A Flexible Modeling Framework for Overdispersed Hierarchical Data

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1 A Flexible Modeling Framework for Overdispersed Hierarchical Data Geert Molenberghs Interuniversity Institute for Biostatistics and statistical Bioinformatics (I-BioStat) Universiteit Hasselt & Katholieke Universiteit Leuven, Belgium & July 10, 2012 Paris, France

2 Joint Work With Modeling framework: Clarice G.B. Demétrio, Tony Vangeneugden Geert Verbeke, Afrânio Vieira Incomplete data concepts: Geert Verbeke, Michael G. Kenward, Marc Aerts, Niel Hens, An Creemers Ongoing work also with: Achmad Efendi, Mehreteab Fantahun, Aklilu Ghebretinsae, Samuel Iddi, Elasma Milanzi, Thomas Neyens, Edmund Njeri. Paris, July 10,

3 The Epilepsy Data Randomized, double-blind, parallel group multi-center study placebo (45) new anti-epileptic drug (AED; 44) 12-week run-in period & 16 weeks of follow up (some until week 27) outcome: the number of epileptic seizures experienced during the last week research question: reduction in # seizures by new therapy Let Y i now be the total number of seizures for subject i: Y i = n i i=1 Y ij where Y ij was the original (longitudinally measured) weekly outcome. Paris, July 10,

4 The basic Poisson model: Y i Poisson(λ i ) ln(λ i /n i ) = x i β The regression model is equivalent to λ i = n i exp(x i β) = exp(x i β + ln n i ) Paris, July 10,

5 Methodological Context Study Outcome type type N(µ, σ 2 ) N(µ,σ 2 ) Effect Param Effect Param Y i Mean µ Var. σ 2 Corr. Mean µ Var. µ (& θ i ) Corr. (θ i ) Effect Param Effect Param (Y i1,...,y ini ) Mean Var. µ i b i & Σ i Mean µ i Var. b i (& θ i ) Corr. b i & Σ i Corr. b i (& θ i ) Paris, July 10,

6 The Generalized Linear Model Suppose a sample Y 1,...,Y N of independent observations is available All Y i have densities f(y i θ i, φ) which belong to the exponential family: f(y θ i, φ) = exp { φ 1 [yθ i ψ(θ i )] + c(y, φ) } natural parameter θ i = x i β linear predictor Scale parameter (overdispersion parameter): φ Inverse link function: ψ (.) Mean-variance relationship: Var(Y ) = φψ [ ψ 1 (µ) ] = φv(µ) Paris, July 10,

7 Some Examples Element notation continuous binary count time to event Standard univariate exponential family Model normal Bernoulli Poisson Exponential Weibull Model f(y) 1 σ e (y µ) 2 2π 2σ 2 π y (1 π) 1 y e λ λ y ϕe ϕy ϕρy ρ 1 e ϕyρ y! Nat. param η µ ln[π/(1 π)] ln λ ϕ Mean function ψ(η) η 2 /2 ln[1 + exp(η)] λ = exp(η) ln( η) Norm. constant c(y, φ) ln(2πφ) 2 y2 2φ 0 lny! 0 (Over)dispersion φ σ Mean µ µ π λ ϕ 1 ϕ 1/ρ Γ(ρ 1 + 1) Variance φv(µ) σ 2 π(1 π) λ ϕ 2 ϕ 2/ρ [Γ(2ρ 1 + 1) Γ(ρ 1 + 1) 2 ] Paris, July 10,

8 Standard Overdispersion Models φ 1 = Var(Y i ) = φµ i Two-stage approach: Formulation: Y i λ i Poi(λ i ) E(λ i ) = µ i Implications: Var(λ i ) = σ 2 i E(Y i ) = E[E(Y i λ i )] = E(λ i ) = µ i Var(Y i ) = E[Var(Y i λ i )] + Var[E(Y i λ i )] = E(λ i ) + Var(λ i ) = µ i + σ 2 i Paris, July 10,

9 Repeated-measures version: λ i = (λ i1,...,λ ini ) E(λ i ) = µ i Var(λ i ) = Σ i E(Y i ) = µ i Var(Y i ) = M i + Σ i Paris, July 10,

