Abstract
Confidence limits of age–specific mortality and fecundity rates are rarely provided. However, this information is important for judging the reliability of life tables, especially those based on small studbooks. The occurrence of a single death within an age class is a Bernoulli process, i.e. an individual is either dead or living at the end of the age class, with the mortality rate as the probability of dying. Simulation experiments using random sampling introduce the effect of sample size on the confidence limits of mortality rates. Variances of mortality rates can be calculated from binomial theory. Fecundity rates are often assumed to follow the Poisson distribution. The variance is estimated accordingly. Variances, confidence limits and coefficients of variation are estimated for both mortality and fecundity in young and older age classes. Bootstrap is applied to measure variance (and associated statistics) in both mortality and fecundity rates in female snow leopards. The results of χ 2 tests show that theoretical and bootstrap variances do not differ for mortality (except for the oldest age class). However, these variances do differ for fecundity, indicating that the Poisson distribution is not the correct assumption. Reliability of life tables can be improved by increasing sample sizes through pooling older age classes and applying wider age classes (at the cost of detail). Smoothing techniques to reduce gaps and peaks in life tables and application of the Gompertz(–Makeham), Weibull and Siler parametric models to fit mortality curves, are illustrated with real studbook data.
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Notes
- 1.
The Sturges algorithm is implemented as default method in R statistics to estimate bin (class) width.
References
Akçakaya HR (2002) Estimating the variance of survival rates and fecundities. Anim Conserv 5:333–336
Ballou JD, Lees C, Faust LJ, Long S, Lynch C, Bingaman Lackey L, Foose TJ (2010) Demographic and genetic management of captive populations. In: Kleiman DG, Thompson KV, Kirk Baer C (eds) Wild mammals in captivity: principles and techniques for zoo management, 2nd edn. The University of Chicago Press, Chicago, pp 219–252
Ballou JD, Lacy RC, Pollak JP (2011) PMx: software for demographic and genetic analysis and management of pedigreed populations (version 1.0). Technical report, Chicago Zoological Society, Brookfield
Carriere JF (1992) Parametric models for life tables. Trans Soc Actuar 44:77–99
Caswell H (2001) Matrix populations models: construction, analysis, and interpretation, 2nd edn. Sinnauer Associates, Sunderland
Caughley G (1977) Analyses of vertebrate populations. The Blackburn Press, Caldwell
Crowley PH (1992) Resampling methods for computation-intensive data analysis in ecology and evolution. Ann Rev Ecol Syst 23:405–447
Davidson AC, Hinkly DV (1997) Bootstrap methods and their application. Cambridge series in statistical and probabilistic mathematics. Cambridge University Press, Cambridge
Davidson R, MacKinnon JG (2000) Bootstrap tests: how many bootstraps? Econom Rev 19(1):55–68
Daw RH (1974) A study on the variance of mortality rates. J Inst Actuar 101:415–434
Devenish–Nelson ES, Stephens PA, Harris S, Soulsbury C, Richards SA (2013) Does litter size variation affect models of terrestrial carnivore extinction risk and management? PLoS ONE 8(2):e58060
Efron B (1979) Bootstrap methods: another look at the jackknife. Ann Stat 7(1):1–26
Efron B (1981) Censored data and the bootstrap. J Am Stat Assoc 76:312–319
Efron B, Tibshirani R (1986) Bootstrap methods for standard error, confidence intervals, and other measures of statistical accuracy. Stat Sci 1(1):54–75
Fay MP (2010) Two-sided exact tests and matching confidence intervals for discrete data. R J 2(1):53–58
Gage TB (1995) A robust estimator of the intrinsic rate of increase and generation length. In: Ballou JD, Gilpin M, Foose TJ (eds) Population management for survival and recovery: analytical methods and strategies in small population conservation. Methods and cases in conservation science series. Columbia University Press, New York, pp 9–23
Gage TB (2001) Age-specific fecundity of mammalian populations: a test of three mathematical models. Zoo Biol 20:487–497
Gompertz B (1825) On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philos Trans R Soc Lond 115:513–583
Horton NJ, Kleiman K (2011) Using R for data management and statistical analysis and graphics. CRC Press, Boca Raton
Keilman N, Pham DQ (2000) Predictive intervals for age–specific fertility. Eur J Popul 16:41–66
Kendall BE, Wittman ME (2010) A stochastic model for annual reproductive success. Am Nat 175(4):461–468
Klein JP, Moeschberger ML (1997) Survival analysis: techniques for censored and truncated data, 1st edn. Statistics for biology and health. Springer, New York
Kohler IV, Preston SH, Lackey LB (2006) Comparative mortality levels among selected species of captive animals. Demogr Res 15:413–434
Makeham WM (1860) On the law of mortality and the construction of annuity tables. Assur Mag J Inst Actuar 8(6):301–310
Meyer JS, Ingersoll CG, McDonald LL, Boyce MS (1986) Estimating uncertainty in population growth rates: jackknife versus bootstrap techniques. Ecology 67(5):1156–1166
Moorad JA, Promislow DEL, Flesness N, Miller RA (2012) A comparative assessment of univariate longevity measures using zoological animal records. Aging Cell 11(6):940–948
Nakazawa M (2015) fmsb: functions for medical statistics book with some demographic data. http://CRAN.R-project.org/package=fmsb. R package version 0.5.2
Patil VV, Kulkarni HV (2012) Comparision of confidence intervals for the poisson mean: some new aspects. Revstat Stat J 10(2):211–227
Peristera P, Kostaki A (2007) Modeling fertility in modern populations. Demogr Res 16(6):141–194
Pinder III JE, Wiener JG, Smith MH (1978) The Weibull distribution: a new method of summarizing survivorship data. Ecology 59(1):175–179
Princée FPG (2014) Population Management Library (PML) software and manual version 1.0. East Rudham, United Kingdom, http://www.princee.com
Quenouille MH (1949) Approximate test of correlation in time–series. J R Stat Soc Ser B (Method) 11(2):68–84
R Core Team (2015) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna. https://www.R-project.org/
Ricklefs R, Scheuerlein A (2001) Comparison of aging-related mortality among birds and mammals. Exp Gerontol 36(4–6):845–857
Ricklefs RE, Scheuerlein A (2002) Biological implications of the Weibull and Gompertz models of aging. J Gerontol Ser A Biol Sci Med Sci 57(2):B69–B76
Scobie P, Bingaman Lackey L (1997–2012) Single Population Animal Record Keeping System (SPARKS) Versions 1.42–1.66. International Species Information System, Eagan -15pt]Please provide publisher details for “Scobie and Bingaman Lackey (1997–2012)”.
Shaffer ML (1981) Minimum population sizes for species conservation. BioScience 31(2):131–134
Shryock HS, Siegel J (1980) U.S. Bureau of the census – the methods and materials of demography, 4th edn. U.S. Government Printing Office, Washington, DC
Siler W (1979) A competing-risk model for animal mortality. Ecology 60(4):750–757
Sokal RR, Rohlf FJ (2012) Biometry, 4th edn. W.H. Freeman and Company, New York
Sturges HA (1926) The choice of class interval. J Am Stat Assoc 21:65–66
Traylor–Holzer K (ed) (2011) PMx users manual, Version 1.0. IUCN SSC Conservation Breeding Specialist Group, Apple Valley, MN
Tukey JW (1958) Bias and confidence in not–quite large samples. (abstract of preliminary report). Ann Math Stat 29(2):614
Tukey JW (1977) Exploratory data analysis. Behavioural science: quantitative methods. Addison-Wesley, Reading
Weibull W (1951) A statistical distribution function of wide applicability. J Appl Mech 18(3):293–297
Wiese RJ, Willis K (2004) Calculation of longevity and life expectancy in captive elephants. Zoo Biol 23:365–373
Williamson L, Norman P (2011) Developing strategies for deriving small population fertility rates. J Popul Res 28(2):129–148
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Princée, F.P.G. (2016). Confidence in Life Tables. In: Exploring Studbooks for Wildlife Management and Conservation. Topics in Biodiversity and Conservation, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-50032-4_8
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