Response surface design, degrees

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... 

Replication is often included in central composite designs. If the response surface is thought to be reasonably homoscedastic, only one of the factor combinations (commonly the center point) need be replicated, usually three or four times to provide sufficient degrees of freedom for s. If the response surface is thought to be heteroscedastic, the replicates can be spread over the response surface to obtain an average purely experimental uncertainty. 

In response surface methodology, it is frequently assumed that / can be approximated in some region of the design variables by a low-degree polynomial. For example, if p=2, and a first-order model is assumed appropriate then... 

In Section 2.2 it was shown that response surface methodology can be applied to enable a researcher to model the effect of multiple quantitative variables on a response with a low-degree polynomial. Frequently, response surface techniques have focused on the mean response as the only response of interest. However, by regarding the variation in the response as an additional response of interest, the researcher can investigate how to achieve a mean response that is on target with minimum variation. In particular, if a researcher replicates each design point in an experiment, then an estimate of the standard deviation at each point can be calculated and used to model the effect of the variables on the variability of the response. 

Fig. 3 Application of the Doehlert experimental design to optimize a MIP for propranolol with respect to the type of cross-linker (EDMA or TRIM) and the degree of cross-linking, (a) Three-dimensional representation of response surfaces for the percentage of bound [3H]propanolol to the molecularly imprinted polymer (MIP) and the corresponding non-imprinted control polymer (NIP), (b) Contour plot of the function describing binding of [3H]propanolol to MIPs relative to the degree and the kind (bi or trifunctional) cross-linking. The values were corrected for non-specific binding to the non-imprinted control polymer. Adapted from [31] with kind permission from Springer Science + Business Media...

A researcher is therefore recommended to use the design of experiments or to achieve an optimum in an experimental way. A researcher who designs an experiment does not know beforehand where in the studied response surface the optimum is located and what the shape of the surface is. Therefore he uses two approaches to reach the optimum. By one approach, he approximates in the given experimental region his experimental data by an assumed empirical model, or fits the response surface to the degree of the needed polynomial accuracy. Based on such an analytical model, he performs analytical optimization. Reaching an optimum in this case is more efficient if the obtained analytical model is adequate. By another approach, the researcher does not form an analytical model, but he does his experiments iteratively by prior established rules until he reaches the optimum. 

Assume that you have run experiments by a factorial design (with Np runs) with a view to assessing the significance of the experimental variables fi om estimates, hj, of the coefficients in a linear response surface model. Assume also that you have made Nq repeated runs of one experiment to obtain an estimate of the experimental error standard deviation. From the average response, J, in repeated runs, an estimate of the experimental error standard deviation, Sq, with (Nq - 1) degrees of freedom is obtained as... 

In a sense, this is a direct optimization method. Only when the domain has been established with a reasonable degree of certainty can the final experimental design be set up to enable response surface modelling and optimization. Very often the initial experiments to situate the region of interest will be part of a screening study, hence used to determine a first-order model. 

The experimental design of the uniform shell design allowing to calculate the coefficients of a second-degree response surface model is ... 

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