Response surface models

A Box-Behnken design was employed to investigate statistically the main and interactive effects of four process variables (reaction time, enzyme to substrate ratio, surfactant addition, and substrate pretreatment) on enzymatic conversion of waste office paper to sugars. A response surface model relating sugar yield to the four variables was developed on the basis of the experimental results. The model could be successfully used to identify the most efficient combination of the four variables for maximizing the extent of sugar production. 

Since only 20 data records were collected from the system during the execution of the designed experiments conducted by Reece et al. (1989), we used their response surface models, deliberately contaminated with small Gaussian noise terms, to generate a total of 500 (x, z) pairs (assuming that the three variables, jCj, Xj, Xj, have independent and uniform... 

Walker, S. P., Demirci, A., Graves, R. E., Spencer, S. B., and Roberts, R. F. (2005b). Response surface modelling for cleaning and disinfection materials used in milking systems with electrolysed oxidizing water. Int. ]. Dairy Technol. 58, 65-73. 

Mixture designs are used to supply data for fitting continuous response surface models, either first-order models, such as... 

Bratchell, N. (1989), Multivariate Response Surface Modelling by Principal Components Analysis, J. Chemom., 3, 579-588. 

A response surface model of the effects of HA protein concentration (gliadin, the wheat prolamin), HA polyphenol concentration (tannic acid, TA), alcohol, and pH on the amount of haze formed was constructed using a buffer model system (Siebert et al., 1996a). Figure 2.12 shows the effects of protein and polyphenol on haze predicted by the model at fixed levels of pH and alcohol. The model indicates that as protein increases at fixed polyphenol levels, the haze rises to a point and then starts to decline. Similarly, when polyphenol increases at a fixed protein level, the haze increases to a maximum and then declines. 

FIGURE 2.12 Response surface model predictions of the effects of HA protein (gliadin) and HA polyphenol (TA) on the haze intensity in a model system at fixed levels of pH and alcohol. Reprinted with permission from Siebert et at. (1996a). Copyright 1996 American Chemical Society. 

A more detailed study was carried out with many more levels of protein and polyphenol than were used to construct the initial response surface model (Siebert and Lynn, 2000). The results (see Fig. 2.15) indicated that... 

Fig. 4. Response surface modeling of removal rate and nonuniformity [4]. Copyright 1996 IEEE.

T. Smith, C. Oji, D. Boning, and J. Chung, Bias and Variance in Multiple Response Surface Modeling, Third International Workshop on Statistical Metrology, Honolulu, HI, June 1998. 

Furthermore, optimal design theory assumes that the model is true within the region defined by the candidate design points, since the designs are optimal in terms of minimizing variance as opposed to bias due to lack-of-fit of the model. In reality, the response surface model is only assumed to be a locally adequate polynomial approximation to the truth it is not assumed to be the truth. Consequently, the experimental design chosen should reflect doubt in the validity of the model by allowing for model lack-of-fit to be tested. 

Response surface modeling of the mean and standard deviation... 

It can be seen that both V(y ) (equation (27)) and E(y ) (equation (28)) are essentially response surface models. From an experiment, estimates of y, D, cTg, Pq, and p can be derived. Suppose, also that the elements of V are known, or can be estimated. Then the search for a choice of design variables that yields a response that is robust to the environmental variation and close to target will involve an examination of these two response surfaces. At this point, the scientist might proceed by following... 

In the response surface strategy that was discussed in Section 2.3 standard response surface techniques are used to generate two response surface models, one for the mean response and one for the standard deviation of the response (or some function of the standard deviation). The standard deviation measures the stability of the response to the environmental variation. Standard analysis can reveal which factors affect the mean only, which only affect the variability, and which affect both the mean and the variability. The researcher can then apply optimization methods or construct contour plots of the mean and standard deviation response surfaces to determine settings of the design variables that will give a mean response that is close to the target with minimum variation. 

Coenegracht et al. [3] have introduced a four solvent system to compose mobile phases for the separation of the parent alkaloids in different medicinal dry plant materials, like Cinchona bark and Opium. Through the use of mixture designs and response surface modeling an optimal mobile phase was found for each type of plant material. These new mobile phases resulted in equally good or better separations than obtained by the procedures of the Pharmacopeias. Although separations were as predicted, the accuracy of the quantitative predictions needed to be improved. 

Doehlert designs and response surface models were developed to evaluate the effects and interactions of flve variables on the extraction of Mn in a flow-injection device. 

There are various mathematical models that can be used to describe and analyze experimental data (Scholze et al. 2001). In addition to these curve-fitting approaches, response surface models are also available (e.g., Greco et al. 1995), but these are suitable primarily for the analyses of experimental data, rather than for predictive purposes. As an example, Altenburger et al. (2004) applied both concentration addition and response addition and observed that the combined effect of a 3-compound mixture out of 10 identified sediment toxicants was sufficient to explain the observed combined effect of the more complex mixture. For identifying remediation priorities in site-specific assessment of complex contamination, this approach has great potential. 

Chapter 12 Development and Evaluation of QSARs for Ecotoxic Endpoints The Benzene Response-Surface Model for Tetrahymena Toxicity T. Wayne Schultz and Tatiana I. Netzeva... 

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