Penalized likelihood methods improve parameter estimates in occupancy models

Summary Occupancy models are employed in species distribution modelling to account for imperfect detection during field surveys. While this approach is popular in the literature, problems can occur when estimating the model parameters. In particular, the maximum likelihood estimates can exhibit bias and large variance for data sets with small sample sizes, which can result in estimated occupancy probabilities near 0 and 1 (‘boundary estimates’). In this paper, we explore strategies for estimating parameters based on maximizing a penalized likelihood. Penalized likelihood methods augment the usual likelihood with a penalty function that encodes information about what parameter values are undesirable. We introduce penalties for occupancy models that have analogues in ridge regression and Bayesian approaches, and we compare them to a penalty developed for occupancy models in prior work. We examine the bias, variance and mean squared error of parameter estimates obtained from each method on synthetic data. Across all of the synthetic data sets, the penalized estimation methods had lower mean squared error than the maximum likelihood estimates. We also provide an example of the application of these methods to point counts of avian species. Penalized likelihood methods show similar improvements when tested using empirical bird point count data. We discuss considerations for choosing among these methods when modelling occupancy. We conclude that penalized methods may be of practical utility for fitting occupancy models with small sample sizes, and we are releasing R code that implements these methods.