The algebra of community temperature indices

The Community Temperature Index (CTI) was developed as a way to measure changes in a community over time in a way that reflects the temperature preferences of species. Several different forms of CTI are in widespread use, but their properties have not been systematically investigated. We argue that a CTI should preserve the algebraic structure of relative abundances given by the Aitchison geometry used in compositional data analysis. We show that this requirement leads to a new Aitchison CTI. The CTI has been used to classify species as warm- or cold-affinity, depending on whether an increase in their relative abundance increases or decreases the CTI. However, such classifications depend on relative abundances as well as on temperature preferences, which is undesirable. We show that the Aitchison CTI leads to a classification of species as relative warm- or cold-affinity that does not depend on relative abundances, with species contributions to changes in CTI that are consistent with the principles of population dynamics. We show that the Aitchison CTI is approximately linearly related to the most popular CTI in current use if relative abundances are approximately equal and temperature preferences for all species are close to their geometric mean. We provide a quantity analogous to an R2 for the Aitchison CTI, that tells us whether the CTI contains useful information. We illustrate our approach using data from a hard-substrate macrobenthos community.