Variance dispersion graph

A variance dispersion graph (VDG) is a graph capable of displaying the

minimum, maximum, and average prediction variances for a specific design and response model versus the distance between the design point and the center of the design space. The distance, or radius, usually differs from zero (the design center) in that in a spherical design, the radius is the distance to the farthest point from the center. Normally, one plots the scaled prediction variance (SPV) as

𝑁 𝑉[𝒚̂(𝒙)] 𝜎⁄ 2 = 𝑁 𝒙𝑇(𝑿𝑇𝑿)−1𝒙 (2.12)

Note that the SPV is the prediction variance in Equation (2.12) multiplied by the number of design runs (𝑁) and divided by the error variance 𝜎2. Dividing by 𝜎2 makes the quantity scale-free and multiplying by 𝑁 often helps to facilitate the comparison of designs of different sizes.

Figure 2.8.(a) is a VDG for the rotatable CCD with 𝑘 = 3 variables and five center runs. Since the design is rotatable, the minimum, maximum, and average SPV are identical for all points that are equidistant from the center of the design. As a result, there is only one line on the VDG. Next, observe how the graph displays the behavior of the SPV over the design space, with nearly constant variance out to a radius, and then it increases steadily from there to the boundary of the design. Figure 2.8.(b) is a VDG for a spherical CCD with 𝑘 = 3 variables and five center runs. Notice that there is little to no difference between the three lines for minimum, maximum, and average SPV; we can therefore conclude that any practical differences between the two types of central composite designs (the rotatable and spherical) versions of this design are minimal. Figures 2.9 (a) and (b) are the VDGs for a rotatable CCD with 𝑘 = 3 variables and 𝛼 =

1.68, √2. In this VDG, the number of center points in the design varies from 𝑛0 = 1

to 𝑛0 = 5. The VDG clearly shows us that a design with too few center points will have

an unstable distribution of prediction variance; the prediction variance quickly stabilizes, however, with increasing values of 𝑛₀. The use of either four or five center runs will give reasonably stable prediction variance over the design space. These recommendations are based on VDG studies on the effects of changing the number of center points in the response surface design [4].

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Figure 2.8. VDG with three factor, five center point with α = 1.68 and α = 1.732.

Figure 2.9. VDG with k = 3 and α = 1.68 with α = √2 (noe to five center point).

An additional benefit of using center points can be found when a factorial

experiment is performed for an ongoing process [4]. Consider using the current operating

conditions (or recipe) as the center point in the design; this is done to assure the operating personnel that at least some of the runs in the experiment are going to be performed under familiar conditions. As such, the results obtained are not likely to be worse than those typically obtained. When the center point in a factorial experiment corresponds to a real- life process, the experimenter can use the observed responses at the center point to note whether anything unusual occurred during the experiment. That is, the responses of the center point should be very similar, if not identical, to any responses observed historically in the routine process. Often times, operating personnel will keep a control chart for monitoring process performance [3]. When they do so, the center point responses can be plotted directly on the control chart to check the behavior of the process within the experiment. Consider running the replicates at the central in nonrandom order, run one or two center points at or close to the beginning of the experiment, one or two near the center, and one or two near the end. By spreading the center points out over the course of the experiment, the experimenter obtains a rough check on the stability of the process. For instance, if a trend appears during the performance of the experiment, plotting the center point responses against the time order may reveal this.

In other cases, experiments must be performed in situations where there is little or no previous information regarding variability in the process. In such cases, running two or three center points in the design will be helpful for the first few runs. These runs provide a preliminary estimate of variance. If the magnitude of the variability appears to be reasonable, then additional runs can be done. On the other hand, if variability is larger than anticipated, no further runs should be done. It is then prudent to determine why the variability is so large before performing additional experiments. Usually, center points

are applied when the design variables are quantitative. However, there will sometimes be one or more qualitative or categorical factors among several quantitative ones. Center points can still be utilized in these cases. For example, consider an experiment with two quantitative variables, each variable at two levels, and a single qualitative variable, also with two levels. In such a case, the central points should be located on the opposing faces of the cube that includes the quantitative variables. In other words, center points can be employed at the high- and low levels for treatment combinations of the qualitative variables, so long as those subspaces include only quantitative variables [4].