D -optimal designs for logistic regression in two variables Linda Haines Department of Statistical Sciences University of Cape Town South Africa Statistical Sciences, UCT mODa8, 2007 – 1 / 19
Joint Work with Statistical Sciences, UCT mODa8, 2007 – 2 / 19
Joint Work with Ga¨ etan Kabera, University of KwaZulu-Natal, South Africa ● Statistical Sciences, UCT mODa8, 2007 – 2 / 19
Joint Work with Ga¨ etan Kabera, University of KwaZulu-Natal, South Africa ● Principal Ndlovu, University of KwaZulu-Natal, South Africa ● Statistical Sciences, UCT mODa8, 2007 – 2 / 19
Joint Work with Ga¨ etan Kabera, University of KwaZulu-Natal, South Africa ● Principal Ndlovu, University of KwaZulu-Natal, South Africa ● Tim O’Brien, Loyola University, Chicago, USA ● Statistical Sciences, UCT mODa8, 2007 – 2 / 19
Outline Statistical Sciences, UCT mODa8, 2007 – 3 / 19
Outline ● Problem Statistical Sciences, UCT mODa8, 2007 – 3 / 19
Outline ● Problem ● Results Statistical Sciences, UCT mODa8, 2007 – 3 / 19
Outline ● Problem ● Results ● Conclusions Statistical Sciences, UCT mODa8, 2007 – 3 / 19
Scope Statistical Sciences, UCT mODa8, 2007 – 4 / 19
Scope ● Drug synergy Statistical Sciences, UCT mODa8, 2007 – 4 / 19
Scope ● Drug synergy ● Models Statistical Sciences, UCT mODa8, 2007 – 4 / 19
Scope ● Drug synergy ● Models ● Optimal Design Statistical Sciences, UCT mODa8, 2007 – 4 / 19
Problem Statistical Sciences, UCT mODa8, 2007 – 5 / 19
Problem ● In search of algebraic solutions! Statistical Sciences, UCT mODa8, 2007 – 5 / 19
Problem ● In search of algebraic solutions! ● Model logit ( p ) = β 0 + β 1 x 1 + β 2 x 2 Statistical Sciences, UCT mODa8, 2007 – 5 / 19
Problem ● In search of algebraic solutions! ● Model logit ( p ) = β 0 + β 1 x 1 + β 2 x 2 ● Locally D -optimal designs Statistical Sciences, UCT mODa8, 2007 – 5 / 19
Early Work Statistical Sciences, UCT mODa8, 2007 – 6 / 19
Early Work ● Sitter and Torsney, 1995 Statistical Sciences, UCT mODa8, 2007 – 6 / 19
Early Work ● Sitter and Torsney, 1995 ● Atkinson and Haines, 1996 Statistical Sciences, UCT mODa8, 2007 – 6 / 19
Early Work ● Sitter and Torsney, 1995 ● Atkinson and Haines, 1996 ● Jia and Myers, 2001 Statistical Sciences, UCT mODa8, 2007 – 6 / 19
Model Statistical Sciences, UCT mODa8, 2007 – 7 / 19
Model ● Logistic regression logit ( p ) = β 0 + β 1 d 1 + β 2 d 2 d 1 , d 2 ≥ 0 with where β 1 , β 2 ≥ 0 and β 0 < 0 Statistical Sciences, UCT mODa8, 2007 – 7 / 19
Model ● Logistic regression logit ( p ) = β 0 + β 1 d 1 + β 2 d 2 d 1 , d 2 ≥ 0 with where β 1 , β 2 ≥ 0 and β 0 < 0 ● Formulate as logit ( p ) = β 0 + z 1 + z 2 z 1 , z 2 ≥ 0 β 0 < 0 with and Statistical Sciences, UCT mODa8, 2007 – 7 / 19
Response Surface Statistical Sciences, UCT mODa8, 2007 – 8 / 19
