How to cite this WIREs title:
WIREs Comp Stat

Zero‐inflated modeling part II: Zero‐inflated models for complex data structures

Advanced Review

Derek S. Young, Eric S. Roemmele, Xuan Shi

Published Online: Dec 18 2020

DOI: 10.1002/wics.1540

Full article on Wiley Online Library: HTML PDF

Can't access this content? Tell your librarian.

Abstract The prequel to this review provided an extensive treatment of classic zero‐inflated count regression models where a univariate discrete distribution is used for the count regression component of the model. The treatment of zero inflation beyond the classic univariate count regression setting has seen a substantial increase in recent years. This second review paper surveys some of this recent literature and focuses on important developments in handling zero inflation for correlated count settings, discrete time series models, spatial models, and multivariate models. We discuss some of the available computational tools for performing estimation in these settings, while again highlighting the diverse data problems that have been addressed using these methods. This article is categorized under: Statistical Models > Multivariate Models Statistical Models > Generalized Linear Models Statistical and Graphical Methods of Data Analysis > Bayesian Methods and Theory

Coefficient estimates and confidence intervals for the fixed effects from the eight model fits to the owl nestlings data

[ Normal View | Magnified View ]

Estimated random effects for the 27 nest sites for the ZIGP regression fit. The red horizontal line is the average of these 27 random nest location effects, which is at 0.012

[ Normal View | Magnified View ]

