Saturated fractional factorial designs and screening

Sieving (or screening) is a process that separates large from small. Suppose we are just beginning a research project and have no idea which of a large number of potentially important factors really are important. Wouldn t it be nice if we could place all of these factors on a screen and sieve them so the unimportant factors fell through and only the important factors stayed on top  

There is a class of experimental designs, called screening designs, that can be used to sieve factors. Behind almost all of these designs is an implicit linear model that is first-order in each factor. The model is 

Note that for k factors, the model will have k + 1 parameters - one for each of the factors being investigated plus one for the offset term (Pq). After this model has been fitted to the data obtained from a screening design, the P s can be used to determine whether the factor is small and can be discarded, or is large and should be retained. 

Clearly, the effects (slopes) of temperature and pH are small. The effects of pressure and catalyst concentration are much larger - the pressure effect is positive, but the catalyst effect is negative (maybe the catalyst is really an inhibitor). It looks like higher pressure and lower catalyst will produce better yield. If this project is continued, then temperature and pH can probably be held constant and not varied, while pressure and catalyst concentration can be studied in more detail with additional designed experiments. This reduction of four potentially useful factors to 

Most screening designs are based on saturated fractional factorial designs. The firactional factorial designs in Section 14.8 are said to be saturated by the first-order factor effects (parameters) in the four-parameter model (Equation 14.27). In other words, the efficiency E = p/f = 4/4 = 1.0. It would be nice if there were 100% efficient fractional factorial designs for any number of factors, but the algebra doesn t work out that way. 

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