Statistics for Environmental Engineers

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Before moving the experimental settings, consider the available information more carefully. The fitted model describes a plane and the plane is almost horizontal, as indicated by the small coefficients of both C and D. One way we can observe a nearly zero effect for both variables is if the four corners of the 22 experimental design straddle the peak of the response surface. Also, the direction of steepest ascent has changed from Figure 42.2 to 42.3. This suggests that we may be near the optimum. To check on this we need an experimental design that can detect and describe the increased curvature at the optimum. Fortunately, the design can be easily augmented to detect and model curvature.

Third Iteration: Exploring for Optimum Conditions


Design — We anticipate needing to fit a model that contains some quadratic terms, such as R = b0 + b1C + b2D + bnCD + bnC2 + b22D2. The basic experimental design is still a two-level factorial but it will be augmented by adding “star” points to make a composite design (Box, 1999). The easiest way to picture this design is to imagine a circle (or ellipse, depending on the scaling of our sketch) that passes through the four corners of the two-level design.


Rather than move the experimental region, we can use the four points from iteration 2 and four more will be added in a way that maintains the symmetry of the original design. The augmented design has eight points, each equidistant from the center of the design. Adding one more point at the center of the design will provide a better estimate of the curvature while maintaining the symmetric design. The nine experimental settings and the results are shown in Table 43.3 and Figure 43.4. The open circles are the two-level design from iteration 2; the solid circles indicate the center point and star points that were added to investigate curvature near the peak.

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