As we have done in previous chapters when developing specific methods, we begin here by considering a situation in which p = 1. This allows us to focus on describing the basic state process model and developing an analysis framework for the model. The class of models commonly employed for modeling avian nest survival is closely related to survival models of animals, but where individual nests can be observed with probability 1 (an unusual circumstance for most animal population studies). Therefore, we provide a brief development of models for avian nest survival. Historical background and recent developments can be found in Mayfield (1975), Johnson (1979), Hensler and Nichols (1981), Stanley (2000), Dinsmore et al. (2002) and Rotella et al. (2004). We note also that survival models with p =1 are often encountered in the analysis of radio telemetry data (White and Garrott, 1990). We develop models of nest survival using data from a study of American redstarts (Setophaga ruticilla) conducted in Michigan's Upper Peninsula (Hahn and Silverman, 2006, 2007; Hahn, 2007) to evaluate the effects of social behavior on elements of population dynamics. The specific data set considered here includes 115 nests initiated during the 2006 nesting season.
We suppose that nests are found over time and thus enter the sample at various points (days, samples or times) during the nesting season. For these data, we assume that when a nest is first discovered its age can be determined precisely, and the outcome of the nest (success or failure) is determined perfectly, as in Dinsmore et al. (2002). The models can be generalized to accommodate uncertainty in nest age (Stanley, 2004), but we consider only the simpler case here. After each nest is found and enters the sample it is detected with probability 1 during subsequent visits, since the investigator knows the nest location. Each nest is revisited at least once, perhaps irregularly, and all nests might have different revisitation dates. At the revisit, the status 'active' or 'failed' is recorded for each nest. If a nest is determined to have failed then this represents a sort of censored fate, since the day of failure is not always known precisely. Rather, it is only known that failure occurred between the previous and last visits.
Encounter histories of 10 nests are depicted in Panel 11.1. Columns of this data table index successive days of the study. Uncertain states are indicated by '—,' terminal 0 indicates nests that were checked and found to have failed. The interior dashes preceding that zero correspond to uncertain states. Leading x's indicate states prior to nest entry into the sample. Consider the first encounter history in Panel 11.1. This nest was first located on day 2 and known to have fledged by day 25. Let fj and Zj be the day of entry and last check, respectively, for nest i, and let mj be the day at which the nest was checked prior to Zj. In the development of the likelihood (described below), we will only be concerned about mj for failed nests. Thus, for the nest corresponding to row 1 of Panel 11.1, fj = 2 and Zj = 25. For the encounter history in the second row, fj = 3, Zj = 10 and mj = 7.
Let ^ be the probability that at least one egg or individual survives an interval (defined here to be a day). We assume that ^ varies by individual and day in response to measurable covariates. We consider first describing the probability of observing a particular encounter history yj for a nest that entered the sample at fj, was observed alive at m; and then observed to have failed by Z.j. Thus, it is known to have failed between mj and Zj. Here, yj is the vector of encounter observations of length Z; — fj, such as from Panel 11.1, without the leading and trailing dashes. An example is nest 2 in Panel 11.1, for which yj = (1,1,1,1,1, —, —, 0), fj = 3, Z; = 10 and mj = 7. If successive nest states are independent Bernoulli trials, then the probability of observing a string of survival events from fj to mj is simply the product of the interval-specific survival probabilities between fj and mj. However, the terminal portion of the encounter history (that subsequent to mj) is not known with certainty. Rather, it is composed of a number of possible events, and we will have to sum the probabilities of these constituent events in order to obtain the probability of the terminal state, which we will also refer to as \t for a nest last known to have been alive on day t. As such, the probability of an encounter history is
When a nest failure is observed, the terminal component of the encounter history is the event that nest i is 'failed by Z; given that it was known to be active at mj.' Thus, define xmi = Pr(observed to have failed by Zj|known alive at mj). Under the
9 xxxxxxxxx1111111111111111111111 10 x x x x x x x x x x x x x x x x x x x 1 1 1 1 1 1 - - 0
Panel 11.1. Sample of 10 nest encounter histories for the American Redstart (Setophaga ruticilla) data (Hahn and Silverman, 2006). Leading x's indicate days prior to nest entry into the sample. Interior dashes indicate uncertain states prior to a check in which the nest was observed to have failed.
day assumption that successive nest states are independent Bernoulli trials, this equals (Dinsmore et al., 2002):
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