Using lorelograms to measure and model correlation in binary data: Applications to ecological studies

Tools for describing correlation structures in binary data are underrepresented in the ecological literature, and methods commonly applied to continuous data are inappropriate because of intrinsic features of binary data (e.g. variance dependent on the mean). Describing how correlation changes in time and space, or in response to different stimuli, can provide insights into the ecological processes generating observed patterns. Moreover, proper modelling of correlation structures is necessary for reliably estimating covariate effects. We introduce ecologists to the lorelogram (Heagerty & Zeger, 1998), a tool used to identify and describe dependency structures in binary data. Lorelograms can help guide data aggregation efforts to facilitate independence, or alternatively, to inform appropriate structures for modelling correlated data. We developed the r-package, lorelogram, for estimating and plotting lorelograms and show how it can be used to identify various dependency structures using simulated data. We analyse data from the North American Breeding Bird Survey and camera trap data from a carnivore study in Minnesota to demonstrate how the lorelogram can describe spatial and temporal dependencies, respectively. In the latter case, we show how the lorelogram can identify short-term dependencies (e.g. individuals that linger at the camera site for several minutes), longer-term dependencies (i.e. diel activity patterns) and effects of site-specific covariates such as attractants. We then illustrate how this information can be incorporated in a modelling framework that accounts for these correlation structures (e.g. when modelling daily activity patterns). The lorelogram is a promising tool for quantifying correlation in binary data over space or time. In addition to the applications presented in our paper, it could be used to identify independent sampling units for occupancy modelling or to quantify behavioural responses to covariates such as anthropogenic stressors or recent presence of prey, predators or competitors.