- In section 6.7, we said that if we wanted an optimal design, we should place all our test points at the boundaries of factor’s values. This is certainly true when we have a linear factor effect. But what if the linearity assumption doesn’t hold? Which design should we pick?
Figure 1: The main lesson from section 6.7
- At least in the first design, we would know that something is off! We could then include polynomial terms to fit curvature in the factor effect. This motivates the addition of center points.
Figure 2: What if our effect is nonlinear?
Figure 3: Added center points in the \(K = 1\) and \(K = 2\) cases.
Testing
- It’s possible to formally test whether there is significant quadratic curvature in the factor effect. We can add quadratic terms to our model and perform ANOVA on the associated term. Alternatively, we can look at \[\begin{align} SS_{\text{curvature}} &= \frac{n_{F}n_{C}\left(\bar{y}_{F} - \bar{y}_{C}\right)^2}{n_{F} + n_{C}} \end{align}\] where \(\bar{y}_{C}\) is the average of the \(n_{C}\) samples at the center point and \(\bar{y}_{F}\) is the average over all other points. A large value suggests that linearity is not plausible, and it turns out that it can be formally used in a \(t\)-test, but we will not develop that point further, since it’s only in the supplemental material for the chapter.
Figure 4: Geometry of the curvature test statistic.
Central Composite Design
- If we have an unreplicated design, then we cannot fit quadratic terms — we would have more parameters than samples, and the linear system would be underdetermined.
- A fix is to require samples at axial points.
- This design is called a central composite design.
Figure 5: A central composite design.