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

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

Introduction • ❙✉r✈❡② t♦ ❜❡ ❝♦♥❞✉❝t❡❞ X ✿ ✈❛r✐❛❜❧❡ ♦❢ ✐♥t❡r❡st✿ X ∈ X ❂ ❬❈✱❉❪✿ ❙❛♠♣❧❡ ❙♣❛❝❡ • ❈❛t❡❣♦r✐❡s ♦❢ ❘❡s♣♦♥s❡ ❘❡❝♦r❞❡❞ C D x 0 x 1 x 2 x k − 1 x k ❈❛t❡❣♦r② ❧✐♠✐ts ♦r ❈✉t✲♣♦✐♥ts x 1 , x 2 . . . , x k − 1 t♦ ❜❡ ❝❤♦s❡♥ ✐♥ ❛❞✈❛♥❝❡✳ • ❆ ♥♦♥✲❧✐♥❡❛r ❞❡s✐❣♥ ♣r♦❜❧❡♠ ❉❡s✐❣♥ ♣♦✐♥t ❂ x = ( x 1 , x 2 . . . , x k − 1 ) T x i ∈ X = [ C, D ] , x 1 < x 2 < . . . < x k − 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. 2

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✐❛✿ α/ ˆ ⊲ ▼✐♥✐♠✐③❡ V ar (ˆ µ ) , µ = − α/β, ˆ µ = − ˆ β = V ar ( c T ˆ µ ) ∼ λ ) , c = ∂µ/∂λ ∝ − (1 , µ ) T /β V ar (ˆ σ = 1 / ˆ ⊲ ▼✐♥✐♠✐③❡ V ar (ˆ σ ) , σ = 1 /β, ˆ β = V ar ( c T ˆ σ ) ∼ λ ) , c = ∂σ/∂λ ∝ − (0 , 1) T /β 2 V ar (ˆ • ●❡♥❡r❛❧ ❖❜❥❡❝t✐✈❡✿ ▼❛❦❡ C = Cov (ˆ λ ) ✧s♠❛❧❧✧ 3

• ❈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 z 1 , z 2 , . . . , z k − 1 ❙❛t✐s❢②✐♥❣ A = z 0 < z 1 < z 2 < . . . < z k − 1 < z k = B 4

A One-Point Design Problem • ❖♥❡ ❉❡s✐❣♥ P♦✐♥t✿ z = ( z 1 , z 2 , . . . , z k − 1 ) z i ∈ Z = [ A, B ] , z 1 < z 2 < . . . < z k − 1 z j = ( x j − µ ) /σ = α + βx j , j = 0 , 1 , 2 , . . . , k Ford, Torsney and Wu (1992) used this approach for the two-category case. • ❋✐s❤❡r ■♥❢♦r♠❛t✐♦♥ ♠❛tr✐① I Z ✲❚❤❡ ❢♦r♠✉❧❛ I Z = ZQZ T ◆♦♥✲s✐♥❣✉❧❛r ❢♦r k ≥ 3 Z T = (1 k − 1 | z ) 1 n = (1 , 1 , . . . , 1) ∈ ℜ n Q = D f HD − 1 θ H T D f D f = ❞✐❛❣ { f ( z 1 ) , f ( z 2 ) , . . . , f ( z k − 1 ) } , f ( z ) = F ′ ( z ) D θ = ❞✐❛❣ ( θ 1 , θ 2 , . . . , θ k ) ✱ θ i : ❈❡❧❧ ♣r♦❜❛❜✐❧✐t✐❡s✳ H = ( I k − 1 | 0 k − 1 ) − (0 k − 1 | I k − 1 ) 0 n = (0 , 0 , . . . , 0) T ∈ ℜ n I n : ■❞❡♥t✐t② ♠❛tr✐① ♦❢ ♦r❞❡r ♥✳ 5

• ❉❡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 ( I Z ) Z )(1 , 0) T ( e 1 − optimality ) µ ) ∝ (1 , 0)( I − 1 V ar (ˆ Z )(0 , 1) T ( e 2 − optimality ) σ ) ∝ (0 , 1)( I − 1 V ar (ˆ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❝r✐t❡r✐❛✿ ⊲ D ✲❖♣t✐♠❛❧✐t②✿ ▼❛①✐♠✐s❡ ④ log det ( I z ) ⑥ ⊲ A ✲❖♣t✐♠❛❧✐t②✿ ▼❛①✐♠✐s❡ {− tr ( I − 1 z ) } 1 I − 1 ⊲ e 1 ✲❖♣t✐♠❛❧✐t②✿ ▼❛①✐♠✐s❡ {− e T z e 1 } 2 I − 1 ⊲ e 2 ✲❖♣t✐♠❛❧✐t②✿ ▼❛①✐♠✐s❡ {− e T z e 2 } ❲❡ ❝❤♦♦s❡ z ♦r z 1 , z 2 ✭ z 1 < z 2 ✮ t♦ ♠❛①✐♠✐③❡ ♦♥❡ ♦❢ t❤❡ ❛❜♦✈❡ ❝r✐t❡r✐❛ ❛s t❤❡ ❢✉♥❝t✐♦♥ ♦❢ I z 6

