Isoresponse contours

Write a table similar to Table 12.3 for three-factor systems. What do the possible isoresponse contours look like in three-dimensional factor space [See, for example. Box (1954).]... 

This conclusion may be totally wrong if there is an interaction between pH and T. The situation may well be as depicted in Fig. 2.3 where the variations in both variables are given and the function describing the yield variation will be described by a response surface over the plane spanned by the two variables. The topography of the surface is given by the isoresponse contours which show the levels of the yields for different settings of the experimental variables. It is seen that the optimum yield will be 97 %, and that the conditions for this will be at pH = 3.6 and T = 55 °C, which does not at all correspond to the results obtained by the one-variable-at-a-time study. 

Analysis of variance of the regression did not indicate any significant lack of fit, Lack.of-fit 4 -74 pCnt g gj (a = 5 %). The residual plots in Fig. 12.10 do not show abnormal behaviour. It was therefore assumed that the variation in yield was adequately described by the model. The isoresponse contour projections are shown in Fig.12.11... 

Contour or isoresponse lines are parallel to the X2-axis. The response surface in Fig. 3.2 is defined by the regression model y = 90X3 + 80X2 + 80X3. It is evident that the regression coefficients for components X2 and X3 are equal (P2=Pi) and that there are no response changes in the directions of response isolines for X2+X3=const. 

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... 

In a second step, a simplex centroid design was realized with seven design points and six check points. Two responses were optimized consistency and whiteness. Using the second-order equations from this matrix, it is possible to obtain isoresponse graphs. The contour plots are given in Figs. S and 6. The optimal gel must correspond to a gel similar to petrolatum. This optimal zone is reported in Fig. 7 where contour plots of the two responses are superimposed. 

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. 

Instead of using a response surface graph, one often uses a contour plot. This translates the response surface in the same way as a geographical map of a mountainous area. The isoresponse lines can be viewed as the contour lines on the map. 

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