Factorial designs surface

Table 14.5 lists the uncoded factor levels, coded factor levels, and responses for a 2 factorial design. Determine the coded and uncoded empirical model for the response surface based on equation 14.10. 

Curved one-factor response surface showing (a) the limitation of a 2 factorial design for modeling second-order effects and (b) the application of a 3 factorial design for modeling second-order effects. 

The following set of experiments provides practical examples of the optimization of experimental conditions. Examples include simplex optimization, factorial designs used to develop empirical models of response surfaces, and the fitting of experimental data to theoretical models of the response surface. 

A 2 factorial design was used to determine the equation for the response surface in problem lb. The uncoded levels, coded levels, and the responses are shown in the following table. 

Ramirez, J., Guttierez, H., and Gschaedler, A., Optimization of astaxanthin production hy Phaffia rhodozyma through factorial design and response surface methodology, J. BiotechnoL, 88, 259, 2001. 

A unimodal surface has been chosen because we have no way of dealing sequentially with a surface that has two or more peaks or valleys. The only reasonable approach is to start the sequential procedure at a number of widely disparate points and to determine whether the paths converse toward the same optimum. If two or more different peaks or valleys are indicated, the investigator must find the optimum for each possible peak or valley and then select the best. There is no way of knowing that all peaks or valleys have been explored except to map the whole surface finely using some factorial design. 

Note, however, there are two critical limitations to these "predicting" procedures. First, the mathematical models must adequately fit the data. Correlation coefficients (R ), adjusted for degrees of freedom, of 0.8 or better are considered necessary for reliable prediction when using factorial designs. Second, no predictions outside the design space can be made confidently, because no data are available to warn of unexpectedly abrupt changes in direction of the response surface. The areas covered by Figures 8 and 9 officially violate this latter limitation, but because more detailed... 

Based on the obtained response surface, a second roimd of optimization follows, using the steepest ascent method where the direction of the steepest slope indicates the position of the optimum. Alternatively, a quadratic model can be fitted around a region known to contain the optimum somewhere in the middle. This so-called central composite design contains an imbedded factorial design with centre... 

The design, which is illustrated in Figure 5.8, gives the most comprehensive evaluation of the response surface using a given number of experiments. It provides greater efficiency than a three level full factorial design yet essentially obtains the same information. However, as k increases the number of required experiments quickly becomes impractical [22]. 

When the number of factors is too high, or when the equation is too complex to be obtained by few determinations, fractional factorial design and simplex movement on the response surface toward the optimum set of conditions are the methods of... 

Figure 3.8. Possible configurations of nine sets of factor values (experiments) that could be used to discover information about the response of a system. Dotted ellipses indicate the (unknown) response surface, (a) A two-factor, three-level factorial design (b) a two-factor central composite design.

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... 

Index Entries Extractive alcoholic fermentation optimization successive quadratic programming factorial design response surface methodology productivity. 

Finally, the use of factorial designs, concluded by a response-surface analysis, is the latest approach used to select the most appropriate combination doses of two classes of drugs that allows complete pharmacological manipulation of the RAS, namely diuretics and ACE inhibitors (250, 251). 

One limitation of two-level factorial designs is the assumption of linearity of the effects. If it is possible that the effect of one or more of the factors is nonlinear, a response surface design may be selected. A central composite response surface design is a full factorial or fractional factorial design that is supplemented with additional trials to allow for estimation of curvature from the factors of interest. For each factor of interest to be studied for curvature, two additional trials are performed (1) one trial with all of the factors at their middle level except for... 

A potential concern in the use of a two-level factorial design is the implicit assumption of linearity in the true response function. Perfect linearity is not necessary, as the purpose of a screening experiment is to identify effects and interactions that are potentially important, not to produce an accurate prediction equation or empirical model for the response. Even if the linear approximation is only very approximate, usually sufficient information will be generated to identify important effects. In fact, the two-factor interaction terms in equation (1) do model some curvature in the response function, as the interaction terms twist the plane generated by the main effects. However, because the factor levels in screening experiments are usually aggressively spaced, there can be situations where the curvature in the response surface will not be adequately modeled by the two-factor interaction... 

Steven Gilmour is Professor of Statistics in the School of Mathematical Sciences at Queen Mary, University of London. His interests are in the design and analysis of experiments with complex treatment structures, including supersaturated designs, fractional factorial designs, response surface methodology, nonlinear models, and random treatment effects. 

---