Centerpoint Designs Include nc 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) Y=process yield A=Reaction time (150, 155, 160 seconds) B=Temperature (30, 35, 40) Factors can’t be nominal

Centerpoint Plot 41.5 40 +1 40.340.540.740.240.6 B -1 39.3 40.9 -1 +1 -1 39.3 40.9 -1 +1 A

Summary Statistics Statistics used for test of curvature Hold piece of paper up for main effects model, then twist paper for interaction model.

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

T test A test statistic for curvature

T test statistic T has a t distribution with nC-1 df (t.975,4=2.776) T>0 indicates a hilltop or ridge T<0 indicates a valley Effect was positive, but negligible. Compute p-value

T test for factor effects We can use sC to construct t tests (with nC-1 df ) for the factor effects as well E.g., To test H0: effect A = 0 the test statistic would be: Do Minitab example. Show output. Regression approach would “fail”.

Positive curvature test 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

Inconclusive curvature test 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 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) Rotor Length (7 cm, 12 cm) Rotor Width (3 cm, 5 cm) 5 centerpoints (9.5 cm, 4 cm) First or second factorial fit in project. Orthogonality means we can use interaction model too, but need additive model for contour plot. Helicopter II example is Worksheet 2 from 2011.

Contour Plot and Steepest Ascent Show how to make this plot using FITS

Standardized factor levels Helicopter II Example: Uncoded values have *

Steepest ascent vector The coefficients from either the coded or uncoded additive model define the steepest ascent vector (b1 b2)’. 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* Both sets of coordinates are available from the output.

Stepwise testing 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 For the helicopter example, let’s use (1, -1)’ as an ascent vector. The ascent vector points toward a higher aspect ratio

Steepest Ascent steps Helicopter II Example: Run RW* RL* 1 3 12 2 2.5 12.5 13 4 1.5 13.5 5 14

Steepest Ascent follow-up 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) Confirm that

Quadratic mean response surface Add axial points to the design to fully characterize the shape of the response surface and predict the maximum.

Central Composite Design Run in Minitab. It’s a central composite design with nc=3, followed by axial points and two more centerpoints.

Fitted quadratic response surface 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

Canonical response surface We usually translate and rotate X1 and X2 to characterize the response surface (canonical analysis) The signs of B1, B2, B11 and B22 provide information on the shape of the surface.

Response Surface B1 B2 B11 B22 Shape <0 Local Max >0 Local Min <0 Local Max >0 Local Min <0 (>0) >0 (<0) Saddlepoint 0 (<0) <0 (0) Stationary Ridge 0 (>0) >0 (0) Stationary Valley Rising Ridge Descending Valley

Response Surface Example Helicopter Rotor Length* (4 in, 6 in) Body Length* (2 in, 4 in) 22 design for factorial points 5 centerpoints (5 in, 3 in) 4 axial points