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Licensed Unlicensed Requires Authentication Published by De Gruyter April 18, 2019

Double Poisson-Tweedie Regression Models

  • Ricardo R. Petterle , Wagner H. Bonat ORCID logo EMAIL logo , Célestin C. Kokonendji , Juliane C. Seganfredo , Atamai Moraes and Monica G. da Silva


In this paper, we further extend the recently proposed Poisson-Tweedie regression models to include a linear predictor for the dispersion as well as for the expectation of the count response variable. The family of the considered models is specified using only second-moments assumptions, where the variance of the count response has the form μ+ϕμp, where µ is the expectation, ϕ and p are the dispersion and power parameters, respectively. Parameter estimations are carried out using an estimating function approach obtained by combining the quasi-score and Pearson estimating functions. The performance of the fitting algorithm is investigated through simulation studies. The results showed that our estimating function approach provides consistent estimators for both mean and dispersion parameters. The class of models is motivated by a data set concerning CD4 counting in HIV-positive pregnant women assisted in a public hospital in Curitiba, Paraná, Brazil. Specifically, we investigate the effects of a set of covariates in both expectation and dispersion structures. Our results showed that women living out of the capital Curitiba, with viral load equal or larger than 1000 copies and with previous diagnostic of HIV infection, present lower levels of CD4 cell count. Furthermore, we detected that the time to initiate the antiretroviral therapy decreases the data dispersion. The data set and R code are available as supplementary materials.


[1] WHO. Consolidated guidelines on the use of antiretroviral drugs for treating and preventing HIV infection: recommendations for a public health approach. Technical report. World Health Organization, 2016.Search in Google Scholar

[2] UNAIDS, J U. N. P. oH. On the Fast-Track to end AIDS by 2030: Focus on location and population. Technical report. Joint United Nations Programme on HIV/AIDS, 2015.Search in Google Scholar

[3] Landefeld CC, Fomenou LA, Ateba F, Msellati P. Prevention of mother-to-child transmission of HIV in Yaounde: Barrier to care. AIDS care. 2018;30:116–20.10.1080/09540121.2017.1390540Search in Google Scholar PubMed

[4] French CE, Thorne C, Byrne L, Cortina-Borja M, Tookey PA. Presentation for care and antenatal management of HIV in the UK 2009–2014. HIV Med. 2017;18:161–70.10.1111/hiv.12410Search in Google Scholar PubMed PubMed Central

[5] Grover G, Vajala R, Swain PK. On the assessment of various factors effecting the improvement in CD4 count of aids patients undergoing antiretroviral therapy using generalized poisson regression. J Appl Stat. 2015;42:1291–305.10.1080/02664763.2014.999649Search in Google Scholar

[6] Lok JJ, Bosch RJ, Benson CA, Collier AC, Robbins GK, Shafer RW, et al. Long-term increase in CD4+ T-cell counts during combination antiretroviral therapy for HIV-1 infection. AIDS (London, England). 2010;24:1867–76.10.1097/QAD.0b013e32833adbcfSearch in Google Scholar PubMed PubMed Central

[7] Seyoum A, Zewotir T. Quasi-Poisson versus negative binomial regression models in identifying factors affecting initial CD4 cell count change due to antiretroviral therapy administered to HIV-positive adults in North–West Ethiopia. AIDS Res Ther. 2016;13:2–10.10.1186/s12981-016-0119-6Search in Google Scholar PubMed PubMed Central

[8] Helleberg M, Kronborg G, Larsen CS, Pedersen G, Pedersen C, Obel N, et al. CD4 decline is associated with increased risk of cardiovascular disease, cancer, and death in virally suppressed patients with HIV. Clin Infec Dis. 2013;57:314–21.10.1093/cid/cit232Search in Google Scholar PubMed

[9] Cameron AC, Trivedi PK.. Regression analysis of count data, vol. 53 Cambridge: Cambridge University Press, 201310.1017/CBO9781139013567Search in Google Scholar

[10] Zeviani WM, Ribeiro Jr. PJ, Bonat WH, Shimakura SE, Muniz JA. The Gamma-count distribution in the analysis of experimental underdispersed data. J Appl Stat. 2014;41:2616–26.10.1080/02664763.2014.922168Search in Google Scholar

[11] Bonat WH, Jørgensen B, Kokonendji CC, Hinde J, Demétrio CG. Extended Poisson-Tweedie: properties and regression models for count data. Stat Modell. 2018;18:24–49.10.1177/1471082X17715718Search in Google Scholar

[12] El-Shaarawi AH, Zhu R, Joe H. Modelling species abundance using the Poisson-Tweedie family. Environmetrics. 2011;22:152–64.10.1002/env.1036Search in Google Scholar