References

Agarwal,, D. K., Gelfand,, A. E., & Citron‐Pousty,, S. (2002). Zero‐inflated models with application to spatial count data. Environmental and Ecological Statistics, 9, 409–426.
Alqawba,, M., & Diawara,, N. (2020). Copula‐based Markov zero‐inflated count time series models with application. Journal of Applied Statistics (in press). https://doi.org/10.1080/02664763.2020.1748581.
Alqawba,, M., Diawara,, N., & Chaganty,, N. R. (2019). Zero‐inflated count time series models using Gaussian copula. Sequential Analysis: Design Methods and Applications, 38, 342–357.
Arab,, A. (2015). Spatial and spatio‐temporal models for modeling epidemiological data with excess Zeros. International Journal of Environmental Research and Public Health, 12, 10536–10548.
Arab,, A., Holan,, S. H., Wikle,, C. K., & Wildhaber,, M. L. (2012). Semiparametric bivariate zero‐inflated Poisson models with application to studies of abundance for multiple species. Environmetrics, 23, 183–196.
Besag,, J. (1974). Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society, Series B, 36, 192–236.
Bolker,, B., Gardner,, B., Maunder,, M., Berg,, C., Brooks,, M., Comita,, L., … Zipkin,, E. (2013). Strategies for fitting nonlinear ecological models in R, AD Model Builder, and BUGS. Methods in Ecology and Evolution, 4, 501–512.
Brooks,, M. E., Kristensen,, K., van Benthem,, K. J., Magnusson,, A., Berg,, C. W., Nielsen,, A., … Bolker,, B. M. (2017). glmmTMB balances speed and flexibility among packages for zero‐inflated generalized linear mixed modeling. The R Journal, 9, 378–400 https://journal.r-project.org/archive/2017/RJ-2017-066/index.html
Burger,, D. A., Schall,, R., Ferreira,, J. T., & Chen,, D.‐G. (2020). A robust Bayesian mixed effects approach for zero inflated and highly skewed longitudinal count data emanating from the zero inflated discrete Weibull distribution. Statistics in Medicine, 39, 1275–1291.
Bürkner,, P.‐C. (2017). brms: An R package for Bayesian multilevel models using Stan. Journal of Statistical Software, 80, 1–28 http://www.jstatsoft.org/v80/i01/
Cressie,, N. A. (1993). Statistics for spatial data. Hoboken, NJ: John Wiley %26 Sons, Inc.
Czado,, C., & Min,, A. (2005). Consistency and asymptotic normality of the maximum likelihood estimator in a zero‐inflated generalized Poisson regression. Tech. rep., Discussion Paper No. 423, Sonderforschungsbereich 386 der Ludwig‐Maximilians‐Universität München. http://nbn-resolving.de/urn:nbn:de:bvb:19-epub-1792-8
Dagne,, G. A. (2004). Hierarchical Bayesian analysis of correlated zero‐inflated count data. Biometrical Journal, 46, 653–663.
Dempster,, A. P., Laird,, N. M., & Rubin,, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B (Statistical Methodology), 39, 1–38.
Diaz,, J., & Joseph,, M. B. (2019). Predicting property damage from Tornadoes with zero‐inflated neural networks. Weather and Climate Extremes, 25, 1–13.
Diggle,, P. J., Tawn,, J. A., & Moyeed,, R. A. (1998). Model‐based geostatistics (with discussion). Journal of the Royal Statistical Society, Series C, 47, 299–350.
Famoye,, F., & Singh,, K. P. (2006). Zero‐inflated generalized Poisson regression model with an application to domestic violence data. Journal of Data Science, 4, 117–130.
Fan,, J., Samworth,, R., & Wu,, Y. (2009). Ultrahigh dimensional feature selection: Beyond the linear model. Journal of Machine Learning Research, 10, 2013–2038.
Fang,, R., Wagner,, B. D., Harris,, J. K., & Fillon,, S. A. (2016). Zero‐inflated negative binomial mixed model: An application to two microbial organisms important in oesophagitis. Epidemiology and Infection, 144, 2447–2455.
Faroughi,, P., & Ismail,, N. (2017). Bivariate zero‐inflated generalized Poisson regression model with flexible covariance. Communications in Statistics – Theory and Methods, 46, 7769–7785.
Feng,, J., & Zhu,, Z. (2011). Semiparametric analysis of longitudinal zero‐inflated count data. Journal of Multivariate Analysis, 102, 61–72.
Fernandes,, M. V. M., Schmidt,, A. M., & Migon,, H. S. (2009). Modelling zero‐inflated Spatio‐temporal processes. Statistical Modelling, 9, 3–25.
