5.1: Introduction to Multivariate Brownian Motion

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As discussed in Chapter 4, body size is one of the most important traits of an animal. Body size has a close relationship to almost all of an animal’s ecological interactions, from whether it is a predator or prey to its metabolic rate. If that is true, we should be able to use body size to predict other traits that might be related through shared evolutionary processes. We need to understand how the evolution of body size is correlated with other species’ characteristics.

A wide variety of hypotheses can be framed as tests of correlations between continuously varying traits across species. For example, is the body size of a species related to its metabolic rate? How does the head length of a species relate to overall size, and do deviations from this relationship relate to an animal’s diet? These questions and others like them are of interest to evolutionary biologists because they allow us to test hypotheses about the factors in influencing character evolution over long time scales. These types of approaches allow us to answer some of the classic “why” questions in biology. Why are elephants so large? Why do some species of crocodilians have longer heads than others? If we find a correlation between two characters, we might suspect that there is a causal relationship between our two variables of interest - or perhaps that both of our measured variables share a common cause.

In this chapter, we will use the example of home range size, which is the area where an animal carries out its day-to-day activities. We will again use data from Garland (1992) and test for a relationship between body size and the size of a mammal’s home range. I will describe methods for using empirical data to estimate the parameters of multivariate Brownian motion models. I will then describe a model-fitting approach to test for evolutionary correlations. This model fitting approach is simple but not commonly used. Finally, I will review two common statistical approaches to test for evolutionary correlations, phylogenetic independent contrasts and phylogenetic generalized least squares, and describe their relationship to model-fitting approaches.