Casswell, S

Huckle, T

Romeo Nunez, J

English

Massey Research Online

Modelling alcohol-consumption in New Zealand: A Bayesianconditional copula-based regression approachJose Romeo, Taisia Huckle & Sally CasswellSHORE & Whariki Research Centre, College of Healthe-mail: j.romeo@massey.ac.nzIntroductionMost regression models focus on explaining distributional aspects of one single outcome variable.Modelling multiple outcome variables and their dependence structure have become very popular.Examples:• Medicine - Time to relapse of a disease and time of death• Marketing - purchase of different products• Actuarial sciences - age of death of husbands and wives (“widowhood effect”)• Finance - understanding the relationship between some financial stocks (time series)• Ecological community data - species display association with one another• Alcohol research - quantity on a typical occasion and frequency of drinkingê Alcohol consumption can be measure in two dimensions: the amounts consumed in a typical occa-sion and the frequency of drinking (Casswell et al., 2016).Alcohol consumption IAC study New Zealand data: The International Alcohol Control (IAC) studyis an international cohort study of alcohol use and alcohol policy relevant behaviours coordinated atSHORE (Huckle et al., 2018). In 2011 a national stratified sample of households was surveyed in NZ:• Alcohol consumption data were collected using a beverage- and location-specific measure in last 6months• For each place, they were asked how often they drank there and what they would drink on a typicaloccasion at that location• This information was used to calculate the outcomes:Y1 : quantity on a typical occasion (standard drinks)Y2 : frequency of drinking• Covariate information: Gender, age, ethnicity, level of education, log equivalised income andpoverty line.Figure 1: Quantity and frequency of drinking, Kendall’s tau = 0.12• Some previous studies on the factors affecting quantity and frequency of alcohol consumption havetreated these two outcomes independently.• We fit conditional copula-based regression models for explaining the joint distribution of the typicalamount and the frequency of alcohol consumption.ê Copula-based regression models allow such an analysis since they enable the separation of themarginal outcome distributions and the dependence structure modelled by a specific copula function.ê We include socio-demographic factors not only in the marginal distributions through the mean andthe dispersion but also in the copula parameter allowing a direct modelling of the association andflexibility in model specification.Copulas(See e.g. Mai & Scherer, 2012)Suppose we have two continuous variables of interest Y1 and Y2 where• Y1 ∼ F1(·) and Y2 ∼ F2(·), marginal distributions• F (y1, y2) = Pr{Y1 ≤ y1, Y2 ≤ y2}, joint distribution of the pair (Y1, Y2)• Then the joint cdf can be written as a copula function relating the marginal distributions asF (y1, y2) = Cα(F1(y1), F2(y2)), α ∈ A• The dependence parameter α measures the strength of association between Y1 and Y2• The joint pdf is given byf (y1, y2) = cα(F1(y1), F2(y2)) · f1(y1) · f2(y2)Based on a specific copula Cα we can compute Kendall’s tau coefficient that is a measure of concor-dance. Considering uj = Fj(yj), j = 1, 2,τα = 4∫ ∫[0,1]2Cα(u1, u2)dCα(u1, u2)− 1Conditional copula-based regression model(See e.g. Nikoloulopoulos & Karlis, 2010, Klein & Kneib, 2016)• Dependence parameter α is mostly treated as constant• We would like to explain the association between outcomes not just the marginalsSuppose we have two outcomes of interest Y1 and Y2 given covariate information x where• Y1|x ∼ F1(·|x) and Y2|x ∼ F2(·|x), conditional marginal distributions• The conditional joint cdf can be written as a conditional copula function relating the conditionalmarginal distributions asF (y1, y2|x) = Cα(x) (F1(y1|x), F2(y2|x)|x)• Copula parameter depends on covariates α(x) through an appropriate link function h(·) as in GLMSuppose (yi1, yi2) independent realizations of (Y1, Y2), and covariate information xi, with i = 1, . . . , n.Then we