Factorial design and optimization

In many analytical techniques the response of the measurement system depends on a variety of experimental factors under the control of the operator. For example, enzyme assays involve the direct or indirect measurement of reaction rates. In a given experiment the reaction rate will depend on factors such as the temperature, the pH, ionic strength, and chemical composition of the buffer solution, the enzyme concentration, and so on. For a particular application it will be important to set the levels of these factors to ensure that (for example) the reaction rate is as high as possible. The process of finding these optimum factor levels is known as optimization. Several methods of optimization are discussed in detail in subsequent sections. But before an optimization process can begin we must determine which factors, and which interactions between them, are important in affecting the response it is also valuable to know which factors have little or no effect, so that time and resources are not wasted on unnecessary experiments. 

The method by which the effects of the factors and their interactions are estimated is Illustrated by the following example. 

In a high-performance liquid chromatography experiment, the dependence of the retention parameter, k, on three factors was investigated. The factors were pH (factor P), the concentration of a counter-ion (factor T) and the concentration of the organic solvent in the mobile phase (factor C). Two levels were used for each factor and two replicate measurements made for each combination. The measurements were randomized. The table below gives the average value for each pair of replicates. 

The effect of changing the level of P can be found from the average difference in response when P changes from high to low level with the levels of C and T fixed. There are four pairs of responses that give an estimate of the effect of the level of P as shown in the table below. 

The average effects of altering the levels of T and C can be found similarly to be  

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