[PPT] - Optimal Cutpoint Determination The Case of A One Point Design rs PowerPoint Presentation

SLIDE 1

Optimal Cutpoint Determination The Case of A One Point Design

❇❡♥ ❚♦rs♥❡② ❜❡♥t❅st❛ts✳❣❧❛✳❛❝✳✉❦ ❚❤❡ ◆❣✉②❡♥ t❤❡❅st❛ts✳❣❧❛✳❛❝✳✉❦ ❉❡♣❛rt♠❡♥t ♦❢ ❙t❛t✐st✐❝s ❯♥✐✈❡rs✐t② ♦❢ ●❧❛s❣♦✇

1

SLIDE 2

Introduction

❙✉r✈❡② t♦ ❜❡ ❝♦♥❞✉❝t❡❞

X✿ ✈❛r✐❛❜❧❡ ♦❢ ✐♥t❡r❡st✿ X ∈ X ❂ ❬❈✱❉❪✿ ❙❛♠♣❧❡ ❙♣❛❝❡

❈❛t❡❣♦r✐❡s ♦❢ ❘❡s♣♦♥s❡ ❘❡❝♦r❞❡❞

x1 x2 x0 xk xk−1 D C ❈❛t❡❣♦r② ❧✐♠✐ts ♦r ❈✉t✲♣♦✐♥ts x1, x2 . . . , xk−1 t♦ ❜❡ ❝❤♦s❡♥ ✐♥ ❛❞✈❛♥❝❡✳

❆ ♥♦♥✲❧✐♥❡❛r ❞❡s✐❣♥ ♣r♦❜❧❡♠

❉❡s✐❣♥ ♣♦✐♥t ❂ x = (x1, x2 . . . , xk−1)T xi ∈ X = [C, D], x1 < x2 < . . . < xk−1

❙♦♠❡ ❆♣♣❧✐❝❛t✐♦♥s✿

✷✳ ▼❛r❦❡t ❘❡s❡❛r❝❤ ❙t✉❞✐❡s✱ X ✐s ■♥❝♦♠❡ ✭s❡♥s✐t✐✈❡✮✳ ✸✳ ❈♦♥t✐♥❣❡♥t ❱❛❧✉❛t✐♦♥ ❙t✉❞✐❡s✱ X ✐s ❲❚P ♦r ❧♦❣✭❲❚P✮✳ WTP: Willingness To Pay for some non-market product or service.

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SLIDE 3

A Generalized Linear Model

P(X ≤ x) = F((x − µ)/σ), x ∈ X, (2) µ ✿ ❧♦❝❛t✐♦♥ ♣❛r❛♠❡t❡r✱ σ✿ s❝❛❧❡ ♣❛r❛♠❡t❡r✳ ❊q✉✐✈❛❧❡♥t❧②✿ P(X ≤ x) = F(α + βx), x ∈ X α = −(µ/σ), β = 1/σ ❆ ❣❡♥❡r❛❧✐③❡❞ ❧✐♥❡❛r ♠♦❞❡❧ ✐♥ α, β✳ ▲❡t λ = (α, β)T

Design Objectives

❈r✐t❡r✐❛✿

⊲ ▼✐♥✐♠✐③❡ V ar(ˆ

µ), µ = −α/β, ˆ µ = −ˆ α/ˆ β V ar(ˆ µ) ∼ = V ar(cT ˆ λ), c = ∂µ/∂λ ∝ −(1, µ)T/β

⊲ ▼✐♥✐♠✐③❡ V ar(ˆ

σ), σ = 1/β, ˆ σ = 1/ˆ β V ar(ˆ σ) ∼ = V ar(cT ˆ λ), c = ∂σ/∂λ ∝ −(0, 1)T/β2

●❡♥❡r❛❧ ❖❜❥❡❝t✐✈❡✿

▼❛❦❡ C = Cov(ˆ λ) ✧s♠❛❧❧✧

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SLIDE 4

❈r✐t❡r✐❛✿

⊲ ♠✐♥✐♠✐③❡ det(C) (D − opt) ⊲ ♠✐♥✐♠✐③❡ tr(C) (A − opt)

Note: Optimal designs here are locally optimal designs.

