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 compre-
hensive 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 manage-
ment covariates. For example, if siis daily survival probability (probability of
surviving from day ito day i1) and xiis a minimum temperature over the
interval itoi1, then survival can be modeled as a linear-logistic function of
temperature using the following expression:
(5.1)
where 0 and 1 are model parameters to be estimated, with 1 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. 2 xi^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 instant-
aneous risk of mortality over the period itoii, where iis a short time
interval (e.g. 1 day). This hazard, h, is related to the daily survival probability as:
hln(s). Proportional hazard models (Cox 1972; Cox and Oakes 1984) pro-
vide 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.
si e 0 1 xi
1 e^0 ^1 xi,122 |Estimating survival and movement