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 Introduction 1 Age-Period-Cohort Models 2 Auxiliary Mixture Sampling 3 Summary and Outlook 4 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 80 75 70 65 60 55 50 45 age 40 35 30 25 20 15 10 5 0 1900 1905 1910 1915 1920 1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 cohorts periods Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook The Lexis-diagram 80 75 70 65 60 55 50 45 age 40 35 30 25 20 15 10 5 0 1900 1905 1910 1915 1920 1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 cohorts periods Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook The Lexis-diagram 80 75 70 65 60 55 42 50 45 age 40 35 30 25 20 15 10 5 0 1900 1905 1910 1915 1920 1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 cohorts periods Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook The Lexis-diagram 80 75 70 65 60 55 42 50 45 age 40 35 30 25 20 15 10 5 0 1900 1905 1910 1915 1920 1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 cohorts periods Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook The Lexis-diagram 80 75 70 65 60 55 42 50 45 age 40 35 30 25 20 15 10 5 0 1900 1905 1910 1915 1920 1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 cohorts periods Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook The Lexis-diagram 80 75 70 65 60 55 42 50 45 age 40 35 30 25 20 15 10 5 0 1900 1905 1910 1915 1920 1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 cohorts periods 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 y ij : # counts in age group i at period j n ij : population in age group i at period j The APC-model y ij ∼ Po ( n ij · p ij ) � �� � λ ij η ij = log( λ ij ) = log( n ij ) + µ + α i + β j + γ k with age effect α i , period effect β j , cohort effect γ k 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 � � � α i = β j = γ k = 0 i j k The APC parameters are still not identifiable: � � i − I + 1 α i → α i + c · ; 2 ⇒ η ij = log( n ij ) + µ + α i + β j + γ k � � j − J + 1 β j → β j − c · ; 2 is left unchanged � � k − K + 1 γ k → γ k + c · 2 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 � � I − κ � f ( α | κ ) ∝ κ ( I − 1) / 2 exp ( α i − α i − 1 ) 2 2 i =2 � − κ � = κ ( I − 1) / 2 exp 2 α T R (1) α with   1 − 1 − 1 2 − 1     − 1 2 − 1     R (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): � � I − κ � f ( α | κ ) ∝ κ ( I − 2) / 2 exp ( α i − 2 α i − 1 + α i − 2 ) 2 2 i =3 � − κ � = κ ( I − 2) / 2 exp 2 α T R (2) α RW1 penalizes deviations from a model where all parameters are constant. The parameters are identifiable. RW2 penalizes deviations from a linear trend α i = 2 α i − 1 − α i − 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 z ij ∼ N (0 , δ − 1 ) can be introduced ξ ij = log( n ij ) + µ + α i + β j + γ k + z ij � �� � η ij Using a reparameterization, it follows z ij = ξ ij − (log n ij + µ + α i + β j + γ k ) = ξ ij − η ij The implied prior of the linear predictor is   I J  − δ � � f ( ξ | η , δ ) ∝ δ IJ / 2 · exp ( ξ ij − η ij ) 2  . 2 i =1 j =1 Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook Full conditional distributions For all parameters except ξ ij 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: y ij ∼ Po ( n ij · p ij ) � �� � λ ij ξ ij = log( λ ij ) = log( n ij ) + µ + α i + β j + γ k + z ij � �� � η ij 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 τ 1 τ 2 τ 3 τ 4 0 t 1 t 2 t 3 1 t 4 crosses mark the occurrence of an event, t 1 , . . . , t 4 arrival times, τ 1 , . . . , τ 4 inter-arrival times Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook Data Augmentation (1) For each y ij > 0 introduce τ ij = ( τ ij 1 , τ ij 2 ), for y ij = 0 just τ ij = τ ij 1 τ ij 2 denotes the last jump before 1 and is Ga ( y ij , λ ij ) τ ij 2 = ρ ij 2 /λ ij , ρ ij 2 ∼ Ga ( y ij , 1) Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook Data Augmentation (1) For each y ij > 0 introduce τ ij = ( τ ij 1 , τ ij 2 ), for y ij = 0 just τ ij = τ ij 1 τ ij 2 denotes the last jump before 1 and is Ga ( y ij , λ ij ) τ ij 2 = ρ ij 2 /λ ij , ρ ij 2 ∼ Ga ( y ij , 1) τ ij 1 denotes the inter-arrival time between the last jump before and the first jump after 1 and is Ex ( λ ij ) τ ij 1 = ρ ij 1 /λ ij , ρ ij 1 ∼ Ex (1) . Andrea Riebler University of Zurich

Introduction Age-Period-Cohort Models Auxiliary Mixture Sampling Summary and Outlook Data Augmentation (2) Reformulation: − log( τ ij 2 ) = log( λ ij ) + ǫ ij 2 � �� � ξ ij − log( τ ij 1 ) = log( λ ij ) + ǫ ij 1 � �� � ξ ij where ǫ ij 1 = − log( ρ ij 1 ) and ǫ ij 2 = − log( ρ ij 2 ). Andrea Riebler University of Zurich