Gonçalves,, E., Mendes‐Lopes,, N., & Silva,, F. (2016). Zero‐inflated compound Poisson distributions in integer‐valued GARCH models. Statistics: A Journal of Theoretical and Applied Statistics, 50, 558–578.
Greene,, W. H. (1994) Accounting for excess Zeros and sample selection in Poisson and negative binomial regression models (Working Paper No. EC‐94‐10). NYU. http://ssrn.com/paper=990012
Gschlößl,, S., & Czado,, C. (2008). Modelling count data with overdispersion and spatial effects. Statistical Papers, 49, 531–552.
Guo,, J. Q., & Li,, T. (2002). Poisson regression models with errors‐in‐variables: Implications and treatment. Journal of Statistical Planning and Inference, 104, 391–401.
Hadfield,, J. D. (2010). MCMC methods for multi‐response generalized linear mixed models: The MCMCglmm R package. Journal of Statistical Software, 33, 1–22 http://www.jstatsoft.org/v33/i02/
Haghani,, S., Sedehi,, M., & Kheiri,, S. (2017). Artificial neural network to modeling zero‐inflated count data: Application to predicting number of return to blood donation. Journal of Research in Health Sciences, 17, 1–4.
Hall,, D. B. (2000). Zero‐inflated Poisson and binomial regression with random effects: A case study. Biometrics, 56, 1030–1039.
Hall,, D. B., & Zhang,, Z. (2004). Marginal models for zero inflated clustered data. Statistical Modelling, 4, 161–180.
Hedeker,, D., & Gibbons,, R. D. (2006). Longitudinal data analysis. Hoboken, NJ: Wiley.
Hilbe,, J. M. (2011). Negative binomial regression (2nd ed.). Cambridge, England: Cambridge University Press.
Huang,, X.‐F., Tian,, G.‐L., Zhang,, C., & Jiang,, X. (2015). Type I multivariate zero‐inflated generalized Poisson distribution with applications. Statistics and its Interface, 10, 291–311.
Iddi,, S., & Molenberghs,, G. (2013). A marginalized model for zero‐inflated, overdispersed and correlated count data. Electronic Journal of Applied Statistical Analysis, 6, 149–165.
Joe,, H. (2014). Dependence modeling with copulas. Boca Raton, FL: CRC Press.
Johnson,, N. L., & Kotz,, S. (1969). Distributions in statistics: Discrete distributions. Boston, MA: Houghton Mifflin.
Jørgensen,, B. (1987). Exponential dispersion models (with discussion). Journal of the Royal Statistical Society, Series B (Statistical Methodology), 49, 127–162.
Kalktawi,, H. S. (2017). Discrete Weibull regression model for count data (Ph.D. thesis). Brunel University London.
Karlis,, D., & Meligkotsidou,, L. (2005). Multivariate Poisson regression with covariance structure. Statistics and Computing, 15, 255–265.
Karlis,, D., & Ntzoufras,, I. (2003). Analysis of sports data by using bivariate Poisson models. Journal of the Royal Statistical Society, Series D (The Statistician), 52, 381–393.
Karlis,, D., & Ntzoufras,, I. (2005). Bivariate Poisson and diagonal inflated bivariate Poisson regression models in R. Journal of Statistical Software, 14, 1–36.
Kong,, M., Xua,, S., Levy,, S. M., & Datta,, S. (2015). GEE type inference for clustered zero‐inflated negative binomial regression with application to dental caries. Computational Statistics %26 Data Analysis, 85, 54–66.
Lam,, K. F., Xue,, H., & Cheung,, Y. B. (2006). Semiparametric analysis of zero‐inflated count data. Biometrics, 62, 996–1003.
Lambert,, D. (1992). Zero‐inflated Poisson regression, with an application to defects in manufacturing. Technometrics, 34, 1–14.
Lee,, D. (2013). CARBayes: An R package for Bayesian spatial modeling with conditional autoregressive priors. Journal of Statistical Software, 55, 1–24 http://www.jstatsoft.org/v55/i13/
Lee,, K. K. W., & Lee,, A. H. (2001). Zero‐inflated Poisson regression with random effects to evaluate an occupational injury prevention programme. Statistics in Medicine, 20, 2907–2920.
Lee,, S.‐K., & Jin,, S. (2006). Decision tree approaches for zero‐inflated count data. Journal of Applied Statistics, 33, 853–865.
Li,, C.‐S., Lu,, J.‐C., Park,, J., Kim,, K., Brinkley,, P. A., & Peterson,, J. P. (1999). Multivariate zero‐inflated Poisson models and their applications. Technometrics, 41, 29–38.
