Hierarchical Hurdle Models for Zero-In(De)flated Count Data of Complex Designs

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1 for Zero-In(De)flated Count Data of Complex Designs Marek Molas 1, Emmanuel Lesaffre 1,2 1 Erasmus MC 2 L-Biostat Erasmus Universiteit - Rotterdam Katholieke Universiteit Leuven The Netherlands Belgium 26th August 2009 International Society for Clinical Biostatistics 30 Prague, Czech Republic ERASMUSMC - Biostatistics / 22

2 Outline Challenges in the analysis of the motivating example Hurdle models H-likelihood for hurdle models Practical application ERASMUSMC - Biostatistics / 22

3 Motivating Example Randomized clinical trial (General Practice Erasmus MC) Two physical exercises regimes: standard and Tai-Chi Chuan Outcome: number of days elderly patients experienced a fall Each patient is followed for about a year: 5 measurements Baseline covariates: age, gender, BMI, alcohol use Is there a reduction of number of falls in the Tai-Chi Chuan patients? ERASMUSMC - Biostatistics / 22

4 Histogram - Tai-Chi Chuan Period 1 Period 2 Control Group Period 3 Period 4 Period 5 Density Density Density Density Density Falls Falls Falls Falls Falls Period 1 Period 2 Tai Chi Chuan Group Period 3 Period 4 Period 5 Density Density Density Density Density Falls Falls Falls Falls Falls ERASMUSMC - Biostatistics / 22

5 The Zero-Inflated Poisson and the Hurdle Model The zero-inflated Poisson model is a mixture: Point mass at zero Standard Poisson distribution The hurdle model has two components: Point mass at zero Truncated Poisson distribution ERASMUSMC - Biostatistics / 22

6 The Zero-Inflated Poisson and the Hurdle Model Zero-Inflated Poisson Model Hurdle Model P(X = 0) =p Z + (1 p Z )e µ Z P(X = k) =(1 p Z )e µ Z µk Z k! P(X = 0) =p H 1 P(X = k) =(1 p H ) 1 e µ e µ µk H H H k! ERASMUSMC - Biostatistics / 22

7 The Zero-Inflated Poisson and the Hurdle Model Relation between zero-value probabilities: {p H = p Z + (1 p Z )e µ Z } > 0 Inflation part probability: p Z = p H e µ H 1 e µ H Relation between means of the distributional parts: µ Z = µ H Hurdle model allows for zero-deflation and zero-inflation ZIP model allows only zero-inflation ERASMUSMC - Biostatistics / 22

8 Hurdle Model for Clustered Data Add covariates and random effects as follows: logit(p H ) = X T 0 β ( ) 0 + b 0 b0 log(µ H ) = X T 1 β G(µ,Σ) 1 + b 1 b 1 Likelihood factorizes if random effects are independent: Bernoulli model Truncated Poisson model ERASMUSMC - Biostatistics / 22

9 Hurdle Model for Clustered Data Marginal Likelihood - SAS PROC NLMIXED (Min & Agresti 2005) 2-level data Normal random effects Different distribution for random effects (Liu & Yu 2007) Marginal Likelihood - Non-parametric Maximum Likelihood (Min & Agresti 2005) ERASMUSMC - Biostatistics / 22

10 Hurdle Model for Clustered Data What are we looking for? Method to handle multilevel or multi-membership data Method to allow for dispersion parameters to depend on covariates Efficient estimation ERASMUSMC - Biostatistics / 22

11 Hurdle Model for Clustered Data H - Likelihood (Lee & Nelder, 1996, 2001; Noh & Lee 2007) What is it? Computational framework allowing estimation of models involving random effects It offers: Efficient estimation algorithm REML type of inference for dispersion components Implications: Random effects can be not normal Multilevel / multi-membership data is easily handled Dispersion parameters can depend on covariates Overdispersion can depend on covariates ERASMUSMC - Biostatistics / 22

12 H-likelihood Three types of parameters: Fixed effects parameters - β Random effects - v Dispersion parameters - λ ERASMUSMC - Biostatistics / 22

13 H-likelihood Extended likelihood L E (β,λ, v y, v) = Extended log-likelihood N n i f β,λ (y ij v i )f λ (v i ) i=1 j=1 h = log(l E (β,λ, v y, v)) ERASMUSMC - Biostatistics / 22

14 H-likelihood Adjusted profile likelihood: Estimation of β: p v (h) = h(β,λ, v) v=ˆv 0.5 log D(h, v) 2π Laplace approximation to the integral: L M (β,λ y) = N i=1 n i v=ˆv f β,λ (y ij v i )f λ (v i )d v i. j=1 ERASMUSMC - Biostatistics / 22

15 H-likelihood Adjusted profile likelihood: Estimation of λ: p β,v (h) = h(β,λ, v) β=ˆβ,v=ˆv 0.5 log D [h,(β, v)] 2π β=ˆβ,v=ˆv ERASMUSMC - Biostatistics / 22

16 Application to the Hurdle Model - Truncated Poisson Standard exponential family distribution f β (y ij v i ) = exp(y ij θ ij b(θ ij ) + c(y ij )) θ ij = x T ij β + zt ij v i Truncated exponential family distribution f β (y ij v i ) = exp(y ij θ ij b(θ ij ) log(m(θ ij )) + c(y ij )) θ ij = x T ij β + zt ij v i ERASMUSMC - Biostatistics / 22

17 Truncated Poisson Distribution Standard weight matrix Modified weight matrix W = W + W = ( M (θ) M(θ) Standard adjusted dependent variable ( ) 2 µ V 1 (µ) η ( M ) ) 2 (θ) V 1 (µ)w M(θ) z = η + (y µ)( η/ µ) Modified adjusted dependent variable z = η + (W/ W)(y µ M (θ) M(θ) )( η/ µ) ERASMUSMC - Biostatistics / 22

18 Application - Hurdle Model Bernoulli model Truncated Poisson model logit[p(y ij > 0)] = x T ij β + v i log u i = exp(v i) 1 + exp(v i ) ( ) 1 1 u i Beta, λ 10 λ 10 log(λ 10 ) = γ γ 101 Female i ( µij days ij ) = x T ij β + v i + v ij v i N(0, λ 20 ) u ij = exp(v ij ) u ij Gamma ( ) 1, λ 21 λ 21 log(λ 21 ) = γ γ 211 Female i ERASMUSMC - Biostatistics / 22

19 Application Bernoulli Model Effect Estimate P-value Intercept Female Time Time*Trt(Tai-Chi) Age γ γ 101 (Female) ERASMUSMC - Biostatistics / 22

20 Application Truncated Poisson Model Effect Estimate P-value Intercept <0.001 Female Time Time*Trt(Tai-Chi) γ γ γ 211 (Female) ERASMUSMC - Biostatistics / 22

21 Further Research Joint estimation of the binary and truncated Poisson model within H-likelihood framework Correlated random effects Non-normal random compoenents - copula approach How to make the correlation depend on covariates Number of random components greater then 2- copula - covariates? ERASMUSMC - Biostatistics / 22

22 Thank you for your attention! ERASMUSMC - Biostatistics / 22