• ❙♦♠❡ 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 ✶✳✶✻✵✵ ✵✳✼✻✶✸ ✲✺✳✵✶✽✷ ✶✳✼✶✵✵ ✵✳✽✹✻✽ ✲✹✳✸✼✽✾ e 1 ✲♦♣t✐♠❛❧✐t② ✵✳✻✾✵✵ ✵✳✻✻✻✵ ✲✸✳✸✼✺✵ ✶✳✶✵✵✵ ✵✳✼✺✵✸ ✲✸✳✷✵✵✵ e 2 ✲♦♣t✐♠❛❧✐t② ✷✳✶✼✵✵ ✵✳✽✾✼✺ ✲✶✳✵✷✷✻ ✷✳✶✼✵✵ ✵✳✽✾✼✺ ✲✶✳✵✷✷✻ E ✲♦♣t✐♠❛❧✐t② ✵✳✻✾✵✵ ✵✳✻✻✻✵ ✲✸✳✸✼✺✵ ✶✳✶✵✵✵ ✵✳✼✺✵✸ ✲✸✳✷✵✵✵ ❚❛❜❧❡ ✷✿ ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❢♦r ❧♦❣✐st✐❝ ❞✐str✐❜✉t✐♦♥✱ ❦❂✺ z ∗ z ∗ F ( z ∗ F ( z ∗ φ ( z ∗ 1 , z ∗ ❈r✐t❡r✐♦♥ 1 ) 2 ) 2 ) 1 2 D ✲♦♣t✐♠❛❧✐t② ✵✳✽✺✵✵ ✷✳✺✶✵✵ ✵✳✼✵✵✻ ✵✳✾✷✹✽ ✲✶✳✵✼✵✾ A ✲♦♣t✐♠❛❧✐t② ✵✳✻✶✵✵ ✷✳✶✻✵✵ ✵✳✻✹✼✾ ✵✳✽✾✻✻ ✲✹✳✶✷✹✺ e 1 ✲♦♣t✐♠❛❧✐t② ✵✳✹✶✵✵ ✶✳✸✾✵✵ ✵✳✻✵✶✶ ✵✳✽✵✵✻ ✲✸✳✶✷✺✶ e 2 ✲♦♣t✐♠❛❧✐t② ✶✳✺✾✵✵ ✸✳✶✼✵✵ ✵✳✽✸✵✻ ✵✳✾✺✾✼ ✲✵✳✽✷✽✹ E ✲♦♣t✐♠❛❧✐t② ✵✳✹✶✵✵ ✶✳✸✾✵✵ ✵✳✻✵✶✶ ✵✳✽✵✵✻ ✲✸✳✶✷✺✶ 7

❚❛❜❧❡ ✸✿ ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❢♦r ❧♦❣✐st✐❝ ❞✐str✐❜✉t✐♦♥✱ ❦❂✻ ❈r✐t❡r✐♦♥ z ∗ z ∗ F ( z ∗ 1 ) F ( z ∗ 2 ) φ ( z ∗ 1 , z ∗ 2 ) 1 2 D ✲♦♣t✐♠❛❧✐t② ✶✳✸✸✵✵ ✷✳✾✶✵✵ ✵✳✼✾✵✽ ✵✳✾✹✽✸ ✲✵✳✾✼✽✽ A ✲♦♣t✐♠❛❧✐t② ✶✳✵✺✵✵ ✷✳✺✹✵✵ ✵✳✼✹✵✽ ✵✳✾✷✻✾ ✲✸✳✾✾✹✷ e 1 ✲♦♣t✐♠❛❧✐t② ✵✳✻✾✵✵ ✶✳✻✶✵✵ ✵✳✻✻✻✵ ✵✳✽✸✸✹ ✲✸✳✵✽✺✼ e 2 ✲♦♣t✐♠❛❧✐t② ✶✳✺✾✵✵ ✸✳✶✼✵✵ ✵✳✽✸✵✻ ✵✳✾✺✾✼ ✲✵✳✽✷✽✹ E ✲♦♣t✐♠❛❧✐t② ✵✳✻✾✵✵ ✶✳✻✶✵✵ ✵✳✻✻✻✵ ✵✳✽✸✸✹ ✲✸✳✵✽✺✼ ❋✐❣✉r❡ ✶✿ P❧♦ts ♦❢ ❝r✐t❡r✐❛ ✈❛❧✉❡ ✈s✳ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛t❡❣♦r✐❡s k Criteria value vs. k: logistic-D Criteria value vs. k: Normal-e1 -1.0 -1.1 -1.10 -1.2 criteria value criteria value -1.15 -1.3 -1.4 -1.20 -1.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 3.0 3.5 4.0 4.5 5.0 5.5 6.0 k k 8