10 The Standard Poisson-normal Model General formulation: f i (y ij b i, β, φ) = exp { φ 1 [y ij θ ij ψ(θ ij )] + c(y ij, φ) } η(µ ij ) = η[e(y ij b i )] = x ijβ + z ijb i b i N(0, D) Poisson formulation: Y ij Poi(λ ij ) ln(λ ij ) = x ijβ + z ijb i b i N(0,D) Paris, July 10,

11 Marginal moments: µ ij = exp x ijβ z ijdz ij Var(Y i ) = M i + M i ( e Z i DZ i J ni ) Mi Univariate version of the marginal moments: µ i = exp x i β d Var(Y i ) = µ i + µ 2 i (ed 1) Paris, July 10,

12 A Combined Model: The Poisson-(... )-normal Model Model formulation: Y ij Poi(λ ij ) λ ij = θ ij exp ( x ijβ + z ijb i ) b i N(0,D) E(θ i ) = E[(θ i1,...,θ ini ) ] = Φ i Var(θ i ) = Σ i Paris, July 10,

13 Marginal moments: µ ij = φ ij exp x ijβ z ijdz ij Var(Y i ) = M i + M i (P i J ni ) M i p i,jk = exp 1 2 z ij Dz ik σi,jk + φ ij φ ik φ ij φ ik exp 1 2 z ik Dz ij Univariate version of the marginal moments: µ i = φ i exp x iβ d Var(Y i ) = µ i + µ 2 i e d 1 + σ2 i φ 2 i e d Paris, July 10,

14 Fully Parametric Specification So far, random effects are specified only in a moment-based fashion: Mean Variance Fully parametric specification is possible Gamma specification is sensible Univariate case: the negative-binomial model Hierarchical case: the Poisson-gamma-normal model Paris, July 10,

15 The Negative-binomial Model λ Gamma(α,β) f(λ) = P(Y = y) = 1 β α Γ(α) λα 1 e λ/β 1 + β α 0 λ α 1 e λ/βλy e λ Γ(α) y! = α + y 1 α 1 β β + 1 y 1 β α Paris, July 10,

16 The Poisson-gamma-normal Model The marginal probability mass function: P(Y i = y i ) = t n i j=1 y ij + t j y ij α j + y ij + t j 1 α j 1 ( 1) tj β y ij+t j j exp n i (y ij + t j )x ijβ exp j=1 The special case of the Poisson-normal model: P(Y i = y i ) = 1 n i j=1 y ij! t ( 1) n i j=1 t j n i j=1 t j! exp n i exp n i (y ij + t j )z ij D j=1 n i j=1 (y ij + t j )x ijβ (y ij + t j )z ij D j=1 n i n i j=1 (y ij + t j )z ij j=1 (y ij + t j )z ij Paris, July 10,

17 Univariate case for Poisson-gamma-normal model: P(Y i = y i ) = + t=0 y i + t y i α + y i + t 1 α 1 β yi+t ( 1) t exp x i β(y i + t) d(y i + t) 2 Univariate case for Poisson-normal model: P(Y i = y i ) = 1 y i! + t=0 ( 1) t t! exp x i β(y i + t) d(y i + t) 2 Paris, July 10,

18 Estimation Integrate over the gamma random effects only: P(Y ij = y ij b i ) = α j + y ij 1 α j 1 κ ij = exp[x ijβ + z ijb i ] β j 1 + κ ij β j ij y κ ij β j α j y κ ij ij Can be fitted using the SAS procedure NLMIXED Model for the epilepsy data: ln(λ ij ) = (β 00 + b i ) + β 01 t ij if placebo (β 10 + b i ) + β 11 t ij if treated, b i N(0, d) Paris, July 10,

19 Parameter Estimates Poisson Negative-binomial Effect Parameter Estimate (s.e.) Estimate (s.e.) Intercept placebo β (0.0424) (0.1119) Slope placebo β (0.0043) (0.0111) Intercept treatment β (0.0383) (0.1093) Slope treatment β (0.0038) (0.0101) Negative-binomial parameter α (0.0255) Negative-binomial parameter α 2 = 1/α (0.0918) Variance of random intercepts d Poisson-normal Combined Effect Parameter Estimate (s.e.) Estimate (s.e.) Intercept placebo β (0.1677) (0.1755) Slope placebo β (0.0044) (0.0077) Intercept treatment β (0.1701) (0.1782) Slope treatment β (0.0043) (0.0074) Negative-binomial parameter α (0.2113) Negative-binomial parameter α 2 = 1/α (0.0348) Variance of random intercepts d (0.1844) (0.1850) Paris, July 10,