❈❤❛r❛❝t❡r✐③❛t✐♦♥✴❙t❛♥❞❛r❞✐③❛t✐♦♥

▲❡t✿ Z = (X − µ)/σ = α + βX, z = (x − µ)/σ = α + βx A = (C − µ)/σ = α + βC, B = (D − µ)/σ = α + βD ❚❤❡♥✿ P(X ≤ x) = P(Z ≤ z) = F(z), z ∈ Z = [A, B] Z✿ ❙t❛♥❞❛r❞✐③❡❞ ✈❡rs✐♦♥ ♦❢ X

❙t❛♥❞❛r❞✐③❡❞ Pr♦❜❧❡♠✿

❉❡t❡r♠✐♥❡ ❝✉t✲♣♦✐♥ts z1, z2, . . . , zk−1 ❙❛t✐s❢②✐♥❣ A = z0 < z1 < z2 < . . . < zk−1 < zk = B

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SLIDE 5

A One-Point Design Problem

❖♥❡ ❉❡s✐❣♥ P♦✐♥t✿

z = (z1, z2, . . . , zk−1) zi ∈ Z = [A, B], z1 < z2 < . . . < zk−1 zj = (xj − µ)/σ = α + βxj, j = 0, 1, 2, . . . , k Ford, Torsney and Wu (1992) used this approach for the two-category case.

❋✐s❤❡r ■♥❢♦r♠❛t✐♦♥ ♠❛tr✐① IZ✲❚❤❡ ❢♦r♠✉❧❛

IZ = ZQZT ◆♦♥✲s✐♥❣✉❧❛r ❢♦r k ≥ 3 ZT = (1k−1|z) 1n = (1, 1, . . . , 1) ∈ ℜn Q = DfHD−1

θ HTDf

Df =❞✐❛❣{f(z1), f(z2), . . . , f(zk−1)}, f(z) = F ′(z) Dθ =❞✐❛❣(θ1, θ2, . . . , θk)✱ θi : ❈❡❧❧ ♣r♦❜❛❜✐❧✐t✐❡s✳ H = (Ik−1|0k−1) − (0k−1|Ik−1) 0n = (0, 0, . . . , 0)T ∈ ℜn In : ■❞❡♥t✐t② ♠❛tr✐① ♦❢ ♦r❞❡r ♥✳

5

SLIDE 6

❉❡t❡r♠✐♥✐♥❣ ❛♥ ♦♣t✐♠❛❧ z∗ ❢♦r s②♠♠❡tr✐❝ F(z)✿

⊲ k = 3 : z∗ = (−z∗, z∗). ⊲ k = 4 : z∗ = (−z∗, 0, z∗). ⊲ k = 5 : z∗ = (−z∗

2, −z∗ 1, z∗ 1, z∗ 2).

⊲ k = 6 : z∗ = (−z∗

2, −z∗ 1, 0, z∗ 1, z∗ 2).

❈r✐t❡r✐❛ ❝♦♥s✐❞❡r❡❞✿

Det(C−1) ∝ Det(IZ) V ar(ˆ µ) ∝ (1, 0)(I−1

Z )(1, 0)T (e1 − optimality)

V ar(ˆ σ) ∝ (0, 1)(I−1

Z )(0, 1)T (e2 − optimality)

❲❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❝r✐t❡r✐❛✿

⊲ D✲❖♣t✐♠❛❧✐t②✿ ▼❛①✐♠✐s❡ ④log det(Iz)⑥ ⊲ A✲❖♣t✐♠❛❧✐t②✿ ▼❛①✐♠✐s❡ {−tr(I−1

z )}

⊲ e1✲❖♣t✐♠❛❧✐t②✿ ▼❛①✐♠✐s❡ {−eT

1 I−1 z e1}

⊲ e2✲❖♣t✐♠❛❧✐t②✿ ▼❛①✐♠✐s❡ {−eT

2 I−1 z e2}

❲❡ ❝❤♦♦s❡ z ♦r z1, z2 ✭z1 < z2✮ t♦ ♠❛①✐♠✐③❡ ♦♥❡ ♦❢ t❤❡ ❛❜♦✈❡ ❝r✐t❡r✐❛ ❛s t❤❡ ❢✉♥❝t✐♦♥ ♦❢ Iz