Liang,, K.‐Y., Zeger,, S. L., & Qaqish,, B. (1992). Multivariate regression analyses for categorical data. Journal of the Royal Statistical Society, Series B (Statistical Methodology), 54, 3–40.
Little,, R. J. A., & Rubin,, D. B. (2002). Statistical analysis with missing data (2nd ed.). Hoboken, NJ: Wiley.
Liu,, Y., & Tian,, G.‐L. (2015). Type I multivariate zero‐inflated Poisson distribution with applications. Computational Statistics and Data Analysis, 83, 200–222.
Lunn,, D. J., Thomas,, A., Best,, N., & Spiegelhalter,, D. (2000). WinBUGS – A Bayesian modelling framework: Concepts, structure, and extensibility. Statistics and Computing, 10, 325–337.
Musenge,, E., Chirwa,, T. F., Kahn,, K., & Vounatsou,, P. (2013). Bayesian analysis of zero inflated spatiotemporal HIV/TB child mortality data through the INLA and SPDE approaches: Applied to data observed between 1992 and 2010 in rural North East South Africa. International Journal of Applied Earth Observation and Geoinformation, 22, 86–98.
Musgrove,, D., Young,, D. S., Hughes,, J., & Eberly,, L. E. (2018). A sparse areal mixed model for multivariate outcomes, with an application to zero‐inflated census data. In N. Diawara, (Ed.), Modern statistical methods for spatial and multivariate data (pp. 51–74). Cham, Switzerland: Springer.
Neelon,, B., Chang,, H. H., Liang,, Q., & Hastings,, N. S. (2016). Spatiotemporal hurdle models for zero‐inflated count data: Exploring trends in emergency department visits. Statistical Methods in Medical Research, 25, 2558–2576.
Neelon,, B., Ghosh,, P., & Loebs,, P. F. (2013). A spatial Poisson hurdle model for exploring geographic variation in emergency department visits. Journal of the Royal Statistical Society, Series A (Statistics in Society), 176, 389–413.
Neelon,, B. H. (2019). Bayesian zero‐inflated negative binomial regression based on Pólya‐gamma mixtures. Bayesian Analysis, 14, 829–855.
Neelon,, B. H., O`Malley,, A. J., & Normand,, S.‐L. T. (2010). A Bayesian model for repeated measures zero‐inflated count data with application to outpatient psychiatric service use. Statistical Modelling, 10, 421–439.
R Core Team. (2016). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing https://www.R-project.org/
Rathbun,, S. L., & Fei,, S. (2006). A spatial zero‐inflated Poisson regression model for oak regeneration. Environmental and Ecological Statistics, 13, 409–426.
Recta,, V., Haran,, M., & Rosenberger,, J. L. (2012). A two‐stage model for incidence and prevalence in point‐level spatial count data. Environmetrics, 23, 162–174.
Roulin,, A., & Bersier,, L.‐F. (2007). Nestling barn owls beg more intensely in the presence of their mother than in the presence of their father. Animal Behaviour, 74, 1099–1106.
Rue,, H., Martino,, S., & Chopin,, N. (2009). Approximate Bayesian inference for latent Gaussian models using integrated nested Laplace approximations (with discussion). Journal of the Royal Statistical Society, Series B (Statistical Methodology), 71, 319–392.
Sarvi,, F., Moghimbeigi,, A., & Mahjub,, H. (2019). GEE‐based zero‐inflated generalized Poisson model for clustered over or under‐dispersed count data. Journal of Statistical Computation and Simulation, 89, 2711–2732.
SAS Institute Inc. (2013). SAS/STAT® 9.4 user`s guide. Cary, NC: SAS Institute Inc.
Sellers,, K. F., & Raim,, A. (2016). A flexible zero‐inflated model to address data dispersion. Computational Statistics and Data Analysis, 99, 68–80.
Skaug,, H., Fournier,, D., Nielsen,, A., Magnusson,, A., & Bolker,, B. M. (2012). glmmADMB: Generalized linear mixed models using AD model builder. https://glmmadmb.r-forge.r-project.org/
Stasinopoulos,, D. M., & Rigby,, R. A. (2007). Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, 23, 1–46 http://www.jstatsoft.org/v23/i07/
Stroup,, W. W. (2013). Generalized linear mixed models – Modern concepts, methods and applications. Boca Raton, FL: CRC Press.
Ver Hoef,, J. M., & Jansen,, J. K. (2007). Space–time zero‐inflated count models of harbor seals. Environmetrics, 18, 697–712.
Wahlin,, J. F. (2001). Bivariate ZIP Models. Biometrical Journal, 43, 147–160.
