A Systematic-Encounter-Sampling Design for Nesting Studies

Abstract

Encounter-sampling designs are data-collection procedures in which population units are included in the sample as they are detected, or encountered. Perhaps the most familiar examples of encountersampling designs are line-transect (e.g. Burnham et al. 1980) and line-intercept (e.g. De Vries 1979) procedures. One characteristic of encounter-sampling designs is that a sampling frame (e.g. Cochran 1977) of population units is not required. When individuals are mobile, elusive, or possess other characteristics that make it difficult to construct a sampling frame, encounter-sampling designs often provide the only effective means of sampling a population. A second characteristic of encounter-sampling designs is that they lack control over the subset of the population comprising the sample. As a result, data collected by these procedures often are not representative of the population of interest (i.e. a biased sample), and are best viewed as a probability sample. If variables of interest are correlated with the probability of inclusion, the data cannot be treated as a simple random sample, and estimators based upon random sampling theory are biased (Rao 1965). In these cases, designspecific estimators or bias corrections dependent on the probability structure of the data must be employed. A population of bird nests is one example of a population whose study requires the use of an encountersampling technique. In addition to the obvious lack of a sampling frame, the population is demographically open in that nests are initiated and fail through time. The usual sampling design consists of conducting searches for viable nests, including all detected nests in the sample. Data collected under such a design are biased because longer-lived nests are included in the sample with higher probability than are shorter-lived nests (Mayfield 1961). However, the method by which searches are conducted, typically, is not structured and is not helpful in deriving estimators. The parameter most often of interest is the nestsurvival rate (i.e. probability a nest survives to success). Success is often defined as the production of at least one offspring, though other definitions are equally appropriate. Nest-survival rates may be estimated using a number of models. Mayfield (1961, 1975), Johnson (1979), and Bart and Robson (1982) modeled nest survival after detection. Hensler and Nichols (1981), Pollock and Cornelius (1988), and Bromaghin and McDonald (1993) modeled the entire existence of nests by incorporating probabilities of inclusion and partial information of the total lifetime of detected nests in the model. While all of these models attempt to treat the probability structure of the data, only the Pollock-Cornelius and BromaghinMcDonald models do so fully and correctly. Although Heisey and Nordheim (1990) indicated that the Pollock-Cornelius model produces biased estimates of nest survival, recent information (Pollock and Cornelius unpubl. data) suggests that the bias decreases as the time between visits to nests decreases. All of the nest-survival models are designed to produce estimators of a single parameter, the probability of nest-success, while acknowledging the biased nature of the data. To some extent, they have all been successful. However, any number of additional parameters may also be of interest, and their estimation, which is also complicated by the biased nature of the data, has received little attention. We present a sampling design and estimation procedures that directly utilize the unequal probabilities with which nests are included in the sample. The method is developed from a classical sampling approach in that all characteristics of the population of nests are considered as fixed; randomness observed in the data is due solely to the sampling design. The design consists of temporally systematic searches for nests, and nests are included in the sample as they are detected. The model assumes that nests are (approximately) aged at the time of detection and monitored until they either fail or are successful. The systematic design permits probabilities of inclusion to be estimated. The estimates are employed in modified Horvitz-Thompson estimators (Horvitz and Thompson 1952) to obtain estimates of any parameter that can be expressed as a total or a ratio of totals. Such parameters include the probability a nest survives to success, the number of nests which survive to success, and the numbers of nests initiated. The sampling model.-Consider a fixed geographic area in which birds are nesting. Nests are initiated and survive for some period of time. The population of interest consists of all nests which exist within the area for any portion of a specified time frame. For example, the time frame might be constructed to contain all or some interesting portion of the nesting season of the species under consideration. Thus, a geographic area and a time frame are used to define the population. The population size is denoted N and the number of time units in the time frame of interest is denoted D. The time frame of D time units is divided into a