,
with two levels, 100 and 200 C, then 100 C is the low level and
200 C is the high level.

12.3.1 Notation for two-level designs
The standard layout for a two-level design uses 1 and 1 notation to
denote the “high level” and the “low level,” respectively, for each factor.
For example, the following matrix describes an experiment in which
four trials (or runs) were conducted with each factor set to high or low
during a run according to whether the matrix had a 1 or 1 set for
the factor during that trial: