Fitting a survival model to describe the age structure of fossil populations

Abstract

Building mortality profiles to study age-at death patterns is a typical component of most faunal analyses. These age profiles are used to reconstruct aspects of a sample’s taphonomical history, including prey selection or mode of accumulation. The attritional profile, produced by natural deaths by predation, disease, etc., reflects the animals dead by each age class in a living population. Several methods, like histograms, boxplots and triangular plots are currently used to describe and compare mortality profiles and living population profiles. The main goal of this study is to reconstruct the age structure of fossil populations from attritional mortality profiles by assuming that this structure may be mathematically modeled by a survival model. Eventually, demographic parameters as life expectancy at birth, birth rate, mortality rate, etc, may be estimated from the model. Moreover, descriptions and comparisons between fossil and living populations can be developed. Dental eruption, dental attrition, epiphyseal fusion and skeletochronology provide reliable estimations of the age at death from fossil samples. However, although some of these techniques are claimed to provide punctual age estimations, most of them provide interval age estimations. The age intervals may be equal for all life stages but more frequently they are different in length for different life stages. These type of data are called censored data in statistics analysis. The nature of censored data produce limitations in the estimations and require the use of specific statistical techniques. If the age at death is reliably known with high accuracy (no censored data) several models can be easily proposed by estimating the parameters using the maximum likelihood method. However, censored data make it more difficult to obtain a likelihood function. Several models are fitted to a sample with interval censored data and the best one is selected. The set of models used are called Exponential, Weitbull, Log- Normal, Extreme value, and Gamma and Rayleigh. These are typical models used in survival analysis. All the tested models fit the data. However, Exponential and Weibull present better performance. Both of them exhibit a similarly good adjustment to the data, but the use of the exponential model is recommended because of its simplicity. Note that the Exponential model has only one parameter whilst Weibull has two. The parameter to be estimated in the case of the exponential model is λ. This can be interpreted like the mean time of death is 1/λ years. Moreover, the Survival Function, i.e. the proportion of individuals alive at time t, or the Lifetime Function, i.e. proportion of individuals with more than t years, can be easily obtained. These functions, besides the mortality profile, are useful to compare populations. Mathematical and statistical techniques have been found to be useful for the description, comparison and analysis of the age structure of fossil populations. In addition, the features of the fossil record, as the limitations in the size of the sample, and the nature of censored data, are properly dealt with using these techniques.