D -optimal designs for logistic regression in two variables Linda - - PowerPoint PPT Presentation

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d optimal designs for logistic regression in two variables
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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

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SLIDE 1

Statistical Sciences, UCT mODa8, 2007 – 1 / 19

D-optimal designs for logistic regression in two variables

Linda Haines Department of Statistical Sciences University of Cape Town South Africa

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SLIDE 2

Joint Work with

Statistical Sciences, UCT mODa8, 2007 – 2 / 19

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SLIDE 3

Joint Work with

Statistical Sciences, UCT mODa8, 2007 – 2 / 19

  • Ga¨

etan Kabera, University of KwaZulu-Natal, South Africa

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SLIDE 4

Joint Work with

Statistical Sciences, UCT mODa8, 2007 – 2 / 19

  • Ga¨

etan Kabera, University of KwaZulu-Natal, South Africa

  • Principal Ndlovu, University of KwaZulu-Natal, South Africa
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SLIDE 5

Joint Work with

Statistical Sciences, UCT mODa8, 2007 – 2 / 19

  • Ga¨

etan Kabera, University of KwaZulu-Natal, South Africa

  • Principal Ndlovu, University of KwaZulu-Natal, South Africa
  • Tim O’Brien, Loyola University, Chicago, USA
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SLIDE 6

Outline

Statistical Sciences, UCT mODa8, 2007 – 3 / 19

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SLIDE 7

Outline

Statistical Sciences, UCT mODa8, 2007 – 3 / 19

  • Problem
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SLIDE 8

Outline

Statistical Sciences, UCT mODa8, 2007 – 3 / 19

  • Problem
  • Results
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SLIDE 9

Outline

Statistical Sciences, UCT mODa8, 2007 – 3 / 19

  • Problem
  • Results
  • Conclusions
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SLIDE 10

Scope

Statistical Sciences, UCT mODa8, 2007 – 4 / 19

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SLIDE 11

Scope

Statistical Sciences, UCT mODa8, 2007 – 4 / 19

  • Drug synergy
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SLIDE 12

Scope

Statistical Sciences, UCT mODa8, 2007 – 4 / 19

  • Drug synergy
  • Models
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SLIDE 13

Scope

Statistical Sciences, UCT mODa8, 2007 – 4 / 19

  • Drug synergy
  • Models
  • Optimal Design
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SLIDE 14

Problem

Statistical Sciences, UCT mODa8, 2007 – 5 / 19

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SLIDE 15

Problem

Statistical Sciences, UCT mODa8, 2007 – 5 / 19

  • In search of algebraic solutions!
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SLIDE 16

Problem

Statistical Sciences, UCT mODa8, 2007 – 5 / 19

  • In search of algebraic solutions!
  • Model

logit(p) = β0 + β1x1 + β2x2

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SLIDE 17

Problem

Statistical Sciences, UCT mODa8, 2007 – 5 / 19

  • In search of algebraic solutions!
  • Model

logit(p) = β0 + β1x1 + β2x2

  • Locally D-optimal designs
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SLIDE 18

Early Work

Statistical Sciences, UCT mODa8, 2007 – 6 / 19

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SLIDE 19

Early Work

Statistical Sciences, UCT mODa8, 2007 – 6 / 19

  • Sitter and Torsney, 1995
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SLIDE 20

Early Work

Statistical Sciences, UCT mODa8, 2007 – 6 / 19

  • Sitter and Torsney, 1995
  • Atkinson and Haines, 1996
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SLIDE 21

Early Work

Statistical Sciences, UCT mODa8, 2007 – 6 / 19

  • Sitter and Torsney, 1995
  • Atkinson and Haines, 1996
  • Jia and Myers, 2001
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SLIDE 22

Model

Statistical Sciences, UCT mODa8, 2007 – 7 / 19

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SLIDE 23

Model

Statistical Sciences, UCT mODa8, 2007 – 7 / 19

  • Logistic regression

logit(p) = β0 + β1d1 + β2d2

with

d1, d2 ≥ 0

where

β1, β2 ≥ 0 and β0 < 0

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SLIDE 24

Model

Statistical Sciences, UCT mODa8, 2007 – 7 / 19

  • Logistic regression

logit(p) = β0 + β1d1 + β2d2

with

d1, d2 ≥ 0

where

β1, β2 ≥ 0 and β0 < 0

  • Formulate as

logit(p) = β0 + z1 + z2

with

z1, z2 ≥ 0

and

β0 < 0

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SLIDE 25

Response Surface

Statistical Sciences, UCT mODa8, 2007 – 8 / 19

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SLIDE 26

Response Surface

Statistical Sciences, UCT mODa8, 2007 – 8 / 19

β0 = −4

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SLIDE 27

Locally D-optimal Designs

Statistical Sciences, UCT mODa8, 2007 – 9 / 19

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SLIDE 28

Locally D-optimal Designs

Statistical Sciences, UCT mODa8, 2007 – 9 / 19

  • Approximate

ξ =

  • (z11, z21),

. . . , (z1r, z2r) w1, . . . , wr

  • where 0 < wi < 1 and r

i=1 wi = 1

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SLIDE 29

Locally D-optimal Designs

Statistical Sciences, UCT mODa8, 2007 – 9 / 19

  • Approximate

ξ =

  • (z11, z21),

. . . , (z1r, z2r) w1, . . . , wr

  • where 0 < wi < 1 and r

i=1 wi = 1

  • Information matrix

M(β0; z) = eu (1 + eu)2

  

