Regular fractional factorial design

Box and Meyer also derived a useful result (which is applied in some of the subsequent methods in this chapter) that relates dispersion effects to location effects in regular 2k p designs. We present the result first for 2k designs and then explain how to extend it to fractional factorial designs. First, fit a fully saturated regression model, which includes all main effects and all possible interactions. Let /3, denote the estimated regression coefficient associated with contrast i in the saturated model. Based on the results, determine a location model for the data that is, decide which of the are needed to describe real location effects. We now compute the Box-Meyer statistic associated with contrast j from the coefficients 0, that are not in the location model. Let i o u denote the contrast obtained by elementwise multiplication of the columns of +1 s and—1 s for contrasts i and u. The n regression coefficients from the saturated model can be decomposed into n/2 pairs such that for each pair, the associated contrasts satisfy i o u = j that is, contrast i o u is identical to contrast j . Then Box and Meyer proved that equivalent expressions for the sums of squares SS(j+) and SS(j-) in their dispersion statistic are... 

The starting simplex is usually regular, but does not have to be so. It is quite possible to select points from a factorial or fractional factorial design, with reduction of the number of factors (taking only those that are active) and using either the basic or extended sequential simplex methods to move from the region of the factorial design to a more favourable point. However under such circumstances the steepest ascent method could equally well be used, and this should normally be preferred. 

Equation (2.173) is used for rotatable designs of second and third order when trials are replicated only in null point. In the case of a full factorial experiment or regular fractional replicas, we use ... 

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