Factorial design effects, supersaturation

Standard methods for analyzing data from fractional factorial designs cannot be used with data from supersaturated designs, because the least squares estimates are not unique and, given any reasonable assumptions, there is no way to estimate all the main effects simultaneously. 

Supersaturated designs, and likewise grouping screening designs, provide very little information about the effects of the factors studied, unless they are followed up with further experiments. If it is possible to use a fractional factorial design... 

Steven Gilmour is Professor of Statistics in the School of Mathematical Sciences at Queen Mary, University of London. His interests are in the design and analysis of experiments with complex treatment structures, including supersaturated designs, fractional factorial designs, response surface methodology, nonlinear models, and random treatment effects. 

If the experimental runs are completely randomized, then randomization theory (see Hinkelmann and Kempthorne, 1994) tells us that least squares gives us unbiased estimators of any pre-chosen set of n — 1 linearly independent contrasts among the n combinations of factor levels (treatments). In most factorial experiments the pre-chosen treatment contrasts would be main effects and, perhaps, interactions. However, in supersaturated designs there is no rational basis for choosing a set of n — 1 contrasts before the analysis. Any model selection method will lead to selection biases, perhaps large biases, in the estimators of effects. If a2 is assumed known, then we can test the null hypothesis that all n treatment populations have equal means. This would not be of great interest, because even if this null hypothesis were true it would not imply that all main effects are zero, only that a particular set of n - 1 linear combinations of treatment means are zero. Of course, in practice, a2 is not known. 

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