Distinguishing within- from between-individual effects: How to use the within-individual centring method for quadratic patterns.

Any average pattern observed at the population level (cross-sectional analysis) 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. The within-individual centring method allows distinguishing within- from among-individual processes and this method has been widely 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 the different assumptions and constraints of both equations. Reviewing the literature, we found that mainly one of these two equations has been used in studies investigating quadratic patterns. Yet this equation might not be the most appropriate in all circumstances leading to bias and imprecision. We show that these two alternative equations make different assumptions about the shape of the within-individual pattern. One equation assumes that the within-individual effect is related to an absolute process whereas the other assumes the effect arises from an individual relative process. The choice of using one equation instead of the other should depend upon the biological process investigated. Using simulations, we showed that a mismatch between the assumptions made by the equation used to analyse the data and the biological process investigated might led to flawed inference affecting output of model selection and accuracy of estimates. We stress that the equation used should be chosen carefully. We provide step by step guidelines for choosing an equation when studying quadratic pattern with the within-individual centring approach. We encourage the use of the within-individual centring method, promoting its relevant application for nonlinear relationships.