1. Centerpoint Designs Include n c center points (0,…,0) in a factorial design Include n c center points (0,…,0) in a factorial design –Obtains estimate of pure error (at center of region of interest) –Tests of curvature –We will use C to subscript center points and F to subscript factorial points Example (Lochner & Mattar, 1990) Example (Lochner & Mattar, 1990) –Y=process yield –A=Reaction time (150, 155, 160 seconds) –B=Temperature (30, 35, 40)

2. Centerpoint Designs 41.5 39.3 40 40.9 A +1 B 0 0 40.3 40.5 40.7 40.2 40.6

3. Centerpoint Designs Statistics used for test of curvature Statistics used for test of curvature

4. Centerpoint Designs When do we have curvature? When do we have curvature? For a main effects or interaction model, For a main effects or interaction model, Otherwise, for many types of curvature, Otherwise, for many types of curvature,

5. Centerpoint Designs A test statistic for curvature

6. Centerpoint Designs T has a t distribution with n C -1 df (t.975,4 =2.776) T has a t distribution with n C -1 df (t.975,4 =2.776) T>0 indicates a hilltop or ridge T>0 indicates a hilltop or ridge T<0 indicates a valley T<0 indicates a valley

7. Centerpoint Designs We can use s C to construct t tests (with n C -1 df ) for the factor effects as well We can use s C to construct t tests (with n C -1 df ) for the factor effects as well E.g., To test H 0 : effect A = 0 the test statistic would be: E.g., To test H 0 : effect A = 0 the test statistic would be:

8. Follow-up Designs If curvature is significant, and indicates that the design is centered (or near) an optimum response, we can augment the design to learn more about the shape of the response surface If curvature is significant, and indicates that the design is centered (or near) an optimum response, we can augment the design to learn more about the shape of the response surface Response Surface Design and Methods Response Surface Design and Methods

9. Follow-up Designs If curvature is not significant, or indicates that the design is not near an optimum response, we can search for the optimum response If curvature is not significant, or indicates that the design is not near an optimum response, we can search for the optimum response Steepest Ascent (if maximizing the response is the goal) is a straightforward approach to optimizing the response Steepest Ascent (if maximizing the response is the goal) is a straightforward approach to optimizing the response

10. Steepest Ascent The steepest ascent direction is derived from the additive model for an experiment expressed in either coded or uncoded units. The steepest ascent direction is derived from the additive model for an experiment expressed in either coded or uncoded units. Helicopter II Example (Minitab Project) Helicopter II Example (Minitab Project) –Rotor Length (7 cm, 12 cm) –Rotor Width (3 cm, 5 cm) –5 centerpoints (9.5 cm, 4 cm)

11. Steepest Ascent Helicopter II Example: Helicopter II Example:

12. Steepest Ascent The coefficients from either the coded or uncoded additive model define the steepest ascent vector (b 1 b 2 ) ’. The coefficients from either the coded or uncoded additive model define the steepest ascent vector (b 1 b 2 ) ’. Helicopter II Example Helicopter II Example 2.775+.425RL-.175RW= 2.775+.425(RL*-9.5)/2.5-.175(RW*-3)= (2.775-1.615+.525) +.17RL* -.175RW*= 1.685+.17RL*-.175RW*

14. Steepest Ascent With a steepest ascent direction in hand, we select design points, starting from the centerpoint along this path and continue until the response stops improving. With a steepest ascent direction in hand, we select design points, starting from the centerpoint along this path and continue until the response stops improving. If the first step results in poorer performance, then it may be necessary to backtrack If the first step results in poorer performance, then it may be necessary to backtrack For the helicopter example, let’s use (1, -1) ’ as an ascent vector. For the helicopter example, let’s use (1, -1) ’ as an ascent vector.

15. Steepest Ascent Helicopter II Example: Helicopter II Example: RunRW*RL* 1312 22.512.5 3213 41.513.5 5114

16. Steepest Ascent The point along the steepest ascent direction with highest mean response will serve as the centerpoint of the new design The point along the steepest ascent direction with highest mean response will serve as the centerpoint of the new design Choose new factor levels (guidelines here are vague) Choose new factor levels (guidelines here are vague) Confirm that Confirm that Add axial points to the design to fully characterize the shape of the response surface and predict the maximum. Add axial points to the design to fully characterize the shape of the response surface and predict the maximum.

18. Steepest Ascent The axial points are chosen so that the response at each combination of factor levels is estimated with approximately the same precision. The axial points are chosen so that the response at each combination of factor levels is estimated with approximately the same precision. With 9 distinct design points, we can comfortably estimate a full quadratic response surface With 9 distinct design points, we can comfortably estimate a full quadratic response surface We usually translate and rotate X1 and X2 to characterize the response surface (canonical analysis) We usually translate and rotate X1 and X2 to characterize the response surface (canonical analysis)