9. Two-Level Factorial Designs

We now change our focus from the number of factors in the experiment to the number of levels those factors have. Specifically, in this and the next several chapters, we consider designs in which all factors have two levels. Many experiments are of this type. This is because two is the minimum number of levels a factor can have and still be studied, and by having the minimum number of levels (2), an experiment of a certain size can include the maximum number of factors. After all, an experiment with five factors at two levels each contains 32 combinations of levels of factors (2⁵), whereas an experiment with these same five factors at just one more level, three levels, contains 243 combinations of levels of factors (3⁵) – about eight times as many combinations! Indeed, studying five factors at three levels each (3⁵ = 243 combinations) requires about the same number of combinations as are needed to study eight factors at two levels each (2⁸ = 256). As we shall see in subsequent chapters, however, one does not always carry out (that is, “run”) each possible combination; nevertheless, the principle that fewer levels per factor allows a larger number of factors to be studied still holds.