centerpoint designs include n c center points (0,…,0) in a factorial design include n c center...

Centerpoint DesignsCenterpoint Designs Include nInclude ncc center points (0,…,0) in a center points (0,…,0) in a

factorial designfactorial design– Obtains estimate of pure error (at center of Obtains estimate of pure error (at center of

region of interest)region of interest)– Tests of curvatureTests of curvature– We will use C to subscript center points and F We will use C to subscript center points and F

to subscript factorial pointsto subscript factorial points Example (Lochner & Mattar, 1990)Example (Lochner & Mattar, 1990)

– Y=process yieldY=process yield– A=Reaction time (150, 155, 160 seconds)A=Reaction time (150, 155, 160 seconds)– B=Temperature (30, 35, 40)B=Temperature (30, 35, 40)

Centerpoint DesignsCenterpoint Designs

41.5

39.3

40

40.9

A

-1

+1

B

+1

-1

0

0

40.340.540.740.240.6


Statistics used for test of curvatureStatistics used for test of curvature

20736.

425.404/)5.41409.403.39(

46.405/)6.402.407.405.403.40(

C

F

C

s

y

y


When do we have curvature?When do we have curvature? For a main effects or interaction For a main effects or interaction

model,model,

Otherwise, for many types of Otherwise, for many types of curvature,curvature,

y c y F

y c y F


FCC

FC

nns

yyT

11

•A test statistic for curvature


T has a t distribution with nT has a t distribution with nCC-1 df -1 df (t(t.975,4.975,4=2.776)=2.776)

T>0 indicates a hilltop or ridgeT>0 indicates a hilltop or ridge T<0 indicates a valleyT<0 indicates a valley

25.139.

035.

41

51

20736.

425.4046.40

T


We can use sWe can use sCC to construct t tests to construct t tests (with n(with nCC-1 df ) for the factor effects -1 df ) for the factor effects as well as well

E.g., To test HE.g., To test H00: effect A = 0: effect A = 0the test statistic would be:the test statistic would be:

22

k

CsA

T

Follow-up DesignsFollow-up Designs

If curvature is significant, and If curvature is significant, and indicates that the design is centered indicates that the design is centered (or near) an optimum response, we (or near) an optimum response, we can augment the design to learn can augment the design to learn more about the shape of the more about the shape of the response surfaceresponse surface

Response Surface Design and Response Surface Design and MethodsMethods

Follow-up DesignsFollow-up Designs

If curvature is If curvature is notnot significant, or significant, or indicates that the design is indicates that the design is notnot near near an optimum response, we can search an optimum response, we can search for the optimum responsefor the optimum response

Steepest AscentSteepest Ascent (if maximizing the (if maximizing the response is the goal) is a response is the goal) is a straightforward approach to straightforward approach to optimizing the responseoptimizing the response

Steepest AscentSteepest Ascent

The steepest ascent direction is The steepest ascent direction is derived from the additive model for an derived from the additive model for an experiment expressed in either coded experiment expressed in either coded or uncoded units.or uncoded units.

Helicopter II Example (Minitab Project)Helicopter II Example (Minitab Project)– Rotor Length (7 cm, 12 cm)Rotor Length (7 cm, 12 cm)– Rotor Width (3 cm, 5 cm)Rotor Width (3 cm, 5 cm)– 5 centerpoints (9.5 cm, 4 cm)5 centerpoints (9.5 cm, 4 cm)


Helicopter II Example:Helicopter II Example:

RWRWRW

RW

RLRLRL

RL

4*1

4*

5.25.9*5.2

5.9*


The coefficients from either the coded or The coefficients from either the coded or uncoded additive model define the steepest uncoded additive model define the steepest ascent vector (bascent vector (b11 b b22))’’..

Helicopter II ExampleHelicopter II Example

2.775+.425RL-.175RW2.775+.425RL-.175RW==

2.775+.425(RL*-9.5)/2.5-.175(RW*-3)=2.775+.425(RL*-9.5)/2.5-.175(RW*-3)=

(2.775-1.615+.525) + .17RL* -.175RW*=(2.775-1.615+.525) + .17RL* -.175RW*=

1.685+.17RL*-.175RW*1.685+.17RL*-.175RW*

3.2

3.0

2.8

2.6

2.4

RotorLength

Roto

rWid

th

121110987

5.0

4.5

4.0

3.5

3.0

Contour Plot of Flight Time (Seconds) vs Rotor Width, Rotor Length


With a steepest ascent direction in With a steepest ascent direction in hand, we select design points, starting hand, we select design points, starting from the centerpoint along this path from the centerpoint along this path and continue until the response stops and continue until the response stops improving.improving.

If the first step results in poorer If the first step results in poorer performance, then it may be performance, then it may be necessary to backtracknecessary to backtrack

For the helicopter example, let’s use For the helicopter example, let’s use (1, -1)(1, -1)’’ as an ascent vector. as an ascent vector.


Helicopter II Helicopter II Example:Example:

Run RW* RL*

1 3 12

2 2.5 12.5

3 2 13

4 1.5 13.5

5 1 14


The point along the steepest ascent The point along the steepest ascent direction with highest mean response will direction with highest mean response will serve as the centerpoint of the new designserve as the centerpoint of the new design

Choose new factor levels (guidelines here Choose new factor levels (guidelines here are vague)are vague)

Confirm that Confirm that Add Add axialaxial points to the design to fully points to the design to fully

characterize the shape of the response characterize the shape of the response surface and predict the maximum.surface and predict the maximum.

FC yy

1.51.00.50.0-0.5-1.0-1.5

1.5

1.0

0.5

0.0

-0.5

-1.0

-1.5

X1

X2

12

Blocks

Central Composite Design

and 3 Block 2 runs(0,0) includes 3 Block 1 runs


The axial points are chosen so that the response The axial points are chosen so that the response at each combination of factor levels is estimated at each combination of factor levels is estimated with approximately the same precision.with approximately the same precision.

With 9 distinct design points, we can comfortably With 9 distinct design points, we can comfortably estimate a full quadratic response surfaceestimate a full quadratic response surface

We usually translate and rotate X1 and X2 to We usually translate and rotate X1 and X2 to characterize the response surface (canonical characterize the response surface (canonical analysis)analysis)

21122222

211122110)|( xxbxbxbxbxbbXYE

centerpoint designs include n c center points (0,…,0) in a factorial design include n c center...

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