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# Factorial designs

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# Factorial designs

Factorial designs are typically used when a set of factors or treatments are to be examined and each can be coded to two levels, for example High and Low, or +1 and -1. With k factors to examine this would require at least 2k runs (or 3k runs for a 3-level factor coding), and for k>5 the number of such runs may be considered excessive or impractical. For this reason, instead of using a full factorial design, a partial or fractional factorial design is generally selected. These fractional designs are carefully constructed to maximize the amount of information gathered on the principal or main effects, whilst accepting some loss of information (or confounding), particularly in respect of (higher order) interactions between factors. This is often the case for industrial applications of such designs, where the number of factors may be very large and the objective is to identify, or screen for, those main effects that are of greatest importance (or most significant) in the process under examination. In these situations the factors that have a lower level of importance and interactions between factors (especially higher order interactions) are of less interest and sometimes referred to as nuisance factors. The aim is to screen out these factors and focus on the main effects so that a simpler experiment can be run for which the main effects alone are the subject of more detailed study and optimization. This is very different from many other areas of scientific research, where the number of factors being considered may be much lower and both main effects and interactions are of interest. The approach to the analysis of these designs varies, but most computer software performs an analysis of variance (ANOVA) as with other experimental design problems. Box et al. (2005, p188, [BOX1]) note that ANOVA is neither the most effective nor meaningful approach in many cases, since it is the size of (main) effects and their standard errors that are often of greater interest. Reflecting this software packages now tend to provide ANOVA and estimates of the effect sizes (using, for example, the ratio of the sum of squares attributable to the treatment divided by the total sum of squares), with their standard errors and a t-distribution statistic with associated probability value, p. For factorial designs that are defined for two-levels (the most common situation) the effect size values are twice the value of the regression parameters that provides the underlying statistical model being considered. Again, many packages will provide the regression model intercept and parameter estimates, with confidence intervals, which may then be used for modeling purposes.

References

[BOX1] Box G E P, Hunter J S, Hunter W G (1978) Statistics for Experimenters: An Introduction to Design, Data Analysis and Model Building. J Wiley & Sons, New York. The second, extended edition was published in 2005