6

SLIDE 7

❙♦♠❡ r❡s✉❧ts✿ ❲❡ ✉s❡ s❡❛r❝❤ ♠❡t❤♦❞ ❛♥❞ t❤❡ s②♠♠❡tr✐❝ ❞✐str✐❜✉t✐♦♥s✳

❚❛❜❧❡ ✶✿ ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❢♦r ❧♦❣✐st✐❝ ❞✐str✐❜✉t✐♦♥✱ ❦❂✸ ❛♥❞ ❦❂✹ ❦❂✸ ❦❂✹ ❈r✐t❡r✐♦♥ z∗ F(z∗) φ(z∗) z∗ F(z∗) φ(z∗) D✲♦♣t✐♠❛❧✐t② ✶✳✹✼✵✵ ✵✳✽✶✸✶ ✲✶✳✺✺✻✼ ✶✳✾✽✵✵ ✵✳✽✼✽✼ ✲✶✳✷✹✽✸ A✲♦♣t✐♠❛❧✐t② ✶✳✶✻✵✵ ✵✳✼✻✶✸ ✲✺✳✵✶✽✷ ✶✳✼✶✵✵ ✵✳✽✹✻✽ ✲✹✳✸✼✽✾ e1✲♦♣t✐♠❛❧✐t② ✵✳✻✾✵✵ ✵✳✻✻✻✵ ✲✸✳✸✼✺✵ ✶✳✶✵✵✵ ✵✳✼✺✵✸ ✲✸✳✷✵✵✵ e2✲♦♣t✐♠❛❧✐t② ✷✳✶✼✵✵ ✵✳✽✾✼✺ ✲✶✳✵✷✷✻ ✷✳✶✼✵✵ ✵✳✽✾✼✺ ✲✶✳✵✷✷✻ E✲♦♣t✐♠❛❧✐t② ✵✳✻✾✵✵ ✵✳✻✻✻✵ ✲✸✳✸✼✺✵ ✶✳✶✵✵✵ ✵✳✼✺✵✸ ✲✸✳✷✵✵✵ ❚❛❜❧❡ ✷✿ ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❢♦r ❧♦❣✐st✐❝ ❞✐str✐❜✉t✐♦♥✱ ❦❂✺ ❈r✐t❡r✐♦♥ z∗

1

z∗

2

F(z∗

1)

F(z∗

2)

φ(z∗

1, z∗ 2)

D✲♦♣t✐♠❛❧✐t② ✵✳✽✺✵✵ ✷✳✺✶✵✵ ✵✳✼✵✵✻ ✵✳✾✷✹✽ ✲✶✳✵✼✵✾ A✲♦♣t✐♠❛❧✐t② ✵✳✻✶✵✵ ✷✳✶✻✵✵ ✵✳✻✹✼✾ ✵✳✽✾✻✻ ✲✹✳✶✷✹✺ e1✲♦♣t✐♠❛❧✐t② ✵✳✹✶✵✵ ✶✳✸✾✵✵ ✵✳✻✵✶✶ ✵✳✽✵✵✻ ✲✸✳✶✷✺✶ e2✲♦♣t✐♠❛❧✐t② ✶✳✺✾✵✵ ✸✳✶✼✵✵ ✵✳✽✸✵✻ ✵✳✾✺✾✼ ✲✵✳✽✷✽✹ E✲♦♣t✐♠❛❧✐t② ✵✳✹✶✵✵ ✶✳✸✾✵✵ ✵✳✻✵✶✶ ✵✳✽✵✵✻ ✲✸✳✶✷✺✶

7

SLIDE 8

❚❛❜❧❡ ✸✿ ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❢♦r ❧♦❣✐st✐❝ ❞✐str✐❜✉t✐♦♥✱ ❦❂✻ ❈r✐t❡r✐♦♥ z∗

1

z∗

2

F(z∗

1)

F(z∗

2)

φ(z∗

1, z∗ 2)

