Bayesian D s -Optimal Designs for Generalized Linear Models with - PowerPoint PPT Presentation

Bayesian D s -Optimal Designs for Generalized Linear Models with Varying Dispersion Parameter Edmilson Rodrigues Pinto - (FAMAT-UFU, Brazil) Antônio Ponce de Leon - (IMS-UERJ, Brazil) MODA 8, Almagro 4 th to 8 th June 2007

Outline • I ntroduction – JMMD (Joint Modeling of Mean and Dispersion) – Review of D-optimality for the JMMD • “Pseudo-Bayesian” criterion of D-optimality • Information matrix and the standardized variance • General Equivalence Theorem • D s -Optimality – D s -optimality for the JMMD – Information matrix and the standardized variance – General Equivalence Theorem • I llustration • Final Considerations

JMMD (Joint Modeling of Mean and Dispersion) � JMMD ⇒ proposed by Nelder & Lee (1991) and it is linked with the criterion of extended quasi likelihood. � Two interlinked GLM ´s ( ) ( ) = µ = φ µ Model for the mean : E y Var y ( ) V ; i i i i i ∑ ( ) η = µ = β = β k x X i i ij j i j ( ) ( ) = φ = ε φ Model for the dispersion : ; ( ) E d Var d V i i i D i ( ) ∑ τ = φ = γ = γ h z Z i i ij j i j � Non correlated observations. � Covariates { z i } may be a subset of covariates { x i }.

JMMD Summary Table 1. Summary of the joint modeling of mean and dispersion Component M odel for mean M odel for dispersion ( ) = − * d d 1 h Response y µ φ M ean ( ) φ µ 2 φ 2 V Variance ( ) ( ) η = µ τ = φ k log Link η = β τ = γ X Z Linear Preditor { } ( ) ( ) − = − ∫ − φ + − φ φ µ y t * * 2 log d d d 2 dt Deviance component ( ) V t y 1 φ − 1 Prior weight h ( ) − 1 = W here h is the i th element in the diagonal of 1 2 T 1 2 H W X X W X XW

The Equivalence Theory (D-optimality) Atkinson & Cook (1995) regard a generic design problem. ∈ χ θ Let be a vector of parameters and a design vector. u ( ) × t 1 The Fisher information matrix per observation is as follows : = ∑ ( ) k θ T I u h h j j = j 1 = θ where h h u ( , ) . j j = ∫ ( ) ( ) ( ) θ ξ θ ξ M I u d u The information matrix is : χ ( ) ( ) ξ u D-optimality ⇒ find ln M θ ξ to maximize .

� The information matrix depends on the parameters. Two solutions are locally optimal or “pseudo-Bayesian” designs. � A “pseudo- Bayesian” criterion function is : ( ) ⎡ ⎤ ψ = θ ξ E ln M ⎣ ⎦ θ θ E θ where refers to expectation with respect to a prior θ distribution for . � In this case the standardized variance is: ( ) ( ) ( ) ( ) ⎡ ⎤ θ ξ = θ − θ ξ θ T 1 d u , , E h u , M h u , ⎣ ⎦ θ j j j

General Equivalence Theorem (GET) (“pseudo-Bayesian” D-optimality) ( ) ξ ∗ ⎡ ⎤ θ ξ Ξ � The design maximizes over E ln M ⎣ ⎦ θ ( ) k k ∑ ∑ ( ) θ ξ ∗ θ ξ = � Min Max d u , , Max d u , , j j Ξ χ χ = = j 1 j 1 ( ) k ∑ θ ξ ∗ = � Max d u , , t j χ = j 1 where t is the number of parameters in the model.

D S -optimality for the JMMD • The researcher is interested only in a reduced number of parameters in the mean model as well as in a reduced number of parameters in the dispersion model. • D S -optimality is an extension of D-optimality. • Discrimination between rival nested models

D S -optimality for the JMMD Two GLM´ s ⇒ mean and dispersion • ( ) ( ) ( ) ( ) η β = β τ γ = γ γ β T T , x f x , z g z and , where and . p × 1 q × 1 ⎡ ⎤ M M 0 0 11 12 ⎢ ⎥ ⎡ ⎤ M 0 0 0 M M ⎢ ⎥ = = × p p 21 22 ⎢ ⎥ M ⎢ ⎥ C 0 D ⎣ ⎦ 0 0 D D × q q 11 12 ⎢ ⎥ ⎣ ⎦ 0 0 D D ( ) ( ) + + 21 22 p q × p q s • Interest lies in parameters in the mean model and m s in parameters in the dispersion model. d ( ) ( ) is , is M s s D s s × × 11 m m 11 d d

