Construction of models for the dependent variables

The construction of models by multiple regression is, in the design space used, somewhat more complicated than usual. This is due to the presence of both mixture and process variables. In this section a short discussion on the consequences of the construction of models with both kind of variables is given. 

Mixture variables, expressing the composition of the mobile phase as fi ac-tions, have the property that they add up to one (the mixture restriction). The consequence is that no intercept can be estimated when the effects of the solvents are evaluated [10,19]. Moreover interactions and quadratic effects, such as used when the independent variables are process variables, can not be estimated independently. Mathematically it is better to use blending effects only. Interpretation of these blending effects, i.e. explicitly stating what components are responsible for the non-linear effects, is not possible. 

When the mixture variables are combined with process variables, the complications caused by the mixture restriction are amplified by the presence of the process variables. For example, when the interactions between all mixture variables and a process variable are estimated, it is not possible to estimate the effect of the process variable itself. This means that the models do not need to contain the more simple model terms, when the more complex model terms are present [19]. 

The terms in the models may be classified by degree of complexity. First order terms cause linear effects, second order terms cause curvature, third order terms more complex curvature etc. When d, m, c, e, t, and h are used for, respectively, the concentrations of DBA, MeOH, CHCI3, EtAc and the setting of temperature and relative humidity, then examples of second order terms are d m, d e, e t and t h. The complete first order model is  

With the design used it is not possible to construct a complete third order model. The most complex third order model which can be used contains 38 terms. This includes third order blending effects (the so called special cubic model terms [10], e.g. d m c) and temperature and relative humidity dependent blending effects (e.g. d m t and e t h). 

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