D✲♦♣t✐♠❛❧✐t② ✶✳✸✸✵✵ ✷✳✾✶✵✵ ✵✳✼✾✵✽ ✵✳✾✹✽✸ ✲✵✳✾✼✽✽ A✲♦♣t✐♠❛❧✐t② ✶✳✵✺✵✵ ✷✳✺✹✵✵ ✵✳✼✹✵✽ ✵✳✾✷✻✾ ✲✸✳✾✾✹✷ e1✲♦♣t✐♠❛❧✐t② ✵✳✻✾✵✵ ✶✳✻✶✵✵ ✵✳✻✻✻✵ ✵✳✽✸✸✹ ✲✸✳✵✽✺✼ e2✲♦♣t✐♠❛❧✐t② ✶✳✺✾✵✵ ✸✳✶✼✵✵ ✵✳✽✸✵✻ ✵✳✾✺✾✼ ✲✵✳✽✷✽✹ E✲♦♣t✐♠❛❧✐t② ✵✳✻✾✵✵ ✶✳✻✶✵✵ ✵✳✻✻✻✵ ✵✳✽✸✸✹ ✲✸✳✵✽✺✼

❋✐❣✉r❡ ✶✿ P❧♦ts ♦❢ ❝r✐t❡r✐❛ ✈❛❧✉❡ ✈s✳ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛t❡❣♦r✐❡s k

Criteria value vs. k: logistic-D

k criteria value 3.0 3.5 4.0 4.5 5.0 5.5 6.0

1.5
1.4
1.3
1.2
1.1
1.0

Criteria value vs. k: Normal-e1

k criteria value 3.0 3.5 4.0 4.5 5.0 5.5 6.0

1.20
1.15
1.10

8

SLIDE 9

z$theta1 z$theta2 0.0 0.5 1.0 0.0 0.5 1.0

6.2
6
6
5.8
5.8
5.6
5.6
5.4
5.4
5.2
5.2
5
5
4.8
4.6
4.4
4.2
4
3.8
3.8
3.6
3.6
3.4
3.4
3.2
3.2
3
3
3
3
3
3
2.8
2.8
2.8
2.8
2.8
2.8
2.8
2.8
2.8
2.6
2.6
2.6
2.6
2.6
2.6
2.4
2.4
2.2
2.2
2.2
2.2
2
1.8
1.6
1.6

theta2=1 theta1=1 theta3=1 max = -1.557168 z$theta1 z$theta2 0.0 0.5 1.0

0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2

5.1
5.1
5
5
4.9
4.9
4.8
4.8
4.7
4.7
4.6
4.6
4.5
4.5
4.4
4.4
4.3
4.2
4.1
4
3.9
3.8
3.7
3.6
3.5
3.4
3.3
3.2
3.1
3
2.9
2.8
2.7
2.6
2.5
2.4
2.3
2.2
2.1
2
1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.3
1.2
1.1
1.1
1
0.9
0.9
0.8
0.8
0.8
0.8
0.8
0.8
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.6
0.6
0.6
0.6 -0.6
0.5
0.5
0.5
0.5
0.5
0.4
0.3

Theta 2=1 theta 1=1 theta 3=1 max = -0.208398 z$theta1 z$theta2 0.0 0.5 1.0 0.0 0.5 1.0

6.6
6.6
6.4
6.4
6.2
6.2
6
6
5.8
5.8
5.6
5.6
5.4
5.4
5.2
5.2
5
5
4.8
4.8
4.6
4.6
4.4
4.4
4.2
4.2
4
4
3.8
3.6
3.4
3.2
3
3
2.8
2.8
2.6
2.6
2.4
2.4
2.4
2.4
2.2
2.2
2.2
2
2
1.8
1.8
1.8
1.8
1.6
1.6
1.6
1.6
1.4
1.4
1.2
1.2

