Distinguishing within from between individual effects: How to use the within-individual centering method for quadratic pattern
O_LIAny average pattern observed at the population level may confound two different types of processes: some processes that occur among individuals and others that occur within individuals. Separating within- from among-individual processes is critical for our understanding of ecological and evolutionary dynamics. C_LIO_LIThe within-individual centering method allows distinguishing within- from among-individual processes and this method has been largely used in ecology to investigate both linear and quadratic patterns. Here we show that two alternative equations could be used for the investigation of quadratic within-individual patterns. We explain under which hypotheses each is valid. Reviewing the literature, we found that mainly one of these two equations has been used by the studies investigating quadratic patterns. Yet this equation could be inappropriate in many cases. C_LIO_LIWe show that these two alternative equations make different assumptions about the shape of the within-individual pattern. The choice of using one equation instead of the other should depend upon the biological process investigated. One equation assumes that all individuals show the same quadratic pattern over the whole range of the explanatory variable whereas the other assumes that the quadratic patterns depend on the average explanatory variable of each individual. We give examples of biological processes corresponding to each equation. C_LIO_LIUsing simulations, we showed that a mismatch between the assumptions made by the equation used to analyze the data and the biological process investigated led to flawed inference affecting both output of model selection and accuracy of estimates. We stress that the equation used should be chosen carefully to ensure that the assumption made about the shape of the within-individual pattern matches the biological process investigated. We hope that this manuscript will encourage the use of the within-individual centering method, promoting its correct application for non-linear relationships. C_LI