20 Implications for Correlation Function Smallest value Largest value Model Arm ρ time pair ρ time pair Poisson-normal placebo & & 2 Poisson-normal treatment & & 2 Combined placebo & & 2 Combined treatment & & 27 Paris, July 10,

21 Implications for Hypothesis Testing p-values Model H 0 : β 11 β 01 = 0 H 0 : β 11 /β 01 = 1 Poisson Poisson-normal negative-binomial combined Paris, July 10,

22 The General Case f i (y ij b i, ξ) = exp { φ 1 [y ij λ ij ψ(λ ij )] + c(y ij, φ) } E(Y ij θ ij, b i ) = µ c ij = θ ijκ ij θ ij G ij (ξ ij,σ 2 ij) κ ij = g(x ij ξ + z ij b i) b i N(0, D) η ij = x ijξ + z ijb i Paris, July 10,

23 Classical Conjugacy f(y θ) = exp { φ 1 [yh(θ) g(θ)] + c(y, φ) } f(θ) = exp {γ[ψh(θ) g(θ)] + c (γ, ψ)} f(y) = exp c(y, φ) + c (γ, ψ) c φ 1 + γ, φ 1 y + γψ φ 1 + γ Paris, July 10,

24 Strong Conjugacy Write: f(y κθ) = exp { φ 1 [yh(κθ) g(κθ)] + c(y, φ) } Apply the transformation theorem: f(θ γ, ψ) = κ f(κθ γ, ψ) Request the form: f(κθ) = exp {γ [ψ h(κθ) g(κθ)] + c (γ, ψ )} Leading to the marginal model: f(y κ) = exp c(y, φ) + c (γ,ψ ) + c φ 1 + γ, φ 1 y + γ ψ φ 1 + γ Paris, July 10,

25 The Weibull Case The Weibull-gamma-normal model: f(y i θ i,b i ) = n i λρθ ijyij ρ 1 j=1 e x ijβ+z ijb i e λyρ ij θ ije x ijβ+z ijb i f(θ i ) = n i j=1 1 j Γ(α j ) θα j 1 ij β α j e θ ij/β j f(b i ) = 1 2b i D 1 b i (2π) q/2 D 1/2e 1 Paris, July 10,

26 Marginal density: f(y i ) = n i (m 1,...,m ni ) j=1 ( 1) m j Γ(α j + m j + 1)β m j+1 j m j! Γ(α j ) exp (m j + 1) x ijβ (m j + 1) z ijdz ij λ m j+1 ρy (m j+1)ρ 1 ij Marginal moments: E(Y k ij) = α jb(α j k/ρ, k/ρ + 1) λ k/ρ β k/ρ j exp k ρ x ijβ + k2 2ρ 2z ijdz ij E(Y ij ) = α jb(α j 1/ρ, 1/ρ + 1) λ 1/ρ β 1/ρ j exp 1 ρ x ijβ + 1 2ρ 2z ijdz ij Paris, July 10,

27 Var(Y ij ) = α j λ 2/ρ β 2ρ j exp 2 ρ x ijβ + 1 ρ ijdz 2z ij B (α j 2/ρ, 2/ρ + 1) exp 1 ρ ijdz 2z ij α j B α j 1 ρ, 1 ρ Cov(Y ij, Y ik ) = α j α k λ 2/ρ β 1/ρ j β 1/ρ k exp 1 ρ (x ijβ + x ikβ) B α j 1 ρ, 1 ρ + 1 B α k 1 ρ, 1 ρ + 1 exp 1 2ρ ijdz 2(z ij + z ikdz ik ) exp 1 ρ ijdz 2z ik 1 Paris, July 10,

28 Identifiability can be achieved in, for example, two ways: For the second case: E(Y k ij) = δk/ρ j f(θ i ) = n i j=1 f(θ i ) = n i k λ Γ k 1 k ρ 1 α j j=1 δ je δ jθ ij Γ Weibull Exponential 1 α j Γ(αj ) θα j 1 k exp ρ ij e α jθ ij k ρ x ijβ + k2 E(Yij k ) = δk jk Γ (1 k) Γ (k) exp kx λk ij β + k2 2ρ 2z ijdz ij 2 z ij Dz ij Paris, July 10,

29 Univariate case, Weibull-exponential: f(y) = ϕρyρ 1 δ (δ + ϕy ρ ) 2 E(Y k ) = k ρ δ ϕ kρ Γ(1 k/ρ) Γ(k/ρ) Univariate case, exponential-exponential: f(y) = ϕδ (δ + ϕy) 2 E(Y k ) = k δ ϕ k Γ(1 k) Γ(k) Paris, July 10,

30 This leads to a class of Cauchy-type distributions, including the log-logistic distribution, of which there are known to be problems. Estimation using the marginal-conditional approach: f(y ij b i ) = λκ ije µ ijρy ij ρ 1 α j β j (1 + λκ ij e µ ijβ j yij) ρ α j+1 Further issues: censorship related functions non-parametric baseline hazard functions Paris, July 10,

31 The Binary Case A logit formulation: Y ij Bernoulli(π ij = θ ij κ ij ) κ ij = exp ( x ijβ + z ijb i 1 + exp ( x ij β + z ij b i ) ) No closed forms Conjugacy no strong conjugacy Marginal-conditional version (for NLMIXED implementation) f(y ij b i ) = 1 α j + β j (κ ij α j ) y ij [(1 κ ij )α j + β j ] 1 y ij Paris, July 10,

32 A probit formulation: κ ij = Φ 1 (x ijβ + z ijb i ) θ ij Beta(α,β) f ni (y i = 1) = α α + β n i Φni (X iβ; L 1 n i ) L ni = I ni Z i ( D 1 + Z iz i ) 1 Z i E(Y ij ) = Var(Y ij ) = α α + β Φ 1(x ijβ; L 1 1 ) = α α + β Φ 1( I + Dz ij z ij 1/2 x ijβ) α α + β Φ 1(x ijβ; L 1 1 ) 1 α α + β.φ 1(x ijβ; L 1 1 ) Paris, July 10,

33 Cov(Y ij, Y ik ) = α α + β 2 Φ 2 x ij x ik β, L 1 2jk Φ 1 (x ijβ; L 1 1j )Φ 1 (x ikβ; L 1 1k ) L 2jk = I 2 z ij z ik D 1 + z ij z ik (z ij z ik ) 1 (z ij z ik ) The above gives the success probabilities; all other joint probabilities can be derived in a similar fashion Paris, July 10,

34 Joint Modeling of a Poisson and Weibull Variable The combined model lends itself naturally to a joint modeling approach Example: Y is Poisson with gamma random effect θ y Z is Weibull with gamma random effect θ z The joint normal random effect is b Moments and joint marginal distribution can be derived Paris, July 10,

35 Joint distribution: P(Y = n, Z = z) = + + ( 1) m+k m=0 k=0 n!m!k! β n+m y Γ(n + m + α y) Γ(α y ) β k+1 z Γ(k α z) Γ(α z ) λ k+1 e (n+m)γ ρz ρ(k+1) 1 exp µ(k + 1) d(n + m + k + 1)2 Other joint distributions: similar Longitudinal versions: similar Estimation: using similar SAS procedure NLMIXED implementation Paris, July 10,

36 Joint Models with Longitudinal Data Missing Data f(y i,r i X i θ,ψ) Selection Models: f(y i X i,θ) f(r i X i, y o i, y m i, ψ) MCAR MAR MNAR f(r i X i, ψ) f(r i X i,y o i,ψ) f(r i X i, y o i, ym i, ψ) Pattern-mixture Models: f(y i X i, r i, θ) f(r i X i,ψ) Shared-parameter Models: f(y i X i, b i, θ) f(r i X i, b i,ψ) Paris, July 10,

37 MAR in 3 Frameworks Selection models f(r i y i, ψ) = f(r i y o i, ψ) Pattern-mixture models f(y m i y o i,r i,θ) = f(y m i y o i, θ) Shared-parameter models? Paris, July 10,

38 MAR in Selection Models: Natural Diggle and Kenward (ApStat 1994) f(r i y i,ψ) = f(r i y o i,ψ) Longitudinal data: logit [P(D i = j D i j,y ij, y i,j 1 )] = ψ 0 + ψ 1 y i,j 1 + ψ 2 y ij ψ 2 0 MNAR ψ 2 = 0 MAR ψ 1 = ψ 2 = 0 MCAR No dependence on the future (NFD): built in Paris, July 10,