1 z1 z2 z1 z2

1

z1z2 z2 z1z2 z2

2

   where u = β0 + z1 + z2

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SLIDE 30

Locally D-optimal Designs

Statistical Sciences, UCT mODa8, 2007 – 9 / 19

  • Approximate

ξ =

  • (z11, z21),

. . . , (z1r, z2r) w1, . . . , wr

  • where 0 < wi < 1 and r

i=1 wi = 1

  • Information matrix

M(β0; z) = eu (1 + eu)2

  

1 z1 z2 z1 z2

1

z1z2 z2 z1z2 z2

2

   where u = β0 + z1 + z2

  • Maximize |M(β0; ξ)|
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SLIDE 31

Four-Point Designs

Statistical Sciences, UCT mODa8, 2007 – 10 / 19

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SLIDE 32

Four-Point Designs

Statistical Sciences, UCT mODa8, 2007 – 10 / 19

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Four-Point Designs

Statistical Sciences, UCT mODa8, 2007 – 10 / 19

ξ⋆

f =

(−u − β0, 0) (0, −u − β0) (u − β0, 0) (0, u − β0) w w

1 2 − w 1 2 − w

  • with 0 < u ≤ −β0
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SLIDE 34

D-optimality

Statistical Sciences, UCT mODa8, 2007 – 11 / 19

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D-optimality

Statistical Sciences, UCT mODa8, 2007 – 11 / 19

  • Maximize
  • M(β; ξ⋆

f)

  • = 2e3uu2w(1 − 2w){(u − β0)2 + 8β0uw}

(1 + eu)6

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SLIDE 36

D-optimality

Statistical Sciences, UCT mODa8, 2007 – 11 / 19

  • Maximize
  • M(β; ξ⋆

f)

  • = 2e3uu2w(1 − 2w){(u − β0)2 + 8β0uw}

(1 + eu)6

  • Optimal u and w

u2(3+3eu+2u−2ueu)+β2

0(1+eu+2u−2ueu)+a(1+eu+u−ueu) = 0

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SLIDE 37

D-optimality

Statistical Sciences, UCT mODa8, 2007 – 11 / 19

  • Maximize
  • M(β; ξ⋆

f)

  • = 2e3uu2w(1 − 2w){(u − β0)2 + 8β0uw}

(1 + eu)6

  • Optimal u and w

u2(3+3eu+2u−2ueu)+β2

0(1+eu+2u−2ueu)+a(1+eu+u−ueu) = 0

with a =

  • u4 + 14β2

0u2 + β4 0 and w = f(u)

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SLIDE 38

D-optimality

Statistical Sciences, UCT mODa8, 2007 – 11 / 19

  • Maximize
  • M(β; ξ⋆

f)

  • = 2e3uu2w(1 − 2w){(u − β0)2 + 8β0uw}

(1 + eu)6

  • Optimal u and w

u2(3+3eu+2u−2ueu)+β2

0(1+eu+2u−2ueu)+a(1+eu+u−ueu) = 0

with a =

  • u4 + 14β2

0u2 + β4 0 and w = f(u)

  • β0 = −4

u∗ = 1.323 and w∗ = 0.189

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SLIDE 39

Directional Derivative

Statistical Sciences, UCT mODa8, 2007 – 12 / 19

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SLIDE 40

Directional Derivative

Statistical Sciences, UCT mODa8, 2007 – 12 / 19

β0 = −4

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SLIDE 41

Optimality?

Statistical Sciences, UCT mODa8, 2007 – 13 / 19

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SLIDE 42

Optimality?

Statistical Sciences, UCT mODa8, 2007 – 13 / 19

  • Trend

β0 = −4 β0 = −3 β0 = −2

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SLIDE 43

Optimality?

Statistical Sciences, UCT mODa8, 2007 – 13 / 19

  • Trend

β0 = −4 β0 = −3 β0 = −2

  • Cut-off

β0 ≈ −1.5434

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SLIDE 44

Three-Point Designs

Statistical Sciences, UCT mODa8, 2007 – 14 / 19

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SLIDE 45

Three-Point Designs

Statistical Sciences, UCT mODa8, 2007 – 14 / 19

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SLIDE 46

Three-Point Designs

Statistical Sciences, UCT mODa8, 2007 – 14 / 19

ξ⋆

t =

  • (0, 0)

(u − β0, 0) (0, u − β0)

1 3 1 3 1 3

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SLIDE 47

Three-Point Designs

Statistical Sciences, UCT mODa8, 2007 – 14 / 19

ξ⋆

t =

  • (0, 0)

(u − β0, 0) (0, u − β0)

