Auxiliary Mixture Sampling for Age-Period-Cohort Models Andrea - - PowerPoint PPT Presentation

Auxiliary Mixture Sampling for Age-Period-Cohort Models

Andrea Riebler & Leonhard Held

University of Zurich

Reisensburg, September 2007

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

Outline

1

Introduction

2

Age-Period-Cohort Models

3

Auxiliary Mixture Sampling

4

Summary and Outlook

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

• 1. Introduction

Goal

Detection of spatial and temporal patterns in epidemiological data Methods: Age-Period-Cohort model: to describe incidence or mortality rates Auxiliary mixture sampling: to estimate the APC-model

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

Age-Period-Cohort Model

Problem

No or just limited data for possible disease factors available Analysis of incidence or mortality rates using three time scales: Age: time between birth and infection or death Period: time of infection or death Cohort: time of birth

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

The Lexis-diagram

cohorts periods age 1900 1905 1910 1915 1920 1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

The Lexis-diagram

cohorts periods age 1900 1905 1910 1915 1920 1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

The Lexis-diagram

cohorts periods age 1900 1905 1910 1915 1920 1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 42 Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

The Lexis-diagram

cohorts periods age 1900 1905 1910 1915 1920 1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 42 Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

The Lexis-diagram

cohorts periods age 1900 1905 1910 1915 1920 1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 42 Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

The Lexis-diagram

cohorts periods age 1900 1905 1910 1915 1920 1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 42 Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

Age,- Period,- Cohort effects

Age effects

Consistent extrinsic factors

Period effects

Factors that influence all persons under risk independent of the age, e.g. improvements of medical treatment

Cohort effects

Factors that influence persons of one generation, e.g. war

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

The cohort index

The cohort index k depends on the age group i and period j: k = k(i, j) = (I − i) + j. For different time grids k = k(i, j) = G · (I − i) + j, where G is the grid factor. (i = 1, . . . , I, j = 1, . . . , J, k = 1, . . . , K and K = G · (I − 1) + J)

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

The age-period-cohort model

yij: # counts in age group i at period j nij: population in age group i at period j

The APC-model

yij ∼ Po(nij · pij

λij

) ηij = log(λij) = log(nij) + µ + αi + βj + γk with age effect αi, period effect βj, cohort effect γk

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

Non-identifiability

To assure identifiability additional constraints are necessary Make the intercept µ identifiable

• i

αi =

• j

βj =

• k

γk = 0 The APC parameters are still not identifiable: αi → αi + c ·

• i − I + 1

2

• ;

βj → βj − c ·

• j − J + 1

2

• ;

γk → γk + c ·

• k − K + 1

2

• ⇒ ηij = log(nij) + µ + αi + βj + γk

is left unchanged

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

Random walks (RW)

Bayesian APC models often use RW priors, e.g. of first order f (α|κ) ∝ κ(I−1)/2 exp

• −κ

2

I

• i=2

(αi − αi−1)2

• = κ(I−1)/2 exp
• −κ

2αTR(1)α

• with

R(1) =            1 −1 −1 2 −1 −1 2 −1 ... ... ... −1 2 −1 −1 2 −1 −1 1           

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

Random walks (RW) II

Random walk of second order (RW2): f (α|κ) ∝ κ(I−2)/2 exp

• −κ

2

I

• i=3

(αi − 2αi−1 + αi−2)2

• = κ(I−2)/2 exp
• −κ

2αTR(2)α

• RW1 penalizes deviations from a model where all parameters

are constant. The parameters are identifiable. RW2 penalizes deviations from a linear trend αi = 2αi−1 − αi−2. The parameters are not identifiable.

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

Heterogeneity

To account for additional“unstructured”heterogeneity, an additional parameter zij ∼ N(0, δ−1) can be introduced ξij = log(nij) + µ + αi + βj + γk

• ηij

+zij Using a reparameterization, it follows zij = ξij − (log nij + µ + αi + βj + γk) = ξij − ηij The implied prior of the linear predictor is f (ξ|η, δ) ∝ δIJ/2 · exp  −δ 2

I

• i=1

J

• j=1

(ξij − ηij)2   .

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

Full conditional distributions

For all parameters except ξij Gibbs sampling possible

Problem

The full conditional distribution f (ξ|y, η, κ, ν, τ, δ) ∝ f (y|ξ)f (ξ|η, δ) is a non-standard distribution. Metropolis-Hastings algorithm, where the proposal could be found using Taylor approximation Auxiliary mixture sampling, where additional auxiliary variables are introduced to enable Gibbs sampling

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

Auxiliary Mixture Sampling for APC models

Model specification:

yij ∼ Po(nij · pij

λij

) ξij = log(λij) = log(nij) + µ + αi + βj + γk

• ηij

+zij

Data augmentation:

Introduce additional auxiliary variables to eliminate nonlinearity and non-normality.

