Survival analysis in telemetry studies: the staggered entry design

The estimation of survival distributions for radio-tagged animals is important to wildlife ecologists. Allowance must be made for animals being lost (or censored) due to radio failure, radio loss, or emigration of the animal from the study area. The Kaplan-Meier procedure (Kaplan and Meier 1958), widely used in medical studies subject to censoring, can be applied to this problem. We developed a simple modification of the Kaplan-Meier procedure that allows for new animals to be added after the study has begun. We present 2 examples using telemetry data collected from northern bobwhite quail (Colinus virginianus) to show the simplicity and utility of the Kaplan-Meier procedure and its modifications. The log rank test used to compare 2 survival distributions can also be modified to allow for additions during the study. Simple computer programs that can be run on a personal computer are available from the authors. J. WILDL. MANAGE. 53(1):7-15 Radio-tagged animals are used to study survival. Present techniques for analyzing data from these studies assume that each survival event (typically an animal surviving a day) is independent and has a constant probability over all animals and all periods (Trent and Rongstad 1974, Bart and Robson 1982, Heisey and Fuller 1985). We believe these assumptions are often unrealistic and restrictive. White (1983) generalized discrete approaches using the same framework as that of band return models (Brownie et al. 1985) and he developed a flexible computer program (SURVIV) for use with his approach. Heisey and Fuller (1985) generalized the Trent and Rongstad (1974) approach to allow mortality from different causes (e.g., predation, starvation) and developed a microcomputer program called MICROMORT. Typically an animal's exact survival time (at least to within 1-2 days) is known unless that survival time is right censored (i.e., only known to be greater than some value). Pollock (1984) and Pollock et al. (1989) suggested a useful approach based on continuous survival models allowing right censoring that is widely used in medicine and engineering (Kalbfleisch and Prentice 1980, Cox and Oakes 1984) and provided examples of the Kaplan-Meier procedure. The Kaplan-Meier procedure does not require specification of a particular parametric continuous distribution; e.g., the exponential or Weibull. Related ecological papers using survival methods include Muenchow (1986), Pyke and Thompson (1986), Kurzejeski et al. (1987), and White et al. (1987). We present a simple description of the Kaplan-Meier procedure with an example using northern bobwhite quail survival data collected by PDC. We then generalize the Kaplan-Meier procedure to allow gradual (or staggered) entry of animals into the study. The calculations are illustrated with an example from the quail data. Finally, we present the log-rank test for comparison of survival distributions (modified for staggered entry of animals) with an example. We also present a discussion of model assumptions and directions for future research. We thank J. D. Nichols and W. L. Link for helpful comments on an earlier draft of this paper. We acknowledge G. C. White and D. M. Heisey for their helpful reviews that improved the final version. THE KAPLAN-MEIER OR PRODUCT LIMIT PROCEDURE The Kaplan-Meier or product limit estimator was developed by Kaplan and Meier (1958) and is d scussed by Cox and Oakes (1984:48) and Kalbfleisch and Prentice (1980:13). The survival function (S[t]) is the probability of an arbitrary animal in a population surviving t units of time from the beginning of the study. A nonparametric estimator of the survival function can be obtained by restricting ourselves to the discrete time points when deaths occur a1, a2, ..., ag. We define r, . . . , rg to be the numbers of an-