## Introduction

The life table is one of the most important tools in demographic and gerontological research because it is used to characterize the mortality and survival properties of cohorts and to quantify the actuarial rate of aging. The historical application of classical life table methods in aging science has been largely restricted to the use of mortality data from either humans or experimental animals maintained in the laboratory, or to life tables based on capture–recapture methods to assess aging in wild populations (Udevitz & Ballachey, 1998). In both applications, it is mandatory that age-at-entry is known. This has limited the use of life tables because in the analysis of field populations one often encounters and marks individuals of unknown age. However, capture–recapture and other current field methods generally require capturing and marking of young individuals, or alternatively of individuals of known age, for monitoring throughout their lives until they die.

The predominance of capture–recapture methods has had a limiting effect on the use of flexible non-parametric statistical methods that make minimal assumptions on survival schemes, and have the desirable property that they do not presume statistical parametric survival models. Because in non-parametric modelling one does not specify the functional form of hazard or survival functions, these methods require the exact recording of lifetimes and therefore are not applicable to usual capture–recapture designs, which correspond to usually coarsely graded life tables (Lebreton *et al*., 1992; Williams *et al*., 2002).

Because of the importance of the life table in aging research and the growing interest in understanding aging in the wild (Austad, 1993; Congdon *et al*., 1994; Finch, 2001; Reznick *et al*., 2001; Tatar & Yin, 2001), the case of life table analysis with unknown age at entry and the analogous situation for continuous lifetimes is clearly of great interest. We describe a life table identity that, by making certain key assumptions, enables us to estimate the age-specific life table rates from data based on the mark, release and monitoring of randomly captured individuals of unknown age from the time of their entry into the study (i.e. marking) to their death. We also discuss an identity for the continuous case in which marked animals are continuously monitored until their death. Continuous monitoring, when feasible, enables the continuous version of the analysis, which provides us with substantially more detailed information about the behaviour of survival functions and hazard rates (force of mortality). Our approach can also be used in conjunction with life tables that are obtained from capture–recapture experiments in those situations in which age-at-entry is unknown. Current designs of capture–recapture experiments, however, are not amenable to continuous lifetime analysis with the preferred flexible non-parametric methods, allowing for the construction of hazard rate estimates.

Consider a population that is assumed to be stable, stationary and closed. Individuals are randomly captured at an unknown age and marked, and their time-to-death recorded. The question we address is this: Can the information on time-to-death for this randomly captured marked subgroup provide the necessary information to construct a life table for the population at large? We will demonstrate that the answer to this question is yes because of a life table identity that reveals a mathematical relationship between the distribution of deaths in the marked cohort and the age structure of the original population. Individuals in the captured and marked sample are assumed to have remaining lifetimes similar to those in the wild. This model may be particularly adequate for some human populations.

The problem of constructing a survival schedule from incomplete data has been studied in anthropology (Müller *et al*., 2002) and has applications to human populations such as the !Kung and the Ache for which only incomplete demographic data are available (Howell, 1979; Hill & Hurtado, 1996; Hawkes *et al*., 1998; Jones *et al*., 2002). An anthropologist may encounter a group of people whose ages are unknown but whose remaining lifetime can be recorded. The key identity, on which the reconstruction of the survival schedule that we propose is based, asserts that for such situations a life table for the population can be obtained, under certain assumptions. Application of the key identity then establishes a new way to construct life tables and estimate survival functions.

We derive this key identity for both discrete life tables and situations that are modelled by continuous survival times. In the continuous case, this identity is a consequence of a close relationship between the density of the remaining lifetimes in a cohort of randomly sampled subjects and the survival schedule of the population from which the subjects were sampled. We provide statistical implementations of this identity by applying suitably adapted non-parametric density estimation methods. The proposed model is developed for a stable, stationary and closed population but possesses sufficient flexibility to allow for modifications of these assumptions.

The concept and techniques that we describe in this paper will help to advance understanding of senescence in the wild in several areas that were outlined by Gaillard *et al*. (1994). First, refinements of this concept have the potential to improve the reliability of survival data because, unlike the approach used in virtually all long-term field studies in which only newborn are marked and their survival monitored throughout their lives, this approach estimates survival using information from individuals first marked at any age. Therefore, for many species it may be possible to mark many more individuals than are available from only a single (newborn) age group and therefore to increase sample size. Second, this approach introduces new biological concepts for measuring senescence in the wild that differ from Nesse's (1988) intensity of selection, Finch's (1990) mortality rate doubling time, Promislow's (1991) log slope mortality and Abrams’ (1993) fitness cost of senescence. Additionally, our method is a useful addition to capture–recapture studies with unknown age-at-entry.

The method we outline in this paper focuses on the information content of wild-caught, living individuals and will ultimately not only include information on survival that can be used to estimate actuarial aging as in conventional approaches, but also information on fertility, behaviour, mating and other life history categories that can be used to shed new light on senescence in the wild. The methods we introduce will provide new techniques for expanding the taxonomic horizons of senescence studies in the wild beyond mammals, to include other vertebrates such as birds, reptiles, fishes and amphibians as well as invertebrates ranging from nematodes to insects. These methods will be especially important for studying aging of invertebrates in the wild, such as nematodes, that cannot be marked and released into the wild for later recapture.