Appendix B. Comparison among survivorship models.
Our choice of the beta-distribution parameterization of survivorship in Eq. 4 was based on our specific desire to focus on survivorship shape and scale and their collective impact on the estimation of r in the Euler equation. However, in this Appendix, we compare the model based on the beta-distribution to two other commonly used models of survivorship, the Weibull,
(B.1) |
and the Gompertz
(B.2) |
where x is age, and ℓ0 (typically fixed at 1), m0, α', and β' are fit parameters. Modeling survivorship ℓ(x), as opposed to mortality rates m(x), is complicated by a lack of independence between different age classes. In this analysis, we model survivorship because survivorship is more commonly reported than age-specific mortality rates and because age-specific mortality rates are particularly difficult to fit when sample sizes are small (Ricklefs 2000, Ricklefs and Scheuerlein 2002). For this reason, model comparison by Akaike’s Information Criteria (AIC) must be interpreted with caution. We compared models using both the AIC of fitting the original data and also using the mean AIC from fits of 1000 bootstrapped samples created by bootstrapping the model residuals. As detailed in the accompanying file, the three-parameter beta-distribution model (Eq. 4 in text) outperforms or is statistically indistinguishable from the three-parameter Weibull model (Eq. B.1) and the two-parameter Gompertz model (Eq. B.2) for 41 of the 58 species in this analysis (37 of 58 if using the bootstrapped mean AIC). The beta-distribution model performed well over the entire range of body sizes and survivorship shapes and scales.
LITERATURE CITED
Ricklefs, R. E. 2000. Intrinsic ageing-related mortality in birds. Journal of Avian Biology 31:103–111.
Ricklefs, R. E., and A. Scheuerlein. 2002. Biological implications of the Weibull and Gompertz models of aging. Journal of Gerontology: Biological Sciences 57A(2):B69–B76.