# Table of Contents

- Page ID
- 25040

Evolution is happening all around us. In many cases – lately, due to technological advances in molecular biology – scientists can now describe the evolutionary process in exquisite detail. We understand the genetics, development, and biomechanical processes that link changes in a Galapagos finches’ diet to the shape of their bill. And, in some cases, we can watch as one species splits to become two.

## 2: Fitting Statistical Models to Data

The best comparative approaches have two features. First, the most useful methods emphasize parameter estimation over test statistics and P-values. Ideal methods fit models that we care about and estimate parameters that have a clear biological interpretation. Second, many useful methods involve model selection, the process of using data to objectively select the best model from a set of possibilities.## 3: Introduction to Brownian Motion

This chapter introduces Brownian motion as a model of trait evolution. I first connected Brownian motion to a model of neutral genetic drift for traits that have no effect on fitness. However, as I demonstrated, Brownian motion can result from a variety of other models, some of which include natural selection.## 4: Fitting Brownian Motion

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.## 5: Multivariate Brownian Motion

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 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.## 6: Beyond Brownian Motion

This chapter consider four ways that comparative methods can move beyond simple Brownian motion models: by transforming the variance-covariance matrix describing trait covariation among species, by incorporating variation in rates of evolution, by accounting for evolutionary constraints, and by modeling adaptive radiation and ecological opportunity.## 7: Models of Discrete Character Evolution

This chapter describes the Mk model, which can be used to describe the evolution of discrete characters that have a set number of fixed states. We can also elaborate on the Mk model to allow more complex models of discrete character evolution (the extended-Mk model). These models can all be used to simulate the evolution of discrete characters on trees.## 8: Fitting Models of Discrete Character Evolution

In this chapter, the Felsenstein’s pruning algorithm was presented and show to be used to calculate the likelihoods of Mk and extended-Mk models on phylogenetic trees. I have also described both ML and Bayesian frameworks that can be used to test hypotheses about character evolution. This chapter also includes a description of the “total garbage” test, which will tell you if your data has information about evolutionary rates of a given character.## 9: Beyond the Mk Model

The simple Mk model provides a useful foundation for a number of innovative methods. These methods capture evolutionary processes that are more complicated than the original model, including models that vary through time or across clades. Modeling more than one discrete character at a time allows us to test for the correlated evolution of discrete characters.## 10: Introduction to Birth-Death Models

This chapter introduces birth-death models and summarized their basic mathematical properties. Birth-death models predict patterns of species diversity over time intervals, and can also be used to model the growth of phylogenetic trees. We can visualize these patterns by measuring tree balance and creating lineage-through-time (LTT) plots.## 11: Fitting Birth-Death Models

In this chapter, I describe how to estimate parameters from birth-death models using data on species diversity and ages, and how to use patterns of tree balance to test hypotheses about changing birth and death rates. I also describe how to calculate the likelihood for birth-death models on trees, which leads directly to both ML and Bayesian methods for estimating birth and death rates.## 12: Beyond Birth-Death models

This chapter discusses models that go beyond constant rate birth-death models. We can fit models where speciation rate varies across clades or through time (or both). In some cases, very different models predict the same pattern in phylogenetic trees, warranting some caution until direct fossil data can be incorporated. I also described a model of protracted speciation, where speciation takes some time to complete.## 13: Characters and Diversification Rates

Many evolutionary models postulate a link between species characteristics and speciation, extinction, or both. These hypotheses can be tested using state-dependent diversification models, which explicitly consider the possibility that species’ characters affect their diversification rates. State-dependent models as currently implemented have some potential problems, but there are methods to deal with these critiques.## 14: What have we learned from the trees?

Comparative methods occupy a central place in evolutionary biology. This is because phylogenies provide an accounting of the historical patterns of evolution and, in turn, give us a natural way to measure long-term evolutionary dynamics. The first phase of comparative methods was focused strongly on adaptation. As discussed in this book, we have now branched out into a wide number of new areas, including diversification, community ecology, quantitative genetics, and more.