[13] Hinde J, Demétrio CG. Overdispersion: models and estimation. Comput Stat Data Anal. 1998;27:151–70.10.1016/S0167-9473(98)00007-3Search in Google Scholar

[14] Kokonendji CC, Demétrio CG, Zocchi SS. On Hinde–Demétrio regression models for overdispersed count data. Stat Method. 2007;4:277–91.10.1016/j.stamet.2006.10.001Search in Google Scholar

[15] Mahmoodi M, Moghimbeigi A, Mohammad K, Faradmal J. Semiparametric models for multilevel overdispersed count data with extra zeros. Stat Method Med Res. 2016;27:1187–201.10.1177/0962280216657376Search in Google Scholar PubMed

[16] Oliveira M, Einbeck J, Higueras M, Ainsbury E, Puig P, Rothkamm K. Zero-inflated regression models for radiation-induced chromosome aberration data: a comparative study. Biometric J. 2016;58:259–79.10.1002/bimj.201400233Search in Google Scholar PubMed

[17] Rigby RA, Stasinopoulos DM, Akantziliotou C. A framework for modelling overdispersed count data, including the Poisson-shifted generalized inverse Gaussian distribution. Comput Stat Data Anal. 2008;53:381–93.10.1016/j.csda.2008.07.043Search in Google Scholar

[18] Sellers KF, Shmueli G. A flexible regression model for count data. Annals Appl Stat. 2010;4:943–61.Search in Google Scholar

[19] Smyth GK. Generalized linear models with varying dispersion. J R Stat Soc Ser B Method. 1989;51:47–60.10.1111/j.2517-6161.1989.tb01747.xSearch in Google Scholar

[20] Andersen DA, Bonat WH. Double generalized linear compound Poisson models to insurance claims data. Electron J Appl Stat Anal. 2017;10:384–407.Search in Google Scholar

[21] Bonat WH, Jørgensen B. Multivariate covariance generalized linear models. J R Stat Soc: Ser C (Appl Stat). 2016;65:649–75.10.1111/rssc.12145Search in Google Scholar

[22] Wedderburn RWM. Quasi-likelihood functions, generalized linear models, and the gauss–newton method. Biometrika. 1974;61:439–47.Search in Google Scholar

[23] Esnaola M, Puig P, Gonzalez D, Castelo R, Gonzalez JR. A flexible count data model to fit the wide diversity of expression profiles arising from extensively replicated rna-seq experiments. BMC Bioinf. 2013;14:254–76.10.1186/1471-2105-14-254Search in Google Scholar PubMed PubMed Central

[24] Kokonendji CC, Dossou-Gbété S, Demétrio CG. Some discrete exponential dispersion models: Poisson-Tweedie and Hinde-Demétrio classes. Stat Oper Res Trans. 2004;28:201–14.Search in Google Scholar

[25] Moria D, Higueras M, Puig P, Oliveira M. hermite: generalized Hermite distribution., r package version 1.1.1. 2015.Search in Google Scholar

[26] Jørgensen B, Kokonendji CC. Discrete dispersion models and their tweedie asymptotics. AStA Adv Stat Anal. 2016;100:43–78.10.1007/s10182-015-0250-zSearch in Google Scholar

[27] Cox DR, Hinkley DV. Theoretical statistics. London, England: Chapman & Hall, 1974.10.1007/978-1-4899-2887-0Search in Google Scholar

[28] Jørgensen B, Knudsen SJ. Parameter orthogonality and bias adjustment for estimating functions. Scand J Stat. 2004;31:93–114.10.1111/j.1467-9469.2004.00375.xSearch in Google Scholar

[29] Bonat WH. Multiple response regression models in R: the mcglm package. J Stat Software. 2018;85:1–30.10.18637/jss.v084.i04Search in Google Scholar

[30] Yu T, Wu L. Robust modelling of the relationship between CD4 and viral load for complex AIDS data. J Appl Stat. 2018;45:367–83.10.1080/02664763.2017.1279594Search in Google Scholar

[31] Liang K-Y, Zeger SL. Longitudinal data analysis using generalized linear models. Biometrika. 1986;73:13–22.10.1093/biomet/73.1.13Search in Google Scholar

[32] Bonat WH, Olivero J, Grande-Vega M, Farfán MA, Fa JE. Modelling the covariance structure in marginal multivariate count models: hunting in bioko island. J Agr Biol Environ Stat. 2017;22:446–64.10.1007/s13253-017-0284-7Search in Google Scholar

Received: 2018-11-14
Revised: 2019-03-18
Accepted: 2019-04-02
Published Online: 2019-04-18

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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