The Assumptions Underlying Factorial Design

A factorial experiment is much more useful, in that its interpretation is simplest and that its efficiency in the sense of giving more information for a given number of experiments than the corresponding classical design is greatest, when the interactions are not significant. 

It may be interesting to note that the non-existence of interactions presupposes that the dependent variable y can be expressed as the sum of a series of functions of the independent variables Xj, Xj, etc., each of the latter involving one and only one independent variable, i.e. 

The functions can be anything, of course, with no limit to the degree of complexity. 

If the two independent variables were at two levels both equal to 1 and 2, we would obtain the results in the table below for y  

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