1. Analysis of the results revealed no evidence of "ure quadratic" curvature in the response of interest (i.e. the respose at the center approximately equals the average of the responses at factorial runs). 
2. The design matrix originally executed included the limits of the factor settings available to run the process. 

If the experimenter has defined factor limits appropriately and/or taken advantage of all the tools available in multiple regression analysis (transformations of responses and factors, for example), then finding an industrial process that requires a third order model is highly unusual. We will only focus on designs that are useful for fitting quadratic models. As we will see, these designs often provide lack of fit detection that will help determine that a higher order model is needed. 

Figures 3.9 to 3.12 identify the general quadratic surface types that an investigator might encounter 
 
 

FIGURE 3.9  A Response Surface "Peak"	FIGURE 3.10   A Response Surface "Hillside"

 

FIGURE 3.11   A Response Surface "Rising Ridge"	FIGURE 3.12  A Response Surface "Saddle"

A two level experiment with center points can detect, but not fit, quadratic effects
 
 
 
 
 
 
 
 
 
 
 

3-level factorial designs can fit quadratic models but they require many runs when there are more than 4 factors
 
 
 
 

Figures 3.13 through 3.15 illustrate possible behaviors of responses as functions of factor settings.  In each case, assume the value of the response increases from the bottom of the figure to the top and that the factor settings increase from left to right. 
 

FIGURE 3.13  Linear Function	FIGURE 3.14 Quadratic Function	FIGURE 3.15 Cubic Function

If a response behaves as shown in Figure 3.13, then the design matrix to quantify that behavior need only contain factors with two levels -- low and high.  This model is a basic assumption of simple two-level factorial and fractional factorial designs.  If a response behaves as shown in Figure 3.14, then the minimum number of levels required for a factor to quantify that behavior is three.  One might logically assume that adding center points to a two-level design would satisfy that requirement, but the arrangement of the treatments in such a matrix confounds all quadratic effects with each other.  While a two-level design with center points cannot estimate individual pure quadratic effects, it can detect them effectively. 

A solution to creating a design matrix that allows estimation of simple curvature as shown in Figure 3.14 would be to use a three-level factorial design. Table 3.21 explores that possibility. 

Finally, in more complex cases such as illustrated in Figure 3.32, the design matrix must contain at least four levels of each factor to characterize the behavior of the response adequately 

Two-level factorial designs quickly become too large for practical application as the number of factors investigated increases.  This problem was the motivation for creating “fractional factorial” designs.  Table 3.21 shows that the number of runs required to satisfy a 3^k factorial matrix becomes unacceptable even more quickly than the case for the two-level design.  The last column in Table 3.21 shows the number of terms present in a quadratic model for each case. 

Even with a modest number of factors, the number of runs becomes very large, even an order of magnitude greater than the number of parameters to be estimated. For example, the absolute minimum number of runs required to estimate all the terms present in a four-factor quadratic model is 15:  the intercept term, 4 main effects, 6 two-factor interactions, and 4 quadratic terms. 

The corresponding 3^k design for k = 4 requires 81 runs. 

Considering a fractional factorial at three-levels is a logical step, given the success of fractional designs when applied to two-level designs.  Unfortunately, the aliasing structure in the three-level fractional factorial designs is considerably more complex and harder to define than in the two-level case. 

Additionally, the three-level factorial designs suffer a major flaw in their lack of “rotatability.”

Central Composite Designs (CCD) are rotatable

In a rotatable design, the variance of the predicted values of y is a function of the distance of a point from the center of the design and is not a function of the direction the point lies from the center.  Before a study begins, little or no knowledge exists about where the optimum responses lie.  Therefore, the experimental design matrix should not bias an investigation in any direction. 

In a rotatable design, the contours associated with the variance of the predicted values are concentric circles.  Figures 3.16 and 3.17 (adapted from Box and Draper, “Empirical Model Building and Response Surfaces,” page 485) illustrate a three-dimensional and contour plot, respectively, of the “information function” associated with a 3² design. 

The information function is: 

where V is the variance of the predicted value . 
Either figure clearly shows that the information content of the design is not only a function of the distance from the center of design space, but also a function of direction. 

Figures 3.18 and 3.19 are the corresponding graphs of the information function for a rotatable quadratic design.  In either of these figures, the value of the information function is a function only of the distance of a point from the center of the space. 
 
 

FIGURE 3.16  Three-Dimensional Illustration of the Information Function of a 32 Design	FIGURE 3.17  Contour Map of the Information Function for a 32 Design

 

FIGURE 3.18  Three-Dimensional Illustration of the Information Function for a Rotatable Quadratic Design for Two Factors	FIGURE 3.19  Contour Map of the Information for a Rotatable Quadratic Design for Two Factors

Classical Quadratic Designs

Introduced during the 1950’s, classical quadratic designs fall into two broad categories:  Box-Wilson central composite designs and Box-Behnken designs.  The next sections describe these design classes and their properties.