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I was just wondering, as I have been using ordinary least squares regression a lot lately, could anyone explain why a factorial design called a "factorial" design?

What part of the calculation is factorial? The number of regressors doesn't seem to increase in a factorial manor... is this just a name?

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Factorial in factorial design doesn't refer to the mathematical operation $n!$ during the analysis but instead is named for using factors (or categories) in the experiment. In R factors are represented as a vector of integers that are displayed with character values (e.g. some descriptive text). A factor usually comes with multiple levels. For example the factor Fertilizer can have 3 levels, such as QuickGrow, SuperGrow and InsaneYield.

From Wikipedia:

In statistics, a full factorial experiment is an experiment whose design consists of two or more factors, each with discrete possible values or "levels", and whose experimental units take on all possible combinations of these levels across all such factors.

The terms factor (in context of experimental design) as well as factorial design were first mentioned in print by Ronald Fisher (see Wikipedia entry here).

The term "factorial" may not have been used in print before 1935, when Fisher used it in his book The Design of Experiments.

So in order to figure out why Fisher chose the term factorial for factorial designs, we should grab a copy of the book. But I am guessing it has something do with multiplying factor levels in order to get to the total number of treatment combinations in an experiment. But I am not sure.

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    $\begingroup$ Incidentally as you add more factors, the number of combinations grow exponentially, like a factorial $\endgroup$ – qwr Feb 20 '19 at 0:17
  • $\begingroup$ Good point @qwr! The total number of treatment combinations in a factorial experiment can simply be calculated by multiplying the levels of each factor in the experiment, whereas the total number of pairwise comparisons is calculated as $\binom{n}{k}$, which is equivalent to $\frac{n!}{(n-k)!k!}$, i.e. combinations without repetition. $\endgroup$ – Stefan Feb 20 '19 at 2:55
  • $\begingroup$ Hmm... thank you... I am still a little confused though - would these not also be called factors when used in any regression design? What makes the use of factors specific to a factorial design? $\endgroup$ – JDoe2 Feb 20 '19 at 12:08
  • $\begingroup$ @JDoe2 by "any regression design" to you mean continuous predictor/independent variables? Those I would call covariates and not factors. Factors to me are discrete experimental manipulations (categories) for which I want to estimate and compare mean values (and not slopes). $\endgroup$ – Stefan Feb 20 '19 at 17:21
  • $\begingroup$ @JDoe2 reading your comment again I should add, yes factors are also called factors in other designs (e.g. split-plot designs, repeated measures design, nested designs, etc.). Again, I think "factorial" refers to the fact that all treatment combinations are equally represented and factors are said to be crossed. The downside of factorial designs is, as mentioned before, that the treatment combinations grow exponentially, i.e. $n^k$, where $n$ is the number of levels and $k$ the number of factors (assuming the same number of levels across factors). $\endgroup$ – Stefan Feb 20 '19 at 17:53

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