Modeling participation duration, with application to the North American Breeding Bird Survey

We consider “participation histories,” binary sequences consisting of alternating finite sequences of 1s and 0s, ending with an infinite sequence of 0s. Our work is motivated by a study of observer tenure in the North American Breeding Bird Survey (BBS). In our analysis, j indexes an observer’s years of service and X_j is an indicator of participation in the survey; 0s interspersed among 1s correspond to years when observers did not participate, but subsequently returned to service. Of interest is the observer’s duration D = max {j: X_j = 1}. Because observed records X = (X₁, X₂,..., X_n)¹ are of finite length, all that we can directly infer about duration is that D ⩾ max {j ⩽n: X_j = 1}; model-based analysis is required for inference about D. We propose models in which lengths of 0s and 1s sequences have distributions determined by the index j at which they begin; 0s sequences are infinite with positive probability, an estimable parameter. We found that BBS observers’ lengths of service vary greatly, with 25.3% participating for only a single year, 49.5% serving for 4 or fewer years, and an average duration of 8.7 years, producing an average of 7.7 counts.