Response Surface β 0 = − 4 Statistical Sciences, UCT mODa8, 2007 – 8 / 19
Locally D -optimal Designs Statistical Sciences, UCT mODa8, 2007 – 9 / 19
Locally D -optimal Designs ● Approximate � � ( z 11 , z 21 ) , ( z 1 r , z 2 r ) . . . , ξ = w 1 , . . . , w r where 0 < w i < 1 and � r i =1 w i = 1 Statistical Sciences, UCT mODa8, 2007 – 9 / 19
Locally D -optimal Designs ● Approximate � � ( z 11 , z 21 ) , ( z 1 r , z 2 r ) . . . , ξ = w 1 , . . . , w r where 0 < w i < 1 and � r i =1 w i = 1 ● Information matrix 1 z 1 z 2 e u z 2 M ( β 0 ; z ) = z 1 z 1 z 2 1 (1 + e u ) 2 z 2 z 2 z 1 z 2 2 where u = β 0 + z 1 + z 2 Statistical Sciences, UCT mODa8, 2007 – 9 / 19
Locally D -optimal Designs ● Approximate � � ( z 11 , z 21 ) , ( z 1 r , z 2 r ) . . . , ξ = w 1 , . . . , w r where 0 < w i < 1 and � r i =1 w i = 1 ● Information matrix 1 z 1 z 2 e u z 2 M ( β 0 ; z ) = z 1 z 1 z 2 1 (1 + e u ) 2 z 2 z 2 z 1 z 2 2 where u = β 0 + z 1 + z 2 ● Maximize | M ( β 0 ; ξ ) | Statistical Sciences, UCT mODa8, 2007 – 9 / 19
Four-Point Designs Statistical Sciences, UCT mODa8, 2007 – 10 / 19
Four-Point Designs Statistical Sciences, UCT mODa8, 2007 – 10 / 19
Four-Point Designs � ( − u − β 0 , 0) � (0 , − u − β 0 ) ( u − β 0 , 0) (0 , u − β 0 ) ξ ⋆ f = 1 1 w w 2 − w 2 − w with 0 < u ≤ − β 0 Statistical Sciences, UCT mODa8, 2007 – 10 / 19
D -optimality Statistical Sciences, UCT mODa8, 2007 – 11 / 19
D -optimality ● Maximize � = 2 e 3 u u 2 w (1 − 2 w ) { ( u − β 0 ) 2 + 8 β 0 uw } � � M ( β ; ξ ⋆ � f ) � � (1 + e u ) 6 Statistical Sciences, UCT mODa8, 2007 – 11 / 19
D -optimality ● Maximize � = 2 e 3 u u 2 w (1 − 2 w ) { ( u − β 0 ) 2 + 8 β 0 uw } � � M ( β ; ξ ⋆ � f ) � � (1 + e u ) 6 ● Optimal u and w u 2 (3+3 e u +2 u − 2 ue u )+ β 2 0 (1+ e u +2 u − 2 ue u )+ a (1+ e u + u − ue u ) = 0 Statistical Sciences, UCT mODa8, 2007 – 11 / 19
D -optimality ● Maximize � = 2 e 3 u u 2 w (1 − 2 w ) { ( u − β 0 ) 2 + 8 β 0 uw } � � M ( β ; ξ ⋆ � f ) � � (1 + e u ) 6 ● Optimal u and w u 2 (3+3 e u +2 u − 2 ue u )+ β 2 0 (1+ e u +2 u − 2 ue u )+ a (1+ e u + u − ue u ) = 0 u 4 + 14 β 2 0 u 2 + β 4 � with a = 0 and w = f ( u ) Statistical Sciences, UCT mODa8, 2007 – 11 / 19
D -optimality ● Maximize � = 2 e 3 u u 2 w (1 − 2 w ) { ( u − β 0 ) 2 + 8 β 0 uw } � � M ( β ; ξ ⋆ � f ) � � (1 + e u ) 6 ● Optimal u and w u 2 (3+3 e u +2 u − 2 ue u )+ β 2 0 (1+ e u +2 u − 2 ue u )+ a (1+ e u + u − ue u ) = 0 u 4 + 14 β 2 0 u 2 + β 4 � with a = 0 and w = f ( u ) ● β 0 = − 4 u ∗ = 1 . 323 and w ∗ = 0 . 189 Statistical Sciences, UCT mODa8, 2007 – 11 / 19
Directional Derivative Statistical Sciences, UCT mODa8, 2007 – 12 / 19
Directional Derivative β 0 = − 4 Statistical Sciences, UCT mODa8, 2007 – 12 / 19
