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

April 22, 2014 Leave a comment

Fractional factorial designs use the 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.

- is the
**number of levels**in each factor; note that the notation assumes that all factors have the same number of levels.- If a factor has 2 levels, then the levels are usually coded as and .
- If a factor has 3 levels, then the levels are usually coded as , , and .

- is the
**number of factors**in the experiment - is the number of times that the full factorial design is
**fractionated by**. This number is badly explained by most textbooks and web sites that I have seen, because they simply say that is the fraction – this is confusing, because a fraction has a numerator and a denominator, and is just 1 number. To clarify,- the fraction is
- the number of treatments in the fractional factorial design is multiplied by the total possible number of treatments in the full factorial design, which is .

If all possible treatments are used in the experiment, then a full factorial design is used. **If a fractional factorial design is used instead, then denotes the fraction of the treatments that is used.**

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

- if , then the fraction of treatments that is used is .
- if , then the fraction of treatments that is used is .

This is why

- a design is often called a
**half-fraction**design. - a design is often called a
**quarter-fraction**design.

However, most sources that I have read do not bother to mention that 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 , then

- a design uses one-third of all possible treatments.
- a design uses one-ninth of all possible treatments.

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