consider• Y1|x ∼ F1(y1|µ1(x), σ1(x)), Y2|x ∼ F2(y2|µ2(x), σ2(x))• (Y1, Y2)|x ∼ Cα(x) (F1(y1|x), F2(y2|x)|x)and for j = 1, 2, linear predictors• hj(µj(xi)) = βj0 + βj1x1i + βj2x2i + . . . + βjpxpi• hσj(σj(xi)) = γj0 + γj1x1i + γj2x2i + . . . + γjpxpi• h(α(xi)) = α0 + α1x1i + α2x2i + . . . + αpxpiApplication: Alcohol consumption IAC study NZ dataMarginals modelling:• Quantity: log yi1 |xi ∼ Normal(µi1(xi), σ2i1(xi))• Frequency: 4√yi2 |xi ∼ Normal(µi2(xi), σ2i2(xi)), where µij(xi) = β′xi and σij(xi) = exp(γ′xi)Copula model DIC Copula model DICIndependent 4343.1 PVF† 4341.5Gaussian 4346.5 Inverse Gaussian 4340.1Clayton 4341.1 Frank 4357.1Table 1: Model selection criterionModelling: Mean Dispersion DependenceY1 Y2 Y1 Y2 αEffect Median SD Median SD Median SD Median SD Median SDIntercept 0.865 0.028 2.877 0.025 -0.962 0.045 -0.675 0.042 0.417 0.391Gender: Male vs Female 0.119 0.019 0.108 0.023 0.083 0.044 0.551 0.319Poverty line: Under vs Over -0.046 0.067Age -0.012 0.002 0.004 0.002 -0.015 0.004 0.035 0.022Education: Low vs High 0.377 0.075Medium vs High 0.169 0.038Ethnicity: Maori vs NZ Euro 0.353 0.067 -0.155 0.079Pacific vs NZ Euro 0.492 0.132 -0.547 0.147Asian vs NZ Euro -0.280 0.089 -0.415 0.099Log equivalised income 0.206 0.032 -0.301 0.059 -0.202 0.057 -0.761 0.466Age × Under poverty line -0.012 0.005Table 2: Parameter estimates, Inverse Gaussian copula modelFigure 2: Estimated Kendall’s tau between Quantity and Frequency by gender, age and incomeê Interpretation: Young women with high income: as they drink more frequently they also drinkmore quantities. Older men with low income: quantity and frequency are almost independent.Discussion and future work• Overall, the association between quantity and frequency of drinking is not strong:Kendall’s τ = 0.15 (95% CI:0.11− 0.19).• Flexibility of models based on copulas: dependence structure and marginals distributionsFuture work• About the marginals (e.g. skew-normal, negative binomial distributions)• A more efficient posterior computation (see e.g. Wichitaksorn et al. 2018)• To include more countries members of the IAC study, e.g., Australia, Thailand, Vietnam, England,Scotland, South Africa: ê Hierarchical modelMain ReferencesCasswell, S., Huckle, T., Wall, M. & Parker, K. (2016). Policy relevant behaviours mediate the relationship between socio-economic status and alcohol consumption - analysis from the IAC study. Alcoholism: Clinical and Experimental Research,40(2), 385-392.Huckle, T., Casswell, S., Mackintosh, A.-M. et al. (2018). The International Alcohol Control Study: methodology andimplementation. Drug and Alcohol Review, 37(2), S10–S17.Klein, N. & Kneib, T. (2016). Simultaneous inference in structured additive conditional copula regression models: aunifying Bayesian approach. Statistics and Computing, 26(4), 841–860.Mai, J. & Scherer, M. (2012). Simulating Copulas: Stochastic Models, Sampling Algorithms, and Applications. ImperialCollege, Boca Raton.Nikoloulopoulos, A.K. & Karlis, D. (2010). Regression in a copula model for bivariate count data. Journal of AppliedStatistics, 37(9), 1555–1568.†Romeo, J.S., Meyer, R. & Gallardo, D.I. (2018). Bayesian bivariate survival analysis using the power variance functioncopula. Lifetime Data Analysis, 24, 355–383.Wichitaksorn, N., Gerlach, R. & Choy, B. (2019). Efficient MCMC estimation of some elliptical copula regression modelsthrough scale mixtures of normals. Applied Stochastic Models in Business and Industry, 35(3), 808–822.MASSEY UNIVERSITYMASSEY RESEARCH ONLINE http://mro.massey.ac.nz/Massey Documents by Type Conference PostersModelling alcohol-consumption in NewZealand: A Bayesian conditionalcopula-based regression approach.Romeo Nunez, J2019-11-25http://hdl.handle.net/10179/1669001/11/2021 - Downloaded from MASSEY RESEARCH ONLINE

Modelling alcohol-consumption in New Zealand: A Bayesian conditional copula-based regression approach.

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Modelling alcohol-consumption in New Zealand: A Bayesian conditional copula-based regression approach.

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