Higher-order models, response surface

Higher-order models are rarely applied. In many cases, the true response surface can be sufficiently well approximated by the second-order model. Occasionally, higher-order models can be used when quadratic models are clearly inadequate, for example, when a sigmoid-like relation between the response and a variable is observed (7). Then, either a third-order model, an appropriate transformation, a mechanistic physical model, nonlinear modeling techniques, or neural networks can be applied (1,7). 

It is also possible to derive the procedure as a constrained maxiinization problem, by using the method of Lagrange s multipliers. This procedure is more elaborate, and can be applied also to second and higher order response surface models. This is in essence the method of Ridge anafysis[2]. We do not go into this here. 

Let us now imagine that the response surface, that is, the relation between the response and the factors in the experimental range we are investigating, is sufficiently smooth, so that small variations in the factors do not cause abrupt variations in the response. If this is so, we can expect the model coefficients to obey a certain hierarchy, where the lower-order terms are in principle more important than the higher-order ones. Returning to oiu- example, we expect the main effects to be more important to the model than, say, the four-factor interaction. In general, we can expect that an interaction term will become less important as the number of factors involved in its definition increases. Otherwise, we would not be able to obtain a satisfactory model for oiu response smface with a finite number of terms. 

Second and higher order designs and response surface modelling... 

Another approach for designing second-order models to be used for process optimisation, without using higher-order factorial design for optimisation, is response surface design. Many different approaches have been developed, many of which require the use of specialised computer software to obtain a tractable solution. 

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