Response surface models contour plot

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. 

After experimentation and calculation of a model, a relation is established between each formulation property separately and the variables in the employed model. When a model adequately describes this relation, predictions of this property can be made by interpolation over the whole range of the boundary values of the used variables, which forms the response surface. In Figure 4.12 the relation between the crushing strength and mixtures of three components (where the factor space can be represented by a triangle) is presented by a contour plot. The composition that gives a desired criterion value can be read directly fi-om the figure. 

A mathematical model was found for each studied response. From the models, the contoured curves and the response surfaces were plotted, and the optimal points were sought and confirmed. 

Figure 2.4 displays a contour plot of this model (representing the response surface) for different levels of the response at pH = 5 (A = 1). It is expected that any point inside the shaded zones will have a yield between 87 and 90%. Thus, we can easily define the working levels for B (temperature) and C (time) to achieve a yield equal to or higher than 87%. 

As stated earlier the model should be obtained for responses such as k (or log< ). These are also the responses that should be predicted from the models and only then responses such as or the global responses of Section 6.2 should be obtained. The optimum is typically derived by first obtaining isorespon.se contour plots from the response surfaces such as those of Fig. 6.20 or directly on the response surface and then visually deciding where the optimum is to be found. For the measured responses, the surfaces are often relatively simple (Fig. 6.21), but for the global responses they can be very complex (Fig. 6.22). If a threshold criterion is applied, then overlapping resolution maps can be obtained similar to those of Fig. 6.4. 

The surface representing the model is called the response surface. Graphically, the response surface can be visualized by drawing 2D contour plots or 3D response surface plots (7). A 2D contour plot shows the isoresponse lines as a function of the levels of two variables, while a 3D response surface plot represents the response, on a third dimension, as a function of the levels of two variables. An example of a 2D contour plot and a 3D response surface plot is shown in Figure 2.18. When more than two factors... 

The elution order of the enantiomers was the same for all experiments. Thus, a modeling of the resolution is meaningful. The 2D contour plot and the 3D response surface plot for this response Rs are shown in Figure 2.18. 

Figure 4.1 Response versus factors surface). By using search methods, plot In the case of response surface the response is measured along a methods (RSMs), the response is search path, here along a simplex described by a mathematical model path (cf. "Analytical Performance dotted contour lines of the response Characteristics" Section).

Based on the mathematical model, the response surfaces can be explored graphically. An example plot of the response rate in dependence on PPD concentration and pH is shown in Figure 4.15a. The curved dependences in the direction of both factors lead to a maximum rate at coded levels of PPD of about 0.4 and of pH at 0.2. This relates to decoded levels of 16.6 mM PPD and a pH value of 5.95. Maxima are best found from the contour plots as represented in Figure 4.15b. 

The response surface representing the model can be visualised graphically by means of 2D contour plots and/or 3D response surface plots. In a 2D contour plot, the isoresponse lines are represented as a function of levels of two factors, while in a 3D plot the response is represented on a third dimension, as a function of the factor levels (see Figure 3.24). When more than two factors are examined and modelled, all but two factors need to be fixed at a given level to draw both plots. The optimal or acceptable experimental conditions can be derived from the graphical representation of the model or by mathematical analysis of its equation. 

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