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Bayesian Analysis of Zero-Truncated Poisson Model: Application to the Self-Controlled Case-series Design

Received: 12 October 2020     Accepted: 28 October 2020     Published: 4 November 2020
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Abstract

A Bayesian Self-Controlled Case-Series (BSCCS) method is proposed and used to estimate the relative risk of an adverse drug event (ADE) given transient exposure to a drug or vaccine. Markov Chain Monte Carlo (MCMC) methods through WinBUGS are used to estimate parameters of the model given different settings and sample sizes. The method explores full posterior distribution for the model to obtain the relative risk estimates which at times is a challenge in likelihood analysis of complex models. Data was simulated for 10, 20 or 50 children aged between 365 and 730 days, and received their first dose of the measles, mumps, and rubella (MMR) vaccine within this follow-up period. Each child had the outcome event – viral-meningitis, in the follow-up period. Results of the data analysis indicated an increased risk of viral meningitis within 14-35 days post vaccination. Results of Bayesian approach are quite similar to the MLE risk estimates, assuming a non-informative prior. However, with more informative priors, BSCCS method produced better results with narrow credible intervals. For the real data, children aged 365 and 730 days, exposed to MMR vaccine, with viral meningitis (single exposure) were considered. While the frequentist approach estimated the incidence rate ratio (IRR) as IRR 12.037 (95% CI (3.002 - 48.259)), the Bayesian estimate was IRR 8.971 (95% CI 2.869 - 27.994). This is similar to the MLE results but with narrow credible intervals. In all cases, there is significantly higher risk of viral meningitis within 14-35 days post MMR vaccination. Results from the simulation study and real data revealed that the BSCCS model fitted better than the SCCS model.

Published in International Journal of Data Science and Analysis (Volume 6, Issue 6)
DOI 10.11648/j.ijdsa.20200606.12
Page(s) 170-182
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Zero-truncated Poisson Distribution, Case-series, Bayesian Self-controlled Case Series, MMR Vaccine, Viral Meningitis