Theta 2=1 theta 1=1 theta 3=1 max = -1.127672 z$theta1 z$theta2 0.0 0.5 1.0

0.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2

11
11
10.8
10.8
10.6
10.6
10.4
10.4
10.2
10.2
10
10
9.8
9.8
9.8
9.6
9.6
9.6
9.4
9.4
9.4
9.2
9.2
9.2
9
9
9
8.8
8.8
8.8
8.6
8.6
8.6
8.4
8.4
8.4
8.2
8.2
8.2
8
8
8
7.8
7.8
7.8
7.6
7.6
7.6
7.4
7.4
7.4
7.4
7.4
7.2
7.2
7.2
7
7
7
6.8
6.8
6.8
6.6
6.6
6.6
6.6
6.6
6.4
6.4
6.4
6.2
6.2
6.2
6
6
6
5.8
5.8
5.8
5.6
5.6
5.6
5.4
5.4
5.4
5.4
5.4
5.2
5.2
5.2
5.2
5.2
5
5
5
4.8
4.8
4.8
4.6
4.6
4.6
4.6
4.6
4.4
4.4
4.4
4.4
4.2
4.2
4.2
4.2
4.2
4
4
4
4
4
4
3.8
3.8
3.8
3.8
3.6
3.6
3.4
3.2
3
2.8
2.6
2.4
2.4
2.2
2.2
2.2
2.2

Theta 2=1 theta 1=1 theta 3=1 max = -2.081043

❋✐❣✉r❡ ✷✿ P❧♦ts ✈❡rs✉s ❝❡❧❧ ♣r♦❜❛❜✐❧✐t✐❡s θ1, θ2, θ3✱ ✭k = 3✮ θ∗ = (.187, .626, .187) ❉✲♦♣t✐♠❛❧✐t② ❛♥❞ ▲♦❣✐st✐❝ ❉✐str✐❜✉t✐♦♥ θ∗ = (.133, .734, .133) ❉✲♦♣t✐♠❛❧✐t② ❛♥❞ ◆♦r♠❛❧ ❉✐str✐❜✉t✐♦♥ θ∗ 1 = (.1, .4, .5)❀ θ∗ 2 = (.5, .4, .1) ❉✲♦♣t✐♠❛❧✐t② ❛♥❞ ❉♦✉❜❧❡✲❊①♣♦♥❡♥t✐❛❧ ❉✐str✐❜✉t✐♦♥ θ∗ 1 = (.25, .25, .5)❀ θ∗ 2 = (.5, .25, .25) ❉✲♦♣t✐♠❛❧✐t② ❛♥❞ ❉♦✉❜❧❡✲❘❡❝✐♣r♦❝❛❧ ❉✐str✐❜✉t✐♦♥ 9

SLIDE 10

Conclusion and Future Work

❆❧❧ ❝r✐t❡r✐❛ ✐♥❝r❡❛s❡ ✇❤❡♥ ✇❡ ✐♥❝r❡❛s❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛t❡❣♦r✐❡s k ❜✉t ❧❡✈❡❧ ♦✛

❢r♦♠ k ∼ 4, 5✳

❲❡ ❝❛♥ ✉s❡ ♦♥❡ ♣♦✐♥t ❞❡s✐❣♥s ✐♥ ✇❤✐❝❤ t❤❡ ♥✉♠❜❡r ♦❢ ❝✉t✲♣♦✐♥ts ♠❛② ❜❡ ✹ ♦r ✺✳
■♥ t❤❡ ❝❛s❡ ♦❢ ❞♦✉❜❧❡ ❡①♣♦♥❡♥t✐❛❧ ❛♥❞ ❞♦✉❜❧❡ r❡❝✐♣r♦❝❛❧ ❞✐str✐❜✉t✐♦♥s✱ t❤❡r❡ ❛r❡

t✇♦ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥s ❢♦r t❤r❡❡ ❝❛t❡❣♦r✐❡s✳

Future Work

❆s②♠♠❡tr✐❝ ❞✐str✐❜✉t✐♦♥✿ ❯s✐♥❣ ❛ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❛❧❣♦r✐t❤♠✳
▼✉❧t✐♣❧❡ ♣♦✐♥t ❞❡s✐❣♥s✳
❚❤❡ ❜✐✈❛r✐❛t❡ ❛♣♣r♦❛❝❤✳
❊①t❡♥s✐♦♥s t♦ ❤✐❣❤❡r ❞✐♠❡♥s✐♦♥s✳

10