Understanding step selection analysis through numerical integration

Step selection functions (SSFs) are flexible models to jointly describe animals' movement and habitat preferences. Their popularity has grown rapidly and extensions have been developed to increase their utility, including various distributions to describe movement constraints, interactions to allow movements to depend on local environmental features, and random effects and latent states to account for within- and among-individual variability. Although the SSF is a relatively simple statistical model, its presentation has not been consistent in the literature, leading to confusion about model flexibility and interpretation. We believe that part of the confusion has arisen from the conflation of the SSF model with the methods used for parameter estimation. Notably, conditional logistic regression can be used to fit SSFs in exponential form, and this approach is often presented interchangeably with the actual model (the SSF itself). However, reliance on conditional logistic regression reduces model flexibility, and suggests a misleading interpretation of step selection analysis as being equivalent to a case-control study. In this review, we explicitly distinguish between model formulation and inference technique, presenting a coherent framework to fit SSFs based on numerical integration and maximum likelihood estimation. We provide an overview of common numerical integration techniques, and explain how they relate to step selection analyses. This framework unifies different model fitting techniques for SSFs, and opens the way for improved inference. In particular, it makes it straightforward to model movement with distributions outside the exponential family, and to apply different SSF formulations to a data set and compare them with AIC. By separating the model formulation from the inference technique, we hope to clarify many important concepts in step selection analysis.