1 3 1 3 1 3

  • where u satisfies

2 − β0 + 2eu + β0eu + u − ueu = 0

with

u > β0

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SLIDE 48

Directional Derivative

Statistical Sciences, UCT mODa8, 2007 – 15 / 19

β0 = −1

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SLIDE 49

Cut-off

Statistical Sciences, UCT mODa8, 2007 – 16 / 19

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SLIDE 50

Cut-off

Statistical Sciences, UCT mODa8, 2007 – 16 / 19

  • Condition

u∗ = −β0

so that

1 + β0 + eβ0 − β0eβ0 = 0

and thus

β0 ≈ −1.5434

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SLIDE 51

Cut-off

Statistical Sciences, UCT mODa8, 2007 – 16 / 19

  • Condition

u∗ = −β0

so that

1 + β0 + eβ0 − β0eβ0 = 0

and thus

β0 ≈ −1.5434

  • Locally D-optimal?
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SLIDE 52

Directional derivative

Statistical Sciences, UCT mODa8, 2007 – 17 / 19

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SLIDE 53

Directional derivative

Statistical Sciences, UCT mODa8, 2007 – 17 / 19

  • Optimality condition

eu1 (1 + eβ0)2 eβ0 (1 + eu1)2

  

β2

0 + u2 1 + (β0 − u1)z1 + z2 1

2 β2

   ≤ 1 where u1 = β0 + z1 + z2

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SLIDE 54

Directional derivative

Statistical Sciences, UCT mODa8, 2007 – 17 / 19

  • Optimality condition

eu1 (1 + eβ0)2 eβ0 (1 + eu1)2

  

β2

0 + u2 1 + (β0 − u1)z1 + z2 1

2 β2

   ≤ 1 where u1 = β0 + z1 + z2

  • Quadratic in z1 for constant u1
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SLIDE 55

Directional derivative

Statistical Sciences, UCT mODa8, 2007 – 17 / 19

  • Optimality condition

eu1 (1 + eβ0)2 eβ0 (1 + eu1)2

  

β2

0 + u2 1 + (β0 − u1)z1 + z2 1

2 β2

   ≤ 1 where u1 = β0 + z1 + z2

  • Quadratic in z1 for constant u1
  • Thus require

(β2

0 + u2 1)

β2 ≤ 2eβ0 (1 + eu1)2 eu1 (1 + eβ0)2

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SLIDE 56

One variable

Statistical Sciences, UCT mODa8, 2007 – 18 / 19

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SLIDE 57

One variable

Statistical Sciences, UCT mODa8, 2007 – 18 / 19

  • Model

logit(p) = β0 + β1x1 = u

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SLIDE 58

One variable

Statistical Sciences, UCT mODa8, 2007 – 18 / 19

  • Model

logit(p) = β0 + β1x1 = u

  • D-optimality

±u∗

where u∗ ≈ 1.5434

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SLIDE 59

One variable

Statistical Sciences, UCT mODa8, 2007 – 18 / 19

  • Model

logit(p) = β0 + β1x1 = u

  • D-optimality

±u∗

where u∗ ≈ 1.5434

  • Directional derivative

((u∗)2 + u2) (u∗)2 ≤ 2eu∗ (1 + eu)2 eu (1 + eu∗)2

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SLIDE 60

One variable

Statistical Sciences, UCT mODa8, 2007 – 18 / 19

  • Model

logit(p) = β0 + β1x1 = u

  • D-optimality

±u∗

where u∗ ≈ 1.5434

  • Directional derivative

((u∗)2 + u2) (u∗)2 ≤ 2eu∗ (1 + eu)2 eu (1 + eu∗)2

SAME!

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SLIDE 61

Conclusions

Statistical Sciences, UCT mODa8, 2007 – 19 / 19

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SLIDE 62

Conclusions

Statistical Sciences, UCT mODa8, 2007 – 19 / 19

  • Logistic with d1, d2 ≥ 0
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SLIDE 63

Conclusions

Statistical Sciences, UCT mODa8, 2007 – 19 / 19

  • Logistic with d1, d2 ≥ 0
  • D-optimality
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SLIDE 64

Conclusions

Statistical Sciences, UCT mODa8, 2007 – 19 / 19

  • Logistic with d1, d2 ≥ 0
  • D-optimality
  • Interaction term
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SLIDE 65

Conclusions

Statistical Sciences, UCT mODa8, 2007 – 19 / 19

  • Logistic with d1, d2 ≥ 0
  • D-optimality
  • Interaction term
  • General case
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SLIDE 66

Conclusions

Statistical Sciences, UCT mODa8, 2007 – 19 / 19

  • Logistic with d1, d2 ≥ 0
  • D-optimality
  • Interaction term
  • General case
  • Taxonomy of designs
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SLIDE 67

Conclusions

Statistical Sciences, UCT mODa8, 2007 – 19 / 19

  • Logistic with d1, d2 ≥ 0
  • D-optimality
  • Interaction term
  • General case
  • Taxonomy of designs
  • Interaction term
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SLIDE 68

Conclusions

Statistical Sciences, UCT mODa8, 2007 – 19 / 19

  • Logistic with d1, d2 ≥ 0
  • D-optimality
  • Interaction term
  • General case
  • Taxonomy of designs
  • Interaction term
  • Other models