Field sampling for survival studies using radiotelemetry usually involves a single study area that is small enough to be traversed within a few days at most. Thus, investigators with radio receivers try to cover the area at specified sample periods

(e.g. once each week), listening for radio signals and identifying birds as alive or dead. Such status identification sometimes requires actually locating and observing the bird, although some transmitters are equipped with "mortality sensors," based on either temperature or motion, that indicate whether the bird is alive or dead and that so do not necessarily require location of the bird.

If all radioed birds are detected when alive and are also detected as dead during the first sampling period following death, then the data needed for each bird are simply the sampling occasion of initial capture and release with a radio, the last sample period of detection as a live bird, and, in the case of death, the sample period during which the bird was encountered dead. We can also summarize the data for each marked bird as an encounter history, using codes 1 = marked and alive, 2 = marked and newly dead (i.e. the bird died following the previous sample period and before the current period) and 0 = not yet marked or died during a previous period. The encounter history is a row of these codes, with an entry for each sample period. Thus the encounter history 0 1 1 1 2 0 would denote a bird marked in period 2, detected alive in periods 3 and 4, found dead in period 5, and so not detected in period 6 of a 6-period study. These data can then be modeled in either of two basically equivalent ways, using either binomial survival models or models based on time at death (Williams etal. 2002).

We will illustrate the binomial survival modeling approach (also see Heisey and Fuller 1985) and define si as the probability that any bird alive at sampling period i is still alive at sample period i + 1. We would model the above capture history, 0 1 1 1 2 0, as:

Thus, S2 denotes the probability associated with the bird surviving from week 2 until week 3, and S3 denotes the probability that the bird survives from week 3 to week 4. The (1 - S4) term indicates the probability that the bird did not survive the interval between weeks 4 and 5 (we found the bird dead in week 5). We would have a similar probability for each observed encounter history. The product of these probabilities over all birds in the study would constitute the model for the entire data set and could be used to estimate the model parameters, the s. Nesting studies described in Chapter 3 use similar encounter histories and similar survival models to estimate daily nest survival probabilities and success.

In general, we could obtain estimates under various models of this sort using a software package such as MARK (White and Burnham 1999). Program MARK can also be used to fit competing binomial models (e.g. interval survival varies over time and sample period or is instead constant; survival differs for two groups of birds such as males and females or is instead the same for both sexes) and to discriminate among them based on model selection procedures or likelihood ratio tests (e.g. Lebreton et al. 1992; Burnham and Anderson 2002). Goodness-of-fit tests should also be conducted as part of the testing or selection procedure (Pollock et al. 1985, 1990; Burnham et al. 1987), as both likelihood ratio tests and model selection procedures assess relative model fit and are therefore strictly appropriate for inference only when the most general model in the pair or model set fits the data adequately. When the general model does not fit well, quasilikelihood methods based on the goodness-of-fit statistic can be used to adjust model test and selection results for lack of fit (e.g. Burnham et al. 1987; Lebreton et al. 1992; Burnham and Anderson 2002). Time at death models (not described here) and associated estimators, such as Kaplan—Meier, can frequently be implemented using comprehensive statistical software packages such as SAS (see Pollock et al. 1989a,b).

In most studies, point estimates themselves are not of primary interest, even if these estimates are of fundamental parameters such as survival probability. Instead, biologists are interested in the relationship between these parameters and such quantities as environmental covariates and management actions. One approach to covariate modeling is to write survival probability for a specific time period as a linear-logistic function of time-specific environmental or management covariates. For example, if s¿ is daily survival probability (probability of surviving from day i to day i + 1) and x¿ is a minimum temperature over the interval i to i + 1, then survival can be modeled as a linear-logistic function of temperature using the following expression:

where and fii are model parameters to be estimated, with fii reflecting the nature of the relationship between temperature and survival. If the relationship between survival and temperature is hypothesized not to be monotonic, but to instead involve higher survival at intermediate temperatures, then an additional quadratic term (e.g. ^ x¿2) can be added to the model. This flexible modeling approach can be implemented using MARK (White and Burnham 1999).

If the linear-logistic model does not provide an adequate parametric structure for the problem of interest, then another approach models the hazard or instantaneous risk of mortality over the period i to i + i, where i is a short time interval (e.g. 1 day). This hazard, h, is related to the daily survival probability as: h = — ln(s). Proportional hazard models (Cox 1972; Cox and Oakes 1984) provide an alternative approach to equation (5.1) for covariate modeling of survival data. Under this approach, the time-specific hazard is modeled as the product of a baseline hazard and an exponential term reflecting the level of the covariate.

Proportional hazards modeling can be implemented in many biomedical statistical packages and in program MARK (White and Burnham 1999). Although the above discussion is focused on time-specific covariates, individual covariates may also be of interest. For example, Pollock et al. (1989b) modeled survival of wintering Black Ducks Anas rubripes as a function of individual body mass at the time of radio attachment.

The questions about survival that are of most interest to scientists and managers require discriminating among competing models. For example, we might model weekly survival as a function of a weekly management action (e.g. different levels of food provisioning), under the hypothesis that increased food improves survival. A competing model is that natural foods are sufficient and that the amount of food provided by managers is not relevant to survival. Under this hypothesis, we might specify a statistical model in which survival varied over time but independently of food (i.e. there would be no food covari-ate or associated parameter in this model). We would fit both models to the data and compute either likelihood ratio tests (under a hypothesis testing approach) or Akaike's Information Criterion (AIC; under a model selection approach) to decide which model is most appropriate for the data and, hence, which hypothesis is supported by the data (e.g. Lebreton et al. 1992; Burnham and Anderson 2002; Williams et al. 2002). Under some study designs (e.g. random selection each week from a small number of management treatments), we could fit models that include both time effects and management effects and thus consider the possibility that management is relevant to weekly survival, but that additional time effects are important as well. The important point is that this sort of modeling, with databased model selection, is a key component of science and science-based management (also see Hilborn and Mangel 1997; Nichols 2001; Williams et al. 2002).