References
[1] Barry, S. C. and T. J. O'Neill, The Bayesian Analysis of Truncated Regression Models. 1994.
[2] Johnson, N. L., A. W. Kemp, and S. Kotz, Univariate discrete distributions. Vol. 444. 2005: John Wiley & Sons.
[3] Hardin, J. W. and J. M. Hilbe, Regression models for count data from truncated distributions. The Stata Journal, 2015. 15 (1): p. 226–246.
[4] Whitaker, H. J., The self-controlled case series method. 2005.
[5] Whitaker, H. J., C. Paddy Farrington, B. Spiessens, and P. Musonda, Tutorial in biostatistics: the self-controlled case series method. Statistics in medicine, 2006. 25 (10): p. 1768–1797.
[6] David, F. N. and N. L. Johnson, The truncated poisson. Biometrics, 1952. 8 (4): p. 275–285.
[7] Tate, R. F. and R. L. Goen, Minimum variance unbiased estimation for the truncated Poisson distribution. The Annals of Mathematical Statistics, 1958: p. 755–765.
[8] Springael, L. and I. Van Nieuwenhuyse, on the sum of independent zero-truncated Poisson random variables. 2006.
[9] Shanker, R., F. Hagos, S. Sujatha, and Y. Abrehe, On zero-truncation of poisson and poisson-lindley distributions and their applications. Biometrics & Biostatistics International Journal, 2015. 2 (6): p. 1–14.
[10] Yahaya, A., N. Dibal, A. Mobolaji, and G. Adegoke, Obtaining parameter estimate from the truncated Poisson probability distribution. 2016.
[11] Lee, C. H., Imprecise Prior for Imprecise Inference on Poisson Sampling Model. 2014.
[12] Gelman, A., J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and D. B. Rubin, Bayesian Data Analysis, Third Edition. 2013: Taylor & Francis.
[13] Casella, G. and R. L. Berger, Statistical Inference. 2002: Duxbury Thomson Learning.
[14] Gamerman, D. and H. F. Lopes, Markov chain Monte Carlo: stochastic simulation for Bayesian inference. 2006: Chapman and Hall/CRC.
[15] Laksono, B. M., R. D. de Vries, S. McQuaid, W. P. Duprex, and R. L. de Swart, Measles Virus Host Invasion and Pathogenesis. Viruses, 2016. 8 (27483301): 210.
[16] Institute of Medicine, Adverse Effects of Vaccines: Evidence and Causality, ed. K. Stratton, et al. 2012: The National Academies Press.
[17] Stratton, K. R., C. J. Howe, and R. B. R. B. Johnston, Institute of Medicine; Measles and mumps vaccines, ed. K. R. Stratton, C. J. Howe, and R. B. R. B. Johnston. 1994: National Academy Press (US).
[18] World Health Organisation, Measles: Disease burden. 2020.
[19] Anderson, R. M., J. A. Crombie, and B. T. Grenfell, The epidemiology of mumps in the UK: a preliminary study of virus transmission, herd immunity and the potential impact of immunization. Epidemiology and infection, 1987. 99 (3609175): p. 65–84.
[20] Bennett, J. E., R. Dolin, and M. J. Blaser, Mandell, douglas, and bennett's principles and practice of infectious diseases: 2-volume set. Vol. 2. 2014: Elsevier Health Sciences.
[21] Mawson, A. R. and A. M. Croft, Rubella Virus Infection, the Congenital Rubella Syndrome, and the Link to Autism. International journal of environmental research and public health, 2019. 16 (31546693): 3543.
[22] Herrera, O. R., T. A. Thornton, R. A. Helms, and S. L. Foster, MMR Vaccine: When Is the Right Time for the Second Dose? The journal of pediatric pharmacology and therapeutics: JPPT: the official journal of PPAG, 2015. 20 (25964732): p. 144–148.
[23] Marlow, M. A., M. Marin, K. Moore, and M. Patel, CDC Guidance for Use of a Third Dose of MMR Vaccine During Mumps Outbreaks. Journal of Public Health Management and Practice, 2020. 26 (2).
[24] Su, S.-B., H.-L. Chang, Chen, and T. Kow, Current Status of Mumps Virus Infection: Epidemiology, Pathogenesis, and Vaccine. International journal of environmental research and public health, 2020. 17 (32150969): 1686.
[25] Furesz, J. and G. Contreras, Vaccine-related mumps meningitis–Canada. Canada diseases weekly report=Rapport hebdomadaire des maladies au Canada, 1990. 16: p. 253-4.
[26] Miller, E., M. Goldacre, S. Pugh, A. Colville, P. Farrington, A. Flower, J. Nash, L. MacFarlane, and R. Tettmar, Risk of aseptic meningitis after measles, mumps, and rubella vaccine in UK children. Lancet (London, England), 1993. 341: p. 979-82.
[27] Miller, E., P. Waight, C. P. Farrington, N. Andrews, J. Stowe, and B. Taylor, Idiopathic thrombocytopenic purpura and MMR vaccine. Archives of Disease in Childhood, 2001. 84 (3): p. 227–229.
[28] France, E. K., J. Glanz, S. Xu, S. Hambidge, K. Yamasaki, S. B. Black, M. Marcy, J. P. Mullooly, L. A. Jackson, J. Nordin, E. A. Belongia, K. Hohman, R. T. Chen, and R. Davis, Risk of Immune Thrombocytopenic Purpura After Measles-Mumps-Rubella Immunization in Children. Pediatrics, 2008. 121 (3): p. e687–e692.
[29] Center for Disease Control Prevention, Meningitis. 2020.
[30] Dubey, A. P. and S. Banerjee, Measles, mumps, rubella (MMR) vaccine. The Indian Journal of Pediatrics, 2003. 70 (7): p. 579–584.
[31] Weldeselassie, Y. G., H. Whitaker, and P. Farrington, SCCS: The Self-Controlled Case Series Method. 2019.
[32] Dourado, I., S. Cunha, M. d. G. Teixeira, C. P. Farrington, A. Melo, R. Lucena, and M. L. Barreto, Outbreak of Aseptic Meningitis associated with Mass Vaccination with a Urabe-containing Measles-Mumps-Rubella Vaccine: Implications for Immunization Programs. American Journal of Epidemiology, 2000. 151 (5): p. 524-530.
[33] Sugiura, A. and A. Yamada, Aseptic meningitis as a complication of mumps vaccination. Pediatr Infect Dis J, 1991. 10 (3): p. 209-13.
[34] Kamali Aghdam, M., M. Sadeghzadeh, S. Fakhimi, and K. Eftekhari, Evaluation of Aseptic Meningitis Following Measles-Mumps-Rubella Vaccine in Children Admitted due to Febrile Convulsion. International Journal of Pediatrics, 2018. 6 (8): p. 8147-8152.
[35] Park, T., M. Ki, and S. G. Yi, Statistical analysis of MMR vaccine adverse events on aseptic meningitis using the case cross-over design. Stat Med, 2004. 23 (12): p. 1871-83.
[36] Ki, M., T. Park, S. G. Yi, J. K. Oh, and B. Choi, Risk Analysis of Aseptic Meningitis after Measles-Mumps-Rubella Vaccination in Korean Children by Using a Case-Crossover Design. American Journal of Epidemiology, 2003. 157 (2): p. 158-165.
[37] Musonda, P., M. N. Hocine, H. J. Whitaker, and C. P. Farrington, Self-controlled case series analyses: Small-sample performance. Computational Statistics & Data Analysis, 2008. 52 (4): p. 1942-1957.
[38] Campos, L. F., D. Şentürk, Y. Chen, and D. V. Nguyen, Bias and estimation under misspecification of the risk period in self-controlled case series studies. Stat (International Statistical Institute), 2017. 6 (1): p. 373-389.
Cite This Article
  • APA Style

    Henry Athiany, Anthony Wanjoya, George Orwa, Samuel Mwalili. (2020). Bayesian Analysis of Zero-Truncated Poisson Model: Application to the Self-Controlled Case-series Design. International Journal of Data Science and Analysis, 6(6), 170-182. https://doi.org/10.11648/j.ijdsa.20200606.12

