Proportional hazards modeling can be implemented in many biomedical statist-
ical 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 rubripesas 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 hypo-
thesis 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).
The above discussion of estimation and modeling has been based on an
ideal field situation in which birds are always detected with probability 1 for the
duration of the study. Detection of radioed animals is seldom perfect in actual
field studies. In reality, some radios typically fail during the study, birds some-
times leave the study area either temporarily or permanently with respect to the
study duration, and birds with functioning radios are sometimes missed despite
searches. These kinds of problems must be dealt with in the modeling of observ-
ation histories (e.g. see White and Garrott 1990; Pollock et al. 1995). A poten-
tially severe problem occurs when radio signals are lost for many birds that die
(e.g. because predators or scavengers destroy the radio when handling the dead
bird). If all such losses of radio contact can be assumed to reflect dead birds,
then there is no problem, but in the usual case of some undetected temporary or
Survival rates| 123