Abstract
In many areas of application, there is increasing interest in modeling multivariate time series of counts on several subjects as a function of subject-specific and time-dependent covariates. We propose a level correlated model (LCM) to account for the association among the components of the response vector, as well as possible overdispersion. The flexible LCM framework allows us to combine different marginal count distributions and to build a hierarchical model for the vector time series of counts. We employ the Integrated Nested Laplace Approximation (INLA) for fast approximate Bayesian modeling using the R package INLA (r-inla.org). We illustrate it by modeling the monthly prescription counts by physicians of a focal drug from a multinational pharmaceutical firm along with monthly counts of other drugs with a sizable market share for the same therapeutic category.
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References
Aitchison, J., and C. Ho. 1989. The multivariate Poisson-log normal distribution. Biometrika 76(4): 643–653.
Carlin, B., N. Polson, and D. Stoffer. 1992. A Monte Carlo approach to nonnormal and nonlinear state-space modeling. Journal of the American Statistical Association 87: 493–500.
Carter, C., and R. Kohn. 1994. On Gibbs sampling for state-space models. Biometrika 81: 541–553.
Chen, M.-H., Q.-M. Shao, and J. Ibrahim. 2000. Monte Carlo methods in Bayesian computation. New York: Springer.
Davis, R., W. Dunsmuir, and S. Streett. 2003. Observation-driven models for Poisson counts. Biometrika 90(4): 777–790.
Gamerman, D. 1998. Markov chain Monte Carlo for dynamic generalized linear models. Biometrika 85(1): 215–227.
Gamerman, D., and H. Migon. 1993. Dynamic hierarchical models. Journal of the Royal Statistical Society Series (B) 55(3): 629–642.
Hu, S. 2012. Dynamic modeling of discrete-valued time series with applications. Ph.D. Thesis, University of Connecticut.
Kalman, R. 1960. A new approach to linear filtering and prediction theory. Transactions of the ASME Series D, Journal of Basic Engineering 82: 35–45.
Kalman, R., and R. Bucy. 1961. New results in filtering and prediction theory. Transactions of the ASME Series D, Journal of Basic Engineering 83: 95–108.
Karlis, D., and L. Meligkotsidou. 2005. Multivariate Poisson regression with covariance structure. Statistics and Computing 15: 255–265.
Karlis, D., and L. Meligkotsidou. 2007. Finite mixtures of multivariate Poisson regression with application. Journal of Statistical Planning and Inference 137: 1942–1960.
Kedem, B., and K. Fokianos. 2002. Regression models for time series analysis. Hoboken: Wiley.
Lambert, D. 1992. Zero-inflated Poisson regression, with an application to defects in manufacturing. Technometrics 34(1): 1–13.
Landim, F., and D. Gamerman. 2006. Dynamic hierarchical models: an extension to matrix-variate observations. Computational Statistics and Data Analysis 35: 11–42.
Ma, J., K. Kockelman, and P. Damien. 2008. A multivariate Poisson-lognormal regression model for prediction of crash counts by severity, using Bayesian methods. Accident Analysis and Prevention 40(3): 964–975.
McCullagh, P., and J. Nelder. 1989. Generalized linear models, 2nd ed. London: Chapman and Hall.
Mizik, N., and R. Jacobson. 2004. Are physicians easy marks? Quantifying the effects of detailing and sampling on new prescriptions. Management Science 50(12): 1704–1715.
Montoya, R., O. Netzer, and K. Jedidi. 2010. Dynamic allocation of pharmaceutical detailing and sampling for long-term profitability. Marketing Science 29(5): 909–924.
Park, E., and D. Lord. 2007. Multivariate Poisson-lognormal models for jointly modeling crash frequency by severity. Transportation Research Record: Journal of the Transportation Research Board 2009(1): 1–6.
Ravishanker, N., V. Serhiyenko, and M. Willig. 2014. Hierarchical dynamic models for multivariate times series of counts. Statistics and Its Interface 7: 559–570.
Ravishanker, N., R. Venkatesan, and S. Hu 2015. Dynamic models for time series of counts with a marketing application. In Handbook of discrete-valued time series, ed. R.A. Davis, S.H. Holan, R.L. Lund, and N. Ravishanker, 1–10.
Rue, H., and L. Held. 2005. Gaussian Markov random fields theory and applications. New York: Chapman & Hall/CRC.
Rue, H., and S. Martino. 2007. Approximate Bayesian inference for hierarchical Gaussian Markov random fields models. Journal of Statistical Planning and Inference 137: 3177–3192.
Rue, H., S. Martino, and N. Chopin. 2009. Approximate Bayesian inference for latent Gaussian models by using Integrated Nested Laplace Approximations. Journal of the Royal Statistical Society Series B 71: 319–392.
Venkatesan, R., W. Reinartz, and N. Ravishanker. 2012. The role of attitudinal information in CLV-based customer management. MSI Working Paper Series, 12: 107.
Venkatesan, R., W. Reinartz, and N. Ravishanker, 2014. ZIP models for CLV based customer management using attitudinal and behavioral data. Technical Report, Department of Statistics, University of Connecticut.
West, M., and P. Harrison. 1989. Bayesian forecasting and dynamic models. New York: Springer.
Zeger, S. 1998. A regression model for time series of counts. Biometrika 75: 621–629.
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We thank the reviewers for their very helpful comments.
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Serhiyenko, V., Ravishanker, N., Venkatesan, R. (2015). Approximate Bayesian Estimation for Multivariate Count Time Series Models. In: Choudhary, P., Nagaraja, C., Ng, H. (eds) Ordered Data Analysis, Modeling and Health Research Methods. Springer Proceedings in Mathematics & Statistics, vol 149. Springer, Cham. https://doi.org/10.1007/978-3-319-25433-3_10
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DOI: https://doi.org/10.1007/978-3-319-25433-3_10
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