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    ACS Style

    Henry Athiany; Anthony Wanjoya; George Orwa; Samuel Mwalili. Bayesian Analysis of Zero-Truncated Poisson Model: Application to the Self-Controlled Case-series Design. Int. J. Data Sci. Anal. 2020, 6(6), 170-182. doi: 10.11648/j.ijdsa.20200606.12

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    AMA Style

    Henry Athiany, Anthony Wanjoya, George Orwa, Samuel Mwalili. Bayesian Analysis of Zero-Truncated Poisson Model: Application to the Self-Controlled Case-series Design. Int J Data Sci Anal. 2020;6(6):170-182. doi: 10.11648/j.ijdsa.20200606.12

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  • @article{10.11648/j.ijdsa.20200606.12,
      author = {Henry Athiany and Anthony Wanjoya and George Orwa and Samuel Mwalili},
      title = {Bayesian Analysis of Zero-Truncated Poisson Model: Application to the Self-Controlled Case-series Design},
      journal = {International Journal of Data Science and Analysis},
      volume = {6},
      number = {6},
      pages = {170-182},
      doi = {10.11648/j.ijdsa.20200606.12},
      url = {https://doi.org/10.11648/j.ijdsa.20200606.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijdsa.20200606.12},
      abstract = {A Bayesian Self-Controlled Case-Series (BSCCS) method is proposed and used to estimate the relative risk of an adverse drug event (ADE) given transient exposure to a drug or vaccine. Markov Chain Monte Carlo (MCMC) methods through WinBUGS are used to estimate parameters of the model given different settings and sample sizes. The method explores full posterior distribution for the model to obtain the relative risk estimates which at times is a challenge in likelihood analysis of complex models. Data was simulated for 10, 20 or 50 children aged between 365 and 730 days, and received their first dose of the measles, mumps, and rubella (MMR) vaccine within this follow-up period. Each child had the outcome event – viral-meningitis, in the follow-up period. Results of the data analysis indicated an increased risk of viral meningitis within 14-35 days post vaccination. Results of Bayesian approach are quite similar to the MLE risk estimates, assuming a non-informative prior. However, with more informative priors, BSCCS method produced better results with narrow credible intervals. For the real data, children aged 365 and 730 days, exposed to MMR vaccine, with viral meningitis (single exposure) were considered. While the frequentist approach estimated the incidence rate ratio (IRR) as IRR 12.037 (95% CI (3.002 - 48.259)), the Bayesian estimate was IRR 8.971 (95% CI 2.869 - 27.994). This is similar to the MLE results but with narrow credible intervals. In all cases, there is significantly higher risk of viral meningitis within 14-35 days post MMR vaccination. Results from the simulation study and real data revealed that the BSCCS model fitted better than the SCCS model.},
     year = {2020}
    }
    

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  • TY  - JOUR
    T1  - Bayesian Analysis of Zero-Truncated Poisson Model: Application to the Self-Controlled Case-series Design
    AU  - Henry Athiany
    AU  - Anthony Wanjoya
    AU  - George Orwa
    AU  - Samuel Mwalili
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    JF  - International Journal of Data Science and Analysis
    JO  - International Journal of Data Science and Analysis
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    PB  - Science Publishing Group
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    UR  - https://doi.org/10.11648/j.ijdsa.20200606.12
    AB  - A Bayesian Self-Controlled Case-Series (BSCCS) method is proposed and used to estimate the relative risk of an adverse drug event (ADE) given transient exposure to a drug or vaccine. Markov Chain Monte Carlo (MCMC) methods through WinBUGS are used to estimate parameters of the model given different settings and sample sizes. The method explores full posterior distribution for the model to obtain the relative risk estimates which at times is a challenge in likelihood analysis of complex models. Data was simulated for 10, 20 or 50 children aged between 365 and 730 days, and received their first dose of the measles, mumps, and rubella (MMR) vaccine within this follow-up period. Each child had the outcome event – viral-meningitis, in the follow-up period. Results of the data analysis indicated an increased risk of viral meningitis within 14-35 days post vaccination. Results of Bayesian approach are quite similar to the MLE risk estimates, assuming a non-informative prior. However, with more informative priors, BSCCS method produced better results with narrow credible intervals. For the real data, children aged 365 and 730 days, exposed to MMR vaccine, with viral meningitis (single exposure) were considered. While the frequentist approach estimated the incidence rate ratio (IRR) as IRR 12.037 (95% CI (3.002 - 48.259)), the Bayesian estimate was IRR 8.971 (95% CI 2.869 - 27.994). This is similar to the MLE results but with narrow credible intervals. In all cases, there is significantly higher risk of viral meningitis within 14-35 days post MMR vaccination. Results from the simulation study and real data revealed that the BSCCS model fitted better than the SCCS model.
    VL  - 6
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Author Information
  • Department of Statistics and Actuarial Sciences, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

  • Department of Statistics and Actuarial Sciences, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

  • Department of Statistics and Actuarial Sciences, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

  • Department of Statistics and Actuarial Sciences, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

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