Wang,, K., Lee,, A. H., Yau,, K. K. W., & Carrivick,, P. J. W. (2003). A bivariate zero‐inflated Poisson regression model for analyzing occupational injuries. Accident Analysis and Prevention, 35, 625–629.
Wang,, K., Yau,, K. K. W., & Lee,, A. H. (2002). A zero‐inflated Poisson mixed model to analyze diagnosis related groups with majority of same‐day hospital stays. Computer Methods and Programs in Biomedicine, 68, 195–203.
Wang,, P. (2003). A bivariate zero‐inflated negative binomial regression model for count data with excess zeros. Economics Letters, 78, 373–378.
Wang,, X., Chen,, M.‐H., Kuo,, R. C., & Dey,, D. K. (2015). Bayesian spatial–temporal modeling of ecological zero‐inflated count data. Statistica Sinica, 25, 189–204.
Wikle,, C. K., & Anderson,, C. J. (2003). Climatological analysis of tornado report counts using a hierarchical Bayesian spatiotemporal model. Journal of Geophysical Research, 108, 1–15.
Wood,, S. (2018). mgcv: Mixed GAM computation vehicle with automatic smoothness estimation. R package version 1.8‐25.
Yang,, H., Li,, R., Zucker,, R. A., & Buu,, A. (2016). Two‐stage model for time varying effects of zero‐inflated count longitudinal covariates with applications in health behaviour research. Journal of the Royal Statistical Society, Series C, 65, 431–444.
Yang,, M., Cavanaugh,, J. E., & Zamba,, G. K. D. (2015). State‐space models for count time series with excess zeros. Statistical Modelling, 15, 70–90.
Yang,, M., Zamba,, G. K. D., & Cavanaugh,, J. E. (2013). Markov regression models for count time series with excess zeros: A partial likelihood approach. Statistical Methodology, 14, 26–38.
Yang,, M., Zamba,, G. K. D., & Cavanaugh,, J. E. (2017). ZIM: Zero‐inflated models for count time series with excess zeros. R package version 1.0.3. https://CRAN.R-project.org/package=ZIM
Yau,, K. K. W., Lee,, A. H., & Carrivick,, P. J. W. (2004). Modeling zero‐inflated count series with application to occupational health. Computer Methods and Programs in Biomedicine, 74, 47–52.
Ye,, P., Tang,, W., He,, J., & He,, H. (2019). A GEE‐type approach to untangle structural and random zeros in predictors. Statistical Methods in Medical Research, 28, 3683–3696.
Young,, D. S., Raim,, A. M., & Johnson,, N. R. (2017). Zero‐inflated modelling for characterizing coverage errors of extracts from the US Census Bureau`s master address file. Journal of the Royal Statistical Society: Series A (Statistics in Society), 180, 73–97.
Zellner,, A. (1986). On assessing prior distributions and Bayesian regression analysis with g‐prior distributions. In P. Goel, & A. Zellner, (Eds.), Bayesian inference and decision techniques: Essays in honor of Bruno de Finetti (pp. 233–243). New York, NY: North Holland Publishing Company.
Zhang,, S., Midthune,, D., Guenther,, P. M., Krebs‐Smith,, S. M., Kipnis,, V., Dodd,, K. W., … Carroll,, R. J. (2011). A new multivariate measurement error model with zero‐inflated dietary data, and its application to dietary assessment. The Annals of Applied Statistics, 5, 1456–1487.
Zhang,, W., Wang,, J., Qian,, F., & Chen,, Y. (2020). A joint mean‐correlation modeling approach for longitudinal zero‐inflated count data. Brazilian Journal of Probability and Statistics, 34, 35–50.
Zhao,, Y., Lee,, A. H., Burke,, V., & Yau,, K. K. W. (2009). Testing for zero‐inflation in count series: Application to occupational health. Journal of Applied Statistics, 36, 1353–1359.
Zhou,, H., Li,, L., & Zhu,, H. (2013). Tensor regression with applications to neuroimaging data analysis. Journal of the American Statistical Association, 108, 540–552.
Zhu,, F. (2012). Zero‐inflated Poisson and negative binomial integer‐valued GARCH models. Journal of Statistical Planning and Inference, 142, 826–839.
Zuur,, A. F., Ieno,, E. N., Walker,, N., Saveliev,, A. A., & Smith,, G. M. (2009). Mixed effects models and extensions in ecology with R. New York, NY: Springer.

Access to this WIREs title is by subscription only.

Recommend to Your
Librarian Now!

Zero‐inflated modeling part II: Zero‐inflated models for complex data structures

Browse by Topic

Access to this WIREs title is by subscription only.

The latest WIREs articles in your inbox