11: Fitting Birth-Death Models
- Page ID
- 21648
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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. In the next chapter, we will explore elaborations on birth-death models, and discuss models that go beyond constant-rates birth-death models to analyze the diversity of life on Earth.
- 11.1: Hotspots of Diversity
- Some parts of the tree of life have more species than others. This imbalance in diversity tells us that speciation is much more common in some lineages than others. Likewise, numerous studies have argued that certain habitats are "hotbeds" of speciation.
- 11.2: Clade Age and Diversity
- There are two different ways that one can measure the age of a clade: the stem age and the crown age. A clade's crown age is the age of the common ancestor of all species in the clade. By contrast, a clade's stem age measures the time that that clade descended from a common ancestor with its sister clade. The stem age of a clade is always at least as old, and usually older, than the crown age.
- 11.3: Tree Balance
- Birth-death trees have a certain amount of "balance," perhaps a bit less than your intuition might suggest. We can look to real trees to see if the amount of balance matches what we expect under birth-death models. A less balanced pattern in real trees would suggest that speciation and/or extinction rate vary among lineages more than we would expect. By contrast, more balanced trees would suggest more even and predictable diversification across the tree of life than expected under birth-death mo
- 11.4: Fitting birth-death models to branching times
- Most modern approaches to fitting birth-death models to phylogenetic trees use the intervals between speciation events on a tree - the "waiting times" between successive speciation - to estimate the parameters of birth-death models. Frequently, information about the pattern of species accumulation in a phylogenetic tree is summarized by a lineage-through-time (LTT) plot, which is a plot of the number of lineages in a tree against time
- 11.5: Sampling and birth-death models
- It is important to think about sampling when fitting birth-death models to phylogenetic trees. If any species are missing from your phylogenetic tree, they will lead to biased parameter estimates. This is because missing species are disproportionally likely to connect to the tree on short, rather than long, branches. If we randomly sample lineages from a tree, we will end up badly underestimating both speciation and extinction rates (and wrongly inferring slowdowns).