⎡ ⎤ 11 12 M M 0 0 ⎢ ⎥ 21 22 M M 0 0 ⎢ ⎥ − = ⎢ 1 • In this case: M ⎥ C 11 12 0 0 D D ⎢ ⎥ 21 22 ⎣ ⎦ 0 0 D D ⎡ ⎤ I 0 0 0 s = ⎢ ⎥ T m • Let: A ⎢ ⎥ 0 0 I 0 ⎣ ⎦ ( ) ( ) s + + d s s s s × m d m d ⎡ ⎤ 11 M 0 − = ⎢ T 1 ⎥ • Then: A M A C 11 ⎣ ⎦ 0 D • The “pseudo-Bayesian” criterion to D s -optimality is: ⎧ ⎫ ⎛ ⎞ ⎛ ⎞ ⎪ ⎪ M D ( ) ⎡ ⎤ = ϕ θ ξ + ⎨ ⎜ ⎟ ⎜ ⎟ ⎬ M E ln ln ⎣ ⎦ ⎜ ⎟ ⎜ ⎟ θ θ C ⎪ ⎪ M D ⎝ ⎠ ⎝ ⎠ ⎩ ⎭ 22 22

• The Frechét derivative at M C1 in the direction of M C2 u at the point is i { ( ) ( ) ( ) ( ) ( ) − − θ ξ = − + T 1 T 1 f x f x f x f x D u , , E w M w M ϕ θ i i i i i i i 2 i i 22 2 i i i ( ) } ( ) ( ) ( ) − − − − 1 1 T T v g z D g z v g z D g z s i i i i i i 2 i i 22 2 i i • The standardized variance is: { ( ) ( ) ( ) ( ) ( ) ( ) − − θ ξ = − T 1 T 1 d u , , E w f x M f x f x M f x + ϕ θ i i i i i i i 2 i i 22 2 i i ) } ( ( ) ( ) ( ) ( ) − − − 1 1 T T v g z D g z g z D g z i i i i i 2 i i 22 2 i i ( ∗ ≤ ) ∗ ξ ξ • For the optimal design , with equality d x , s S ( ) + = s s s at the design support points. Here . m d

Example: Coffee tasting • Aim: Ideal conditions of toasting to enhance quality of coffee. 3 • Design: Complete factorial 2 • Response: quantity of trigoneline. ( ) x • Factors: Temperature of drying , 1 ( ) x Temperature of toasting , and 2 ( ) Air speed in the drying of coffee . x 3

Experimental Setting ( ) ( ) o o x 1 : 300 C high level and 100 C low level ( ) ( ) o o x 2 : 600 C high level and 300 C low level ( ) ( ) x 3 : 1850 rpm high level and 1300 rpm low level x 1 x 2 x 3 Y 1 1 1 0,38 0,45 0,40 1 1 -1 0,63 0,59 0,65 1 -1 1 0,73 0,68 0,66 1 -1 -1 0,69 0,68 0,70 -1 1 1 0,39 0,37 0,40 -1 1 -1 0,65 0,65 0,64 -1 -1 1 0,70 0,71 0,75 -1 -1 -1 0,67 0,68 0,79

( ) V µ = � Model for the mean: link identity and 1 � Model for dispersion: Gamma with logarithmic link � Design matrix with columns: 1, , x x , , x x x , x x , x x , x x x 1 2 3 1 2 1 3 2 3 1 2 3 • Fitted Model using the JMMD µ = − − − ˆ 2.374 0.374 x 0.167 x 0.261 x x 2 3 2 3 { } σ = − − 2 ˆ exp 6.24 1.106 x x 2 3

x , x the interaction x x To verify if variables and 2 3 2 3 are important in the mean model as well as the x x interaction in the model for dispersion. 2 3 = = • Therefore, and . s 3 s 1 m d = + = s s s 4 m d

• Systematic components ( ) ( ) τ x γ = γ + γ η x β = β + β + β + β , x x , x x x x 0 1 2 3 0 1 2 2 3 3 2 3 • Prior distribution evolves around the parameter estimates of the mean and dispersion models. ( ) ( ) β = − − − γ = − − 2.374, 0.374, 0.167, 0.261 6.24, 1.106 α ( ) δ • We add and deduct to the values of the parameters of the mean (dispersion) model. α = δ = 0.25 • For the pseudo-Bayesian D S -optimum design is: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ⎧ ⎫ − − − − − − − 1.0,1.0 1.0,0.7 1.0, 1.0 1.0,1.0 1.0,0.7 1.0, 1.0 0.7, 1.0 0.7,1.0 ξ = ⎨ * ⎬ S ⎩ ⎭ 0.15 0.08 0.15 0.19 0.08 0.19 0.08 0.08

x and x • The graphic of standardized variance as a function of is: 2 3

Final considerations • Optimal designs for subsets of parameters in the mean and in the dispersion models. • Different scenarios showed by Pinto and Ponce de Leon (2004) for local designs and Bayesian designs for parameters in the mean and dispersion models can be considered.

Thank you Acknowledgements � Edmilson Rodrigues Pinto is grateful to FAPEMI G – the research foundation of Minas Gerais – Brazil. � Antonio Ponce de Leon is grateful to CNPq for the travel grant.