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

Poisson Process

1 t1 t2 t3 t4 τ1 τ2 τ3 τ4

crosses mark the occurrence of an event, t1, . . . , t4 arrival times, τ1, . . . , τ4 inter-arrival times

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

Data Augmentation (1)

For each yij > 0 introduce τij = (τij1, τij2), for yij = 0 just τij = τij1 τij2 denotes the last jump before 1 and is Ga(yij, λij) τij2 = ρij2/λij, ρij2 ∼ Ga(yij, 1)

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

Data Augmentation (1)

For each yij > 0 introduce τij = (τij1, τij2), for yij = 0 just τij = τij1 τij2 denotes the last jump before 1 and is Ga(yij, λij) τij2 = ρij2/λij, ρij2 ∼ Ga(yij, 1) τij1 denotes the inter-arrival time between the last jump before and the first jump after 1 and is Ex(λij) τij1 = ρij1/λij, ρij1 ∼ Ex(1).

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

Data Augmentation (2)

Reformulation: − log(τij2) = log(λij)

ξij

+ǫij2 − log(τij1) = log(λij)

ξij

+ǫij1 where ǫij1 = − log(ρij1) and ǫij2 = − log(ρij2).

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

Mixture Approximation (1)

ǫij1 follows the negative of a log Ex(1) distribution: p(ǫij) = exp(−ǫij − exp(−ǫij)) ≈

R

• r=1

wr N(ǫij; mr, s2

r )

with parameters mr and sr, and weight wr for the rth component. ǫij2 follows a negative log Gamma distribution with integer shape parameter yij: p(ǫij; yij) = exp(−yijǫij − exp(−ǫij)) Γ(yij) ≈

R(yij)

• r=1

wr(yij) N(ǫij; mr(yij), s2

r (yij))

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

Mixture Approximation (2)

Introduce the component indicators S = {rij = (rij1, rij2), i = 1, . . . , I, j = 1, . . . , J} as missing data − log(τij1) = ξij + mrij1(1) + ǫij1, ǫij1 ∼ N(0, s2

rij1(1))

− log(τij2) = ξij + mrij2(yij) + ǫij2, ǫij2 ∼ N(0, s2

rij2(yij))

For yij = 0 just the first equation is necessary.

Consequence

The full conditional distribution for ξij now is a normal distribution ⇒ Gibbs-sampling possible

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

Sampling scheme

Select starting values for the hyperparameters, for τ, S and the main effects α, β, γ.

1 Sample ξ conditional on τ, S, µ, α, β, γ, δ and y Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

Sampling scheme

Select starting values for the hyperparameters, for τ, S and the main effects α, β, γ.

1 Sample ξ conditional on τ, S, µ, α, β, γ, δ and y 2 Sample the inter-arrival times τ and the component indicators

S conditional on µ, α, β, γ and y (exponential resp. discrete distributions)

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

Sampling scheme

Select starting values for the hyperparameters, for τ, S and the main effects α, β, γ.

1 Sample ξ conditional on τ, S, µ, α, β, γ, δ and y 2 Sample the inter-arrival times τ and the component indicators

S conditional on µ, α, β, γ and y (exponential resp. discrete distributions)

3 Update the main effects and hyperparameters using

Gibbs-sampling (multivariate Gaussian, gamma distributions)

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

Implementation

in C using the GMRFLib library by H˚ avard Rue (http://www.math.ntnu.no/~hrue/GMRFLib) GMRFLib includes functions: for efficient sampling from age, period and cohort blocks for defining auxiliary mixture sampling for Poisson and binomial observational models

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

Planned application

Chronic obstructive pulmonary disease (COPD) mortality rates for females in England and Wales excluding conurbations

15−24 25−34 35−44 45−54 55−64 65−74 75+ Age group Mortality rate 20 40 60 80 100 120 140 1950 1960 1970 1980 1990 2000 4 5 6 7 Period Mortality rate

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

Summary and Outlook

Summary

Presentation of a Bayesian framework based on Gaussian Markov random fields to estimate APC models by drawing from standard distributions only.

Outlook

Application to multivariate datasets evaluating different APC models with joint or separated main effects.

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

References

Fr¨ uhwirth-Schnatter, S., R. Fr¨ uhwirth, L. Held, and H. Rue (2007). Improved auxiliary mixture sampling for parameter-driven models of non-Gaussian data. IFAS Research Report 2007-25, http://ifas.jku.at, Johannes Kepler University Linz. Rue, H. and L. Held (2005). Gaussian Markov Random Fields: Theory and Applications. Volume 104 of emph Monographs on Statistics and Applied Probability, Boca Ratin, FL:Chapmann & Hall/CRC.

Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook

Thank you for your attention!

Any questions?

Andrea Riebler University of Zurich