# 4: Fitting Brownian Motion

- Page ID
- 21599

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

By fitting a Brownian motion model to phylogenetic comparative data, one can estimate the rate of evolution of a single character. In this chapter, I demonstrated three approaches to estimating that rate: PICs, maximum likelihood, and Bayesian MCMC. In the next chapter, we will discuss other models of evolution that can be fit to continuous characters on trees.

- 4.1: Introduction
- Mammals come in a wide variety of shapes and sizes. Body size is important as a biological variable because it predicts so many other aspect of an animal’s life, from the physiology of heat exchange to the biomechanics of locomotion. Thus, the rate at which body size evolves is of great interest among mammalian biologists. Throughout this chapter, I will discuss the evolution of body size across different species of mammals. The data I will analyze is taken from Garland (1992).

- 4.2: Estimating Rates using Independent Contrasts
- The information required to estimate evolutionary rates is efficiently summarized in the early (but still useful) phylogenetic comparative method of independent contrasts (Felsenstein 1985). Independent contrasts summarize the amount of character change across each node in the tree, and can be used to estimate the rate of character change across a phylogeny. There is also a simple mathematical relationship between contrasts and maximum-likelihood rate estimates that I will discuss below.

- 4.3: Estimating rates using maximum likelihood
- We can also estimate the evolutionary rate by finding the maximum-likelihood parameter values for a Brownian motion model fit to our data. Recall that ML parameter values are those that maximize the likelihood of the data given our model (see Chapter 2).

- 4.4: Bayesian approach to evolutionary rates
- Finally, we can also use a Bayesian approach to fit Brownian motion models to data and to estimate the rate of evolution. This approach differs from the ML approach in that we will use explicit priors for parameter values, and then run an MCMC to estimate posterior distributions of parameter estimates. To do this, we will modify the basic algorithm for Bayesian MCMC.