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Determining the distribution of fitness effects using a generalized Beta-Burr distribution

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  • Joyce, Paul
  • Abdo, Zaid

Abstract

In Beisel et al. (2007), a likelihood framework, based on extreme value theory (EVT), was developed for determining the distribution of fitness effects for adaptive mutations. In this paper we extend this framework beyond the extreme distributions and develop a likelihood framework for testing whether or not extreme value theory applies. By making two simple adjustments to the Generalized Pareto Distribution (GPD) we introduce a new simple five parameter probability density function that incorporates nearly every common (continuous) probability model ever used. This means that all of the common models are nested. This has important implications in model selection beyond determining the distribution of fitness effects. However, we demonstrate the use of this distribution utilizing likelihood ratio testing to evaluate alternative distributions to the Gumbel and Weibull domains of attraction of fitness effects. We use a bootstrap strategy, utilizing importance sampling, to determine where in the parameter space will the test be most powerful in detecting deviations from these domains and at what sample size, with focus on small sample sizes (n<20). Our results indicate that the likelihood ratio test is most powerful in detecting deviation from the Gumbel domain when the shape parameters of the model are small while the test is more powerful in detecting deviations from the Weibull domain when these parameters are large. As expected, an increase in sample size improves the power of the test. This improvement is observed to occur quickly with sample size n≥10 in tests related to the Gumbel domain and n≥15 in the case of the Weibull domain.

Suggested Citation

  • Joyce, Paul & Abdo, Zaid, 2018. "Determining the distribution of fitness effects using a generalized Beta-Burr distribution," Theoretical Population Biology, Elsevier, vol. 122(C), pages 88-96.
    • Handle: RePEc:eee:thpobi:v:122:y:2018:i:c:p:88-96
    DOI: 10.1016/j.tpb.2017.07.001
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    References listed on IDEAS

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    1. Paranaíba, Patrícia F. & Ortega, Edwin M.M. & Cordeiro, Gauss M. & Pescim, Rodrigo R., 2011. "The beta Burr XII distribution with application to lifetime data," Computational Statistics & Data Analysis, Elsevier, vol. 55(2), pages 1118-1136, February.
    2. Leemis, Lawrence M. & McQueston, Jacquelyn T., 2008. "Univariate Distribution Relationships," The American Statistician, American Statistical Association, vol. 62, pages 45-53, February.
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