# Applied Statistics Lesson of the Day – Fractional Factorial Design and the Sparsity-of-Effects Principle

April 9, 2014 Leave a comment

Consider again an experiment that seeks to determine the causal relationships between **factors** and the **response**, where . Ideally, the sample size is large enough for a full factorial design to be used. However, if the sample size is small and the number of possible treatments is large, then a **fractional factorial** design can be used instead. Such a design assigns the experimental units to a select fraction of the treatments; these treatments are chosen carefully to investigate the most significant causal relationships, while leaving aside the insignificant ones.

When, then, are the significant causal relationships? According to the **sparsity-of-effects principle**, it is unlikely that complex, higher-order effects exist, and that the most important effects are the lower-order effects. Thus, assign the experimental units so that **main (1st-order) effects** and the **2nd-order interaction effects** can be investigated. This may neglect the discovery of a few significant higher-order effects, but that is the compromise that a fractional factorial design makes when the sample size available is low and the number of possible treatments is high.

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