Fractional factorial design
[1] This "fraction" of the full design is chosen to reveal the most important information about the system being studied (sparsity-of-effects principle), while significantly reducing the number of runs required.
However, this reduction in runs comes at the cost of potentially more complex analysis, as some effects can become intertwined, making it impossible to isolate their individual influences.
Fractional factorial design was introduced by British statistician David John Finney in 1945, extending previous work by Ronald Fisher on the full factorial experiment at Rothamsted Experimental Station.
[2] Developed originally for agricultural applications, it has since been applied to other areas of engineering, science, and business.
To reduce the number of runs in comparison to a full factorial, the experiments are designed to confound different effects and interactions, so that their impacts cannot be distinguished.
Another consideration is the number of factors, which can significantly change the experimental labor demand.
[4] To save space, the points in a factorial experiment are often abbreviated with strings of plus and minus signs.
Factorial points are typically arranged in a table using Yates’ standard order: 1, a, b, ab, c, ac, bc, abc, which is created when the level of the first factor alternates with each run.
In this case, the defining relation of the fractional design is I = ABD = ACE = BCDE.
The defining relation allows the alias pattern of the design to be determined and includes 2p words.
Notice that in this case, the interaction effects ABD, ACE, and BCDE cannot be studied at all.
To determine how main effect A is confounded, multiply all terms in the defining relation by A:
[4] An important property of a fractional design is its resolution or ability to separate main effects and low-order interactions from one another.
Formally, if the factors are binary then the resolution of the design is the minimum word length in the defining relation excluding (I).
The 25 − 2 design above is resolution III since its defining relation is I = ABD = ACE = BCDE.
Regular designs have run size that equal a power of two, and only full aliasing is present.
Four factors were considered: temperature (A), pressure (B), formaldehyde concentration (C), and stirring rate (D).
The results of that example may be used to simulate a fractional factorial experiment using a half-fraction of the original 24 = 16 run design.
From inspection of the table, there appear to be large effects due to A, C, and D. The coefficient for the AB interaction is quite small.
