Phenological patterns in ecology: Problems using circular statistics and solutions based on simulations

Abstract Quantification of phenological patterns (e.g. migration, hibernation or reproduction) should involve statistical assessments of non‐uniform temporal patterns. Circular statistics (e.g. Rayleigh test or Hermans‐Rasson test) provide useful approaches for doing so based on the number of individuals that exhibit particular activities during a number of time intervals. This study used monthly reproductive activity as an example to illustrate problems in applying circular statistics to data when marginal totals characterize experimental designs (e.g. the number of reproductively active individuals per time interval depends on sampling effort or sampling success). We illustrate the nature of this problem by crafting four exemplar data sets and developing a bootstrapping simulation procedure to overcome complications that arise from the existence of marginal totals. In addition, we apply circular statistics and our bootstrapping simulation to empirical data on the reproductive phenology of six species of Neotropical bats from the Amazon. Because sampling effort or success can differ among time intervals, circular statistics can produce misleading results of two types: those suggesting uniform phenologies when empirical patterns are markedly modal, and those suggesting non‐uniform phenologies when empirical patterns are uniform. The bootstrapping simulation overcomes these limitations: the exemplar phenology in which the percentage of reproductively active individuals is modal is appropriately identified as non‐uniform based on the bootstrapping approach, and the exemplar phenology in which the percentage of reproductively active individuals is invariant is appropriately identified as uniform based on the bootstrapping approach. The reproductive phenology of each of the six empirical examples is non‐uniform based on the bootstrapping approach, and this is true for bats species with unimodal peaks or bimodal peaks. In addition to problems with marginal totals, a review of analyses of phenological patterns in ecology identified two other frequent issues in the application of circular statistics: sampling bias and pseudoreplication. Each of these issues and potential solutions are also discussed. By providing source code for the execution of the Rayleigh test and Hermans‐Rasson test, along with the code for the bootstrapping simulation, we offer a useful tool for assessing non‐random phenologies when marginal totals characterize experimental designs.