A fractional factorial design avoids a costly full-factorial experiment in which all combinations of all inputs (or factors) at different levels are studied ( for factors, each at levels). A fractional factorial experiment is a certain fractional subset (1/2, 1/4, 1/8, etc. for two-level factors and 1/3, 1/9, 1/27, etc. for three-level factors) of the full factorial experiment that is carefully selected to minimize aberrations in the experiment. Fractional factorial experiments are also useful when some factors are independent of each other or when certain interactions can be neglected. In fractional factorial designs the number of columns in the design matrix is less than the number necessary to represent every factor and all interactions of those factors. Instead, columns are “shared” by these quantities, an occurrence known as confounding. The result of confounding is that you cannot determine which quantity in a given column produced the effect on the outputs attributed to that column (from postprocessing analysis). In such a case the designer must make an assumption as to which quantities are insignificant (typically the highest-order interactions) so that a single contributing quantity can be identified. Optimization Process Composer provides fractional factorial designs for two-level and three-level (Xu, 2005) factors. In a generalized polynomial model for responses, we indicate below whether the coefficients for linear, quadratic, or other interaction terms can be clearly estimated from the sample. The following table summarizes the fractions available for two-level designs (all linear main effects are clear):
Apart from the standard generators available (from Box et al., 1978), Optimization Process Composer also allows the use of custom generators for the above fractions. The following table summarizes the fractions available for three-level designs (all linear main effects are clear):
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