Applied Statistics Lesson of the Day – Notation for Fractional Factorial Designs

Fractional factorial designs use the L^{F-p} notation; unfortunately, this notation is not clearly explained in most textbooks or web sites about experimental design.  I hope that my explanation below is useful.

  • L is the number of levels in each factor; note that the L^{F-p} notation assumes that all factors have the same number of levels.
    • If a factor has 2 levels, then the levels are usually coded as +1 and -1.
    • If a factor has 3 levels, then the levels are usually coded as +1, 0, and -1.
  • F is the number of factors in the experiment
  • p is the number of times that the full factorial design is fractionated by L.  This number is badly explained by most textbooks and web sites that I have seen, because they simply say that p is the fraction – this is confusing, because a fraction has a numerator and a denominator, and p is just 1 number.  To clarify,
    • the fraction is L^{-p}
    • the number of treatments in the fractional factorial design is L^{-p} multiplied by the total possible number of treatments in the full factorial design, which is L^F.

If all L^F possible treatments are used in the experiment, then a full factorial design is used.  If a fractional factorial design is used instead, then L^{-p} denotes the fraction of the L^F treatments that is used.

Most factorial experiments use binary factors (i.e. factors with 2 levels, L = 2).  Thus,

  • if p = 1, then the fraction of treatments that is used is 2^{-1} = 1/2.
  • if p = 2, then the fraction of treatments that is used is 2^{-2} = 1/4.

This is why

  • a 2^{F-1} design is often called a half-fraction design.
  • a 2^{F-2} design is often called a quarter-fraction design.

However, most sources that I have read do not bother to mention that L can be greater than 2; experiments with 3-level factors are less frequent but still common.  Thus, the terms half-fraction design and half-quarter design only apply to binary factors.  If L = 3, then

  • a 3^{F-1} design uses one-third of all possible treatments.
  • a 3^{F-2} design uses one-ninth of all possible treatments.