Optimality? Statistical Sciences, UCT mODa8, 2007 – 13 / 19
Optimality? Trend ● β 0 = − 4 β 0 = − 3 β 0 = − 2 Statistical Sciences, UCT mODa8, 2007 – 13 / 19
Optimality? Trend ● β 0 = − 4 β 0 = − 3 β 0 = − 2 Cut-off ● β 0 ≈ − 1 . 5434 Statistical Sciences, UCT mODa8, 2007 – 13 / 19
Three-Point Designs Statistical Sciences, UCT mODa8, 2007 – 14 / 19
Three-Point Designs Statistical Sciences, UCT mODa8, 2007 – 14 / 19
Three-Point Designs � � (0 , 0) ( u − β 0 , 0) (0 , u − β 0 ) ξ ⋆ t = 1 1 1 3 3 3 Statistical Sciences, UCT mODa8, 2007 – 14 / 19
Three-Point Designs � � (0 , 0) ( u − β 0 , 0) (0 , u − β 0 ) ξ ⋆ t = 1 1 1 3 3 3 where u satisfies 2 − β 0 + 2 e u + β 0 e u + u − ue u = 0 u > β 0 with Statistical Sciences, UCT mODa8, 2007 – 14 / 19
Directional Derivative β 0 = − 1 Statistical Sciences, UCT mODa8, 2007 – 15 / 19
Cut-off Statistical Sciences, UCT mODa8, 2007 – 16 / 19
Cut-off ● Condition u ∗ = − β 0 so that 1 + β 0 + e β 0 − β 0 e β 0 = 0 and thus β 0 ≈ − 1 . 5434 Statistical Sciences, UCT mODa8, 2007 – 16 / 19
Cut-off ● Condition u ∗ = − β 0 so that 1 + β 0 + e β 0 − β 0 e β 0 = 0 and thus β 0 ≈ − 1 . 5434 ● Locally D -optimal? Statistical Sciences, UCT mODa8, 2007 – 16 / 19
Directional derivative Statistical Sciences, UCT mODa8, 2007 – 17 / 19
Directional derivative ● Optimality condition e u 1 (1 + e β 0 ) 2 β 2 0 + u 2 1 + ( β 0 − u 1 ) z 1 + z 2 1 ≤ 1 e β 0 (1 + e u 1 ) 2 2 β 2 0 where u 1 = β 0 + z 1 + z 2 Statistical Sciences, UCT mODa8, 2007 – 17 / 19
Directional derivative ● Optimality condition e u 1 (1 + e β 0 ) 2 β 2 0 + u 2 1 + ( β 0 − u 1 ) z 1 + z 2 1 ≤ 1 e β 0 (1 + e u 1 ) 2 2 β 2 0 where u 1 = β 0 + z 1 + z 2 ● Quadratic in z 1 for constant u 1 Statistical Sciences, UCT mODa8, 2007 – 17 / 19
Directional derivative ● Optimality condition e u 1 (1 + e β 0 ) 2 β 2 0 + u 2 1 + ( β 0 − u 1 ) z 1 + z 2 1 ≤ 1 e β 0 (1 + e u 1 ) 2 2 β 2 0 where u 1 = β 0 + z 1 + z 2 ● Quadratic in z 1 for constant u 1 ● Thus require ≤ 2 e β 0 (1 + e u 1 ) 2 ( β 2 0 + u 2 1 ) e u 1 (1 + e β 0 ) 2 β 2 0 Statistical Sciences, UCT mODa8, 2007 – 17 / 19
One variable Statistical Sciences, UCT mODa8, 2007 – 18 / 19
One variable ● Model logit ( p ) = β 0 + β 1 x 1 = u Statistical Sciences, UCT mODa8, 2007 – 18 / 19
One variable ● Model logit ( p ) = β 0 + β 1 x 1 = u ● D -optimality where u ∗ ≈ 1 . 5434 ± u ∗ Statistical Sciences, UCT mODa8, 2007 – 18 / 19
One variable ● Model logit ( p ) = β 0 + β 1 x 1 = u ● D -optimality where u ∗ ≈ 1 . 5434 ± u ∗ ● Directional derivative ≤ 2 e u ∗ (1 + e u ) 2 (( u ∗ ) 2 + u 2 ) e u (1 + e u ∗ ) 2 ( u ∗ ) 2 Statistical Sciences, UCT mODa8, 2007 – 18 / 19
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