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

Bayesian Ds-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 4th to 8th 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
• Ds-Optimality

– Ds-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

Model for the mean:

;

Model for the dispersion:

; Non correlated observations. Covariates { zi} may be a subset of covariates {xi}.

( )

i i

E y µ =

( )

( )

i i i

Var y V φ µ =

( )

i i ij j i j

k x X η µ β β = = =

∑

( )

i i

E d φ =

( )

( )

i D i

Var d V ε φ =

( )

i i ij j i j

h z Z τ φ γ γ = = =

∑

JMMD Summary

Table 1. Summary of the joint modeling of mean and dispersion

Component M odel for mean M odel for dispersion Response y

( )

*

1 d d h = − M ean µ φ Variance

( )

V φ µ

2

2φ Link

( )

k η µ =

( )

log τ φ = Linear Preditor X η β = Z τ γ = Deviance component

( )

2

y

y t d dt V t

µ

− = − ∫

( ) ( )

{ }

* *

2 log d d φ φ φ − + − Prior weight 1 φ 1 h − W here h is the ith element in the diagonal of

( )

1 1 2 1 2 T

H W X X W X XW

−

=

The Equivalence Theory (D-optimality)

Atkinson & Cook (1995) regard a generic design problem. Let be a vector of parameters and a design vector. The Fisher information matrix per observation is as follows: where

.

The information matrix is : D-optimality ⇒ find to maximize

.

( )

1 × t

θ

u χ ∈

( )

1

θ

=

= ∑

k T j j j

I u h h

( , ) θ =

j j

h h u

( ) ( )

( )

M I u d u

χ

θ ξ θ ξ = ∫

( )

ξ u

( )

ln M θ ξ

The information matrix depends on the parameters.

Two solutions are locally optimal or “pseudo-Bayesian” designs.

A “pseudo- Bayesian” criterion function is:

where refers to expectation with respect to a prior distribution for .

In this case the standardized variance is:

( )

ln E M

θ θ

ψ θ ξ ⎡ ⎤ = ⎣ ⎦

( ) ( )

( )

( )

1

, , , ,

T j j j

d u E h u M h u

θ

θ ξ θ θ ξ θ

−

⎡ ⎤ = ⎣ ⎦

θ

Eθ

General Equivalence Theorem (GET)

(“pseudo-Bayesian” D-optimality)

The design maximizes over

• where t is the number of parameters in the model.

ξ ∗

( )

ln E M

θ

θ ξ ⎡ ⎤ ⎣ ⎦ Ξ

( )

( )

1 1

, , , ,

k k j j j j

Min Max d u Max d u

χ χ

θ ξ θ ξ ∗

Ξ = =

=

∑ ∑

( )

1

, ,

k j j

Max d u t

χ

θ ξ ∗

=

=

∑

DS-optimality for the JMMD

• The researcher is interested only in a reduced number
• f parameters in the mean model as well as in a

reduced number of parameters in the dispersion model.

• DS-optimality is an extension of D-optimality.
• Discrimination between rival nested models

DS-optimality for the JMMD

• Two GLM´ s ⇒ mean and dispersion

and , where and .

• Interest lies in parameters in the mean model and

in parameters in the dispersion model. ( ) ( )

,

T

η = x f x β β

( ) ( )

,

T

τ = z g z γ γ

1 p×

β

1 q×

γ

( ) ( )

11 12 21 22 11 12 21 22 p p C q q p q p q

M M M M M M D D D D D

× × ×

+ +

⎡ ⎤ ⎢ ⎥ ⎡ ⎤ ⎢ ⎥ = = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦

m

s

d

s

( ) ( )

11 11

is , is

× ×

m m d d

M s s D s s

• In this case:
• Let:
• Then:
• The “pseudo-Bayesian” criterion to Ds-optimality is:

11 12 21 22 1 11 12 21 22 −

⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

C

M M M M M D D D D

( ) ( )

×

+ +

⎡ ⎤ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

m d m d m d

s T s s s s s

I A I

11 1 11 −

⎡ ⎤ = ⎢ ⎥ ⎣ ⎦

T C

M A M A D

( )

22 22

ln ln ⎧ ⎫ ⎛ ⎞ ⎛ ⎞ ⎪ ⎪ ⎡ ⎤ = + ⎜ ⎟ ⎜ ⎟ ⎨ ⎬ ⎣ ⎦ ⎜ ⎟ ⎜ ⎟ ⎪ ⎪ ⎝ ⎠ ⎝ ⎠ ⎩ ⎭

C

M D M E M D

θ θ

ϕ θ ξ

• The Frechét derivative at MC1 in the direction of MC2

at the point is

• The standardized variance is:
• For the optimal design , with equality

at the design support points. Here .

( ) ( ) ( ) ( ) ( )

{

( ) ( ) ( ) ( )}

1 1 2 22 2 1 1 2 22 2

, ,

− − − −

= − + − −

i

T T i i i i i i i i i i i T T i i i i i i i i i i

D u E w M w M v D v D s

ϕ θ

θ ξ f x f x f x f x g z g z g z g z

( ) ( ) ( ) ( ) ( )

( )

{

( ) ( ) ( ) ( )

( )}

1 1 2 22 2 1 1 2 22 2

, , E +

− − − −

= − −

T T i i i i i i i i i i i T T i i i i i i i i i

d w M M v D D

ϕ θ

θ ξ u f x f x f x f x g z g z g z g z

∗

ξ

( )

,

∗ ≤ S

d x s ξ

( )

+ =

m d

s s s

i

u

Example: Coffee tasting

• Aim: Ideal conditions of toasting to enhance quality of

coffee.

• Design: Complete factorial
• Response: quantity of trigoneline.
• Factors: Temperature of drying ,

Temperature of toasting , and Air speed in the drying of coffee .

3

2

( )

1

x

( )

2

x

( )

3

x

Experimental Setting

( ) ( )

• 1 : 300

high level and 100 low level x C C

( ) ( )

• 2 : 600

high level and 300 low level x C C

( ) ( )

3 : 1850

high level and 1300 low level x rpm rpm

x 1 x 2 x 3

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

Y

Model for the mean: link identity and Model for dispersion: Gamma with logarithmic link Design matrix with columns:

• Fitted Model using the JMMD

( )

1 = V µ

1 2 3 1 2 1 3 2 3 1 2 3

1, , , , , , , x x x x x x x x x x x x

2 3 2 3

ˆ 2.374 0.374 0.167 0.261 = − − − x x x x µ

{ }

2 2 3

ˆ exp 6.24 1.106 = − − x x σ

To verify if variables and are important in the mean model as well as the interaction in the model for dispersion.

• Therefore, and .

2 3

, x x

2 3

x x

3 =

m

s 1 =

d

s

4 = + =

m d

s s s

2 3

the interaction x x

• Systematic components
• Prior distribution evolves around the parameter estimates of the

mean and dispersion models.

• We add and deduct to the values of the parameters of

the mean (dispersion) model.

• For the pseudo-Bayesian DS-optimum design is:

( )

1 2 2 3 3 2 3

, x x x x η β β β β = + + + x β

( )

1 2 3

, x x τ γ γ = + x γ

( )

2.374, 0.374, 0.167, 0.261 = − − − β

( )

6.24, 1.106 = − − γ

α ( ) δ

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

*

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

0.15 0.08 0.15 0.19 0.08 0.19 0.08 0.08

− − − − − − −

⎧ ⎫ = ⎨ ⎬ ⎩ ⎭

S

ξ

0.25 α δ = =

• The graphic of standardized variance as a function of

is:

2 3

and x x

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.

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.

Thank you