39 MAR in Pattern-mixture Models: Feasible Molenberghs, Michiels, Kenward, and Diggle (Statistica Neerlandica 1998) Thijs, Molenberghs, Michiels, Verbeke, and Curran (Biostatistics 2002) f(y m i yo i,r i,θ) = f(y m i yo i,θ) For longitudinal data: ACMV: available case missing value restrictions: t 2, s < t : f(y it y i1,,y i,t 1, d i = s) = f(y it y i1,,y i,t 1,d i t) Practical implementation: doable! Likewise, non-future dependence can be built in Paris, July 10,

40 MAR in Shared-parameter Models?! Creemers, Hens, Aerts, Molenberghs, Verbeke, and Kenward (2008) Conventional f(y i X i, b i, θ) f(r i X i,b i, ψ) Extended f(y o i g i,h i, j i,l i )f(y m i y o i, g i, h i,k i, m i ) f(r i g i, j i, k i n i ) f(y o i g i,h i, j i )f(y m i y o i,g i,h i, k i )f(r i g i, j i, k i )f(b i ) db i f(y o i g i, j i )f(r i g i,j i )f(b i ) db i MAR = f(y o i g i, h i )f(y m i yo i, g i, h i )f(b i ) db i f(y o i) Paris, July 10,

41 MAR in Shared-parameter Models Creemers, Hens, Aerts, Molenberghs, Verbeke, and Kenward (2008) Extended f(y o i g i, h i, j i,l i )f(y m i y o i, g i, h i,k i, m i ) f(r i g i, j i, k i n i ) f(y o i g i,h i, j i )f(y m i yo i, g i,h i, k i )f(r i g i, j i, k i )f(b i ) db i f(y o i g i, j i )f(r i g i,j i )f(b i ) db i MAR = f(y o i g i, h i )f(y m i yo i, g i, h i )f(b i ) db i f(y o i) Sub-class MAR f(y o i j i,l i )f(y m i y o i, m i )f(r i j i, n i ) Paris, July 10,

42 Sensitivity: Every MNAR Model Has Got an MAR Counterpart Molenberghs, Beunckens, Sotto, and Kenward (JRSSB 2008) Creemers, Hens, Aerts, Molenberghs, Verbeke, and Kenward (2008) Fit an MNAR model to a set of incomplete data Change the conditional distribution of the unobserved outcomes, given the observed ones, to comply with MAR Resulting new model has exactly the same fit as the original MNAR model The missing data mechanism has changed This implies that definitively testing for MAR versus MNAR is not possible Paris, July 10,

43 MAR Counterpart to Pattern-mixture Models f(y o i, y m i, r i θ, ψ) = f(y o i r i, θ) f(r i ψ) f(y m i y o i, r i, θ) h(y o i, y m i, r i θ, ψ) = f(y o i r i, θ) f(r i ψ) f(y m i y o i, θ, ψ) Starting from PMM is natural : clear separation into: fully observable components entirely unobserved component Paris, July 10,

44 MAR Counterpart to Selection Models f(y o i,y m i,r i θ, ψ) = f(y o i, y m i θ) f(r i y o i,y m i, ψ) f(y o i,y m i,r i θ, ψ) = f(y o i r i, θ, ψ) f(r i θ, ψ) f(y m i y o i,r i, θ, ψ) h(y o i,y m i,r i θ, ψ) = f(y o i r i, θ, ψ) f(r i θ, ψ) f(y m i y o i, θ, ψ) Paris, July 10,

45 MAR Counterpart to Shared-parameter Models: Important Here! f(y i o, y i m, r i b i ) = f(y o i g i, h i, j i, l i ) f(y m i yo i, g i, h i, k i,m i ) f(r i g i, j i,k i n i ) h(y i o, y i m, r i b i ) = f(y o i g i, h i, j i, l i ) h(y m i yo i, m i) f(r i g i, j i,k i n i ) with h(y m i y o i, m i ) = g i h i k i f(y m i y o i, g i, h i,k i, m i )dg i dh i dk i Paris, July 10,

46 Common Ground for Various Settings Classical survival : time to event & censorship Joint modeling : time to event(s) & longitudinal process Incomplete data : outcome(s) & missigness process Sequential trials : outcomes & sample size Informative cluster size : outcomes & cluster size Paris, July 10,

