Understanding the causes and consequences of animal movement: a cautionary note on fitting and interpreting regression models with time-dependent covariates

New technologies have made it possible to simultaneously, and remotely, collect time series of animal location data along with indicators of individuals' physiological condition. These data, along with animal movement models that incorporate individual physiological and behavioural states, promise to offer new insights into determinants of animal behaviour. Care must be taken, however, when attempting to infer causal relationships from biotelemetry data. The possibility of unmeasured confounders, responsible for driving both physiological measurements and animal movement, must be considered. Further, response values (yt) may be predictive of future covariate values (xt+s,s >= 1). When this occurs, the covariate process is said to be endogenous with respect to the response variable, which has implications for both choosing statistical estimation targets and also estimators of these quantities. We explore models that attempt to relate xt = log(daily movement rate) to yt = log(average daily heart rate) using data collected from a black bear (Ursus americanus) population in Minnesota. The regression parameter for xt was 0.19 and statistically different from 0 (P<0.001) when daily measurements were assumed to be independent, but residuals were highly autocorrelated. Assuming an autoregressive model (ar(1)) for the residuals, however, resulted in a negative slope estimate (-0.001) that was not statistically different from 0. The sensitivity of regression parameters to the assumed error structure can be explained by exploring relationships between lagged and current values of x and y and between parameters in the independence and ar(1) models. We hypothesize that an unmeasured confounder may be responsible for the behaviour of the regression parameters. In addition, measurement error associated with daily movement rates may also play a role. Similar issues often arise in epidemiological, biostatistical and econometrics applications; directed acyclical graphs, representing causal pathways, are central to understanding potential problems (and their solutions) associated with modelling time-dependent covariates. In addition, we suggest that incorporating lagged responses and lagged predictors as covariates may prove useful for diagnosing when and explaining why some conclusions are sensitive to model assumptions.