Mixture designs model, second response

The confidence of the response estimate by a regression model differs in different simplex points. Let the variance of response estimate be Sy, and let the variance of replicated design points be Sy2. An example of determining a variance of response estimate is demonstrated on a second-order regression model for a three-component mixture. 

Two level factorial designs are primarily useful for exploratory purposes and calibration designs have special uses in areas such as multivariate calibration where we often expect an independent linear response from each component in a mixture. It is often important, though, to provide a more detailed model of a system. There are two prime reasons. The first is for optimisation - to find the conditions that result in a maximum or minimum as appropriate. An example is when improving die yield of synthetic reaction, or a chromatographic resolution. The second is to produce a detailed quantitative model to predict mathematically how a response relates to die values of various factors. An example may be how the near-infrared spectrum of a manufactured product relates to the nature of the material and processing employed in manufacturing. 

For example, in the case of the 3 component mixture we may assume that the response is well or at least adequately modelled by a third-order polynomial. However if we are not sure of the best model, we may prefer a step-wise approach, first building up a second-order design, and then augmenting it as necessary. The possibility of such a sequential approach to experimental design is as valuable for mixtures, as it is for independent variables. Thus the ternary test point 7 is added to the 3 component second-order design (figure 9.3) and the special cubic model determined. Extra test points may be added but these points do not allow fitting of the full cubic model. 

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