47 Concluding Table Element notation continuous binary count time to event Standard univariate exponential family Model normal Bernoulli Poisson Exponential Weibull (y µ) 2 1 Model f(y) σ 2π e 2σ 2 π y (1 π) 1 y e λ λ y ϕe ϕy ϕρy ρ 1 e ϕyρ y! Nat. param η µ ln[π/(1 π)] ln λ ϕ Mean function ψ(η) η 2 /2 ln[1 + exp(η)] λ = exp(η) ln( η) Norm. constant c(y, φ) ln(2πφ) 2 y2 2φ 0 lny! 0 (Over)dispersion φ σ Mean µ µ π λ ϕ 1 ϕ 1/ρ Γ(ρ 1 + 1) Variance φv(µ) σ 2 π(1 π) λ ϕ 2 [ ϕ 2/ρ Γ(2ρ 1 + 1) Γ(ρ 1 + 1) 2] Exponential family with conjugate random effects Model normal-normal Beta-binomial Negative binomial Exponential-gamma Weibull-gamma Hier. model f(y θ) (y θ) 2 1 σ 2π e 2σ 2 θ y (1 θ) 1 y e θ θ y y! ϕθe ϕθy ϕθρy ρ 1 e ϕθyρ RE model f(θ) 1 e (θ µ)2 2d d 2π Marg. model f(y) θ α 1 (1 θ) β 1 B(α,β) (y µ) 2 1 σ 2 +d 2π e 2(σ 2 +d) (α + β) Γ(α) Γ(β) Γ(α+y) Γ(β+1 y) θ α 1 e θ/β β α Γ(α) ( Γ(α+y) β y!γ(α) β+1 ) y ( 1 β+1 θ α 1 e θ/β β α Γ(α) ) α ϕαβ (1+ϕβy) α+1 θ α 1 e θ/β β α Γ(α) ϕρy ρ 1 αβ (1+ϕβy ρ ) α+1 h(θ) θ ln[θ/(1 θ)] ln(θ) θ θ g(θ) 1 2 θ2 ln(1 θ) θ ln(θ)/ϕ ln(θ)/ϕ φ σ /ϕ 1/ϕ γ 1/d α + β 2 1/β ϕ(α 1) ϕ(α 1) ψ µ c(y, φ) c (γ, ψ) 1 2 φy2 1 2 ln( 2π φ 1 2 γψ2 1 2 ln( 2π γ Mean E(Y ) µ Variance Var(Y ) σ 2 + d ) ) α 1 α+β 2 β(α 1) [βϕ(α 1)] 1 [βϕ(α 1)] 1 ( 0 ln(y!) ln(ϕ) ln ϕρy ρ 1) γ+ϕ ln B(γψ + 1,γ ψγ + 1) (1 + γψ) ln γ lnγ(1 + γψ) ϕ ln(γψ) lnγ( ) γ+ϕ γ+ϕ ϕ ϕ ln(γψ) lnγ( ) γ+ϕ ϕ α αβ [ϕ(α 1)β] 1 Γ(α ρ 1 )Γ(ρ 1 +1) α+β (ϕβ) 1/ρ Γ(α) [ αβ (α+β) 2 αβ(β + 1) α[ϕ 2 (α 1) 2 (α 2)β 2 ] 1 1 2Γ(α 2ρ 1 )Γ(2ρ 1 ) ρ(ϕβ) 2/ρ Γ(α) ] Γ(α ρ 1 ) 2 Γ(ρ 1 ) 2 ργ(α) Paris, July 10,

48 References Molenberghs, G. and Verbeke, G. (2011). On the Weibull-Gamma frailty model, its infinite moments, and its connection to generalized log-logistic, logistic, Cauchy, and extreme-value distributions. Journal of Statistical Planning and Inference, 141, Molenberghs, G., Verbeke, G., and Demétrio, C. (2007). An extended random-effects approach to modeling repeated, overdispersed count data. Lifetime Data Analysis, 13, Molenberghs, G., Verbeke, G., Demétrio, C.G.B., and Vieira, A. (2010). A family of generalized linear models for repeated measures with normal and conjugate random effects. Statistical Science, 25, Vangeneugden, T., Molenberghs, G., Verbeke, G., and Demétrio, C. (2011). Marginal correlation from an extended random-effects model for repeated and overdispersed counts. Journal of Applied Statistics, 38, Paris, July 10,