# 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.