Phantom interactions: Use odds ratios or risk misinterpreting occupancy models

Abstract

Occupancy models rely on logistic regression approaches that use the logit link function for model fitting. Parameter estimates fitted on the logit scale are frequently back-transformed to the probability scale for interpretation. However, caution is necessary, because back-transformation from the linear logit scale to the bounded probability scale can yield statistical artifacts that could lead to spurious inferences. We discuss one such artifact that we term "phantom interactions"-that is, apparent but false interactions-where the effect size of an additive model parameter, when interpreted on the probability scale, appears to vary depending on the value of other model covariates. A recent paper published in The Condor: Ornithological Applications provides an example of this issue (Lee and Bond 2015). The paper's title and central conclusion state that estimated negative effects of fire and salvage logging on Spotted Owl (Strix occidentalis) site occupancy were large at sites where reproduction did not occur the previous year, but small (or "negligible") at sites where reproduction did occur the previous year, suggesting breeding owls were resistant to disturbances. This implies the explicit modeling of a breeding state × disturbance interaction, but no interactive effects were present in the model used by the authors. A simple calculation of odds ratios (ORs) using data from the paper shows the modeled effects of fire and salvage logging on owls were, in fact, equivalent regardless of the previous year's reproductive state. Lee and Bond (2015) incorrectly concluded differential effects of fire and salvage logging on breeding versus nonbreeding owls because the effect for breeding owls occurred near a boundary on the probability scale, where a change on the logit corresponded to very small changes in probability. To avoid problematic inference, researchers should use ORs to interpret logistic regression and occupancy models.