Multifactor response surfaces

One of the most useful models for approximating a region of a multifactor response surface is the full second-order polynomial model. For two factors, the model is of the form... 

Factorial designs are a popular class of experimental designs that are often used to investigate multifactor response surfaces. The word factorial does not have its usual mathematical meaning of an integer multiplied by all integers smaller than itself (e.g. 5 5x4/3/2 / 1) instead, it simply indicates that many... 

Response Surfaces. 3. Basic Statistics. 4. One Experiment. 5. Two Experiments. 6. Hypothesis Testing. 7. The Variance-Covariance Matrix. 8. Three Experiments. 9. Analysis of Variance (ANOVA) for Linear Models. 10. A Ten-Experiment Example. 11. Approximating a Region of a Multifactor Response Surface. 12. Additional Multifactor Concepts and Experimental Designs. Append- ices Matrix Algebra. Critical Values of t. Critical Values of F, a = 0.05. Index. 

S. N. Deming and S. L. Morgan, in Experimental Design A Chemometric Approach, 2nd ed., Elsevier, Amsterdam, 1993, pp. 227-274. Approximating a Region of a Multifactor Response Surface. 

Experiments that will be used to estimate the behavior of a system should not be chosen in a whimsical or unplanned way, but rather, should be carefully designed with a view toward achieving a valid approximation to a region of the true response surface [Cochran and Cox (1950), Youden (1951), Wilson (1952), Mandel (1964), Fisher (1971)]. In the next several chapters, many of the important concepts of the design and analysis of experiments are introduced at an elementary level for the single-factor single-response case. In later chapters, these concepts will be generalized to multifactor, multiresponse systems. 

Full second-order polynomial models used with central composite experimental designs are very powerful tools for approximating the true behavior of many systems. However, the interpretation of the large number of estimated parameters in multifactor systems is not always straightforward. As an example, the parameter estimates of the coded and uncoded models in the previous section are quite different, even though the two models describe essentially the same response surface (see Equations 12.63 and 12.64). It is difficult to see this similarity by simple inspection of the two equations. Fortunately, canonical analysis is a mathematical technique that can be applied to full second-order polynomial models to reveal the essential features of the response surface and allow a simpler understanding of the factor effects and their interactions. 

Injection molding processes using Stat-Ease Inc. response surface methods for process optimization are developed with design of experiments (DOE) and a multifactor linear constant (MLC) [10]. 

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