A hidden Markov model to identify and adjust for selection bias: an example involving mixed migration strategies

An important assumption in observational studies is that sampled individuals are representative of some larger study population. Yet, this assumption is often unrealistic. Notable examples include online public-opinion polls, publication biases associated with statistically significant results, and in ecology, telemetry studies with significant habitat-induced probabilities of missed locations. This problem can be overcome by modeling selection probabilities simultaneously with other predictor-response relationships or by weighting observations by inverse selection probabilities. We illustrate the problem and a solution when modeling mixed migration strategies of northern white-tailed deer (Odocoileus virginianus). Captures occur on winter yards where deer migrate in response to changing environmental conditions. Yet, not all deer migrate in all years, and captures during mild years are more likely to target deer that migrate every year (i.e., obligate migrators). Characterizing deer as conditional or obligate migrators is also challenging unless deer are observed for many years and under a variety of winter conditions. We developed a hidden Markov model where the probability of capture depends on each individual's migration strategy (conditional versus obligate migrator), a partially latent variable that depends on winter severity in the year of capture. In a 15-year study, involving 168 white-tailed deer, the estimated probability of migrating for conditional migrators increased nonlinearly with an index of winter severity. We estimated a higher proportion of obligates in the study cohort than in the population, except during a span of 3 years surrounding back-to-back severe winters. These results support the hypothesis that selection biases occur as a result of capturing deer on winter yards, with the magnitude of bias depending on the severity of winter weather. Hidden Markov models offer an attractive framework for addressing selection biases due to their ability to incorporate latent variables and model direct and indirect links between state variables and capture probabilities.