The beta-binomial distribution is resulted when the probability of success per trial in the binomial distribution varies in successive trials and the mixing distribution is from the beta family. For experiments with binary outcomes, often it may happen that observations exhibit some extra binomial variation and occur in clusters. In such experiments the beta-binomial distribution can generally provide an adequate fit to the data. Here, we introduce an alternative when the mixing distribution is assumed to be from the log-Lindley family. The properties of this new model are explored and it is shown that similar to the beta-binomial distribution, the log-Lindley binomial distribution can also be applied in modeling clustered binary outcomes. An example with real experimental data from a developmental toxicity experiment is utilized to provide further illustration.

Type of Study: Special Issue, Original Paper |
Subject:
62Exx: Distribution theory

Received: 2020/12/4 | Accepted: 2021/02/7 | Published: 2021/06/20

Received: 2020/12/4 | Accepted: 2021/02/7 | Published: 2021/06/20

1. Berson, E. L., Rosner, B., and Simonoff, E. (1980), An outpatient population of retinitis pigmentosa and their normal relatives: risk factors for genetic typing and detection derived from their ocular examination. American Journal of Ophthalmology, 89, 763-775. [DOI:10.1016/0002-9394(80)90163-4]

2. Chen, J. J., and Kodell, R. L. (1989), Quantitative risk assessment for teratological effects. Journal of the American Statistical Association, 84, 966-971. [DOI:10.1080/01621459.1989.10478860]

3. Gomez-Deniz, E., Sordo, M. A., and Calderin-Ojeda, E. (2014), The log-Lindley distribution as an alternative to the beta regression model with applications in insurance. Insurance: Mathematics and Economics, 54, 49-57. [DOI:10.1016/j.insmatheco.2013.10.017]

4. Grassia, A. (1977). On a family of distributions with argument between 0 and 1 obtained by transformation of the gamma and derived compound distributions. Australian Journal of Statistics, 19, 108-114. [DOI:10.1111/j.1467-842X.1977.tb01277.x]

5. Jorda, P., and Jimenez-Gamero, M. D. (2016). A note on the log-Lindley distribution. Insurance: Mathematics and Economics, 71, 189-194.

6. Lindley, D. V. (1958). Fiducial distribution and Bayes' theorem. Journal of the Royal Statristical Society, Series B, 20, 102-107. [DOI:10.1111/j.2517-6161.1958.tb00278.x]

7. Lindley, D. V. (1965). Introduction to Probability and Statistics from a Bayesian Viewpoint, Part II: Inference. Cambridge University Press, New York.

8. Nocedal, J., and Wright, S. J (1999). Numerical Optimization. Springer, New York. [DOI:10.1007/b98874]

9. R Core Team (2012). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http://www.R-project.org/

10. Razzaghi, M. (2020). Statistical Models in Toxicology. CRC Press, Boca Raton, FL. [DOI:10.1201/9780429155185]

11. Rosner, B. (1982). Statistical methods in ophthalmology: an adjustment for the intra-class correlation between eyes. Biometrics, 38, 105-114. [DOI:10.2307/2530293]

12. Smith, D. M. (1983). Maximum likelihood estimation of the parameters of the beta-binomial distribution. Applied Statistics, 32, 192-204. [DOI:10.2307/2347299]

13. Statistical Analysis System SAS (2014). Version 9.4. Cary, NC: SAS Institute Inc.

14. Wilcox, R. E. (1981), A review of the beta-binomial model and its extensions. Journal of Educational Statistics, 6, 3-32. [DOI:10.3102/10769986006001003]

15. Williams, D. A. (1975), The analysis of binary responses from teratological experiments involving reproduction and teratogenicity. Biometrics, 31, 949-952. [DOI:10.2307/2529820]

16. Zakerzadeh, H., and Dolati, A. (2009), Generalized Lindley distribution. Journal of mathematical Extension, 3, 13-25.

Rights and permissions | |

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. |