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diversitree-tutorial.Rnw
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\documentclass[11pt]{article}
\usepackage{fullpage}
\usepackage{natbib}
\usepackage{amsmath}
\usepackage{xcolor}
\definecolor{navy}{rgb}{0,0,0.4}
\usepackage[colorlinks,citecolor=navy,linkcolor=navy,urlcolor=navy]{hyperref}
\usepackage{dtsweave}
% Needed or Sweave adds it in:
% \usepackage{Sweave}
\SweaveOpts{keep.source=TRUE,eps=FALSE}
\SweaveOpts{prefix.string=cache/diversitree-tutorial}
<<results=hide,echo=FALSE>>=
library(cacheSweave)
## Remove stars, as they don't look nice when typeset
options(show.signif.stars=FALSE, continue=" ")
## This helps add real chi^2 numbers to the text
chi <- function(m1, m2, digits=1)
round(2*(m1$lnLik - m2$lnLik), digits)
@
%\VignetteIndexEntry{Introduction to diversitree}
%\VignettePackage{diversitree}
\title{Analysing diversification with diversitree}
\author{Rich FitzJohn, with GeoSSE by Emma Goldberg}
\date{25 March 2012, version 0.9-2}
\begin{document}
\maketitle
\tableofcontents
\clearpage
\section{Introduction}
The diversitree package includes a number of comparative
phylogenetic methods, mostly focusing on analysing diversification and
character evolution. These methods all share a common set of tools
for performing maximum likelihood (ML) parameter estimation,
hypothesis testing and model comparison, and Bayesian parameter
estimation through Markov chain Monte Carlo (MCMC). Included methods
include:
\begin{itemize}
\item Diversification
\begin{itemize}
\item Constant rate birth-death models \citep{Nee-1994-305}
\end{itemize}
\item Character evolution
\begin{itemize}
\item Discrete trait evolution \citep{Pagel-1994-37}
\item Univariate Brownian motion and Ornstein-Uhlenbeck models of
continuous trait evolution
\end{itemize}
\item Joint character evolution and diversification:
\begin{itemize}
\item BiSSE: Binary trait speciation and extinction
\citep{Maddison-2007-701}, and extensions for incompletely
resolved phylogenies \citep{FitzJohn-2009-595}.
\item MuSSE: Multiple State Speciation and Extinction.
\item QuaSSE: Quantitative State Speciation and Extinction
\citep{FitzJohn-2010-619}.
\end{itemize}
\item Joint character evolution and diversification, with
speciation-triggered changes in character states:
\begin{itemize}
\item GeoSSE: Geographic State Speciation and Extinction
\citep{Goldberg-2011-451}.
\item BiSSEness: BiSSE-Node Enhanced State Shift
\citep{Magnuson-Ford-2012}.
\item ClaSSE: Cladogenetic State Speciation and Extinction
\citep{Goldberg-classe}.
\end{itemize}
\end{itemize}
In addition, variants of these models are available:
\begin{itemize}
\item ``Time dependent'': different time epochs have different
parameters (implemented for BiSSE, MuSSE), and where rates are
arbitrary functions of time (implemented for birth death, BiSSE,
and MuSSE).
\item MEDUSA-style partitioned analyses, where different regions of
the tree have different parameters (implemented for birth-death,
BiSSE, MuSSE, QuaSSE, and GeoSSE).
\item Marginal ancestral state reconstruction for discrete characters
(``Pagel94''), BiSSE, and MuSSE.
\item Stochastic character mapping for discrete traits
\citep{Bollback-2006-88}
\end{itemize}
In the future, new methods will include
\begin{itemize}
\item Reflected Brownian motion for bounded traits
\item Stochastic character mapping for discrete traits that affect
speciation or extinction rates
\end{itemize}
For all methods, inference can be carried out under maximum
likelihood, or in Bayesian analyses via MCMC (Markov Chain Monte
Carlo). Phylogenies can also be simulated under most of the models.
This tutorial is designed to give an overview of the features in
diversitree. It does not aim to be a compete reference to the
package, or claim to always follow best practice. The manual follows
the structure above. Many of the examples are just taken from the
online documentation; further examples can be found there. Most are
fairly contrived -- if you have examples you would rather see here, I
would welcome data sets.
\subsection{Regenerating this file}
This file is written in ``Sweave'', which allows mixing R code and
\LaTeX\ markup. If you want to regenerate the file, or run the
empirical examples, you will need some data files, available at
\url{http://www.zoology.ubc.ca/prog/diversitree/files/}, which must be
present in the directory \code{data}. Some of the code chunks take a
very long time to run (total processing time is currently about 40
hours on a 2.8~GHz MacPro), and use the \code{cacheSweave} package to
speed subsequent runs up.
% This is no longer true -- see the makefile...
% With Gregor Gorjanc's
% \href{http://gregor.gorjanc.googlepages.com/Sweave.sh}{Sweave.sh}, the
% file can be easily regenerated with \code{Sweave.sh -c -ld
% diversitree-tutorial.Rnw}.
All code requires that the diversitree package be loaded.
<<>>=
library(diversitree)
@
\section{Constant-rate birth-death models}
The ape package already has some support for constant-rate
birth-death models, but diversitree duplicates this for
completeness. The major differences are (1) the function is not
constrained to positive diversification rates ($\mu$ can exceed
$\lambda$), (2) support for both random taxon sampling and unresolved
terminal clades (but see ape's \code{bd.ext}), and (3) run both MCMC
and MLE fits to birth death trees.
%
The constant rate birth death model is a special case of the other
diversification models implemented in the package, and is the simplest
model in diversitree.
Start with a simulated phylogeny, with speciation rate $\lambda=0.1$
and extinction rate $\mu=0.03$:
<<>>=
set.seed(2)
phy <- tree.bd(c(.1, .03), max.taxa=100)
@
%
(plotted in figure \ref{fig:bd-tree}). The first step in any analysis
in diversitree is to construct a likelihood function. For constant
rate birth death models, this is done with \code{make.bd}:
<<>>=
lik <- make.bd(phy)
@
To see the argument names of the likelihood function, use the
\code{argnames} function
<<>>=
argnames(lik)
@
This shows that \code{lik} takes a vector of two parameters $(\lambda,
\mu)$. It will return the log likelihood of the parameters, following
the calculations in \citep{Nee-1994-305}.
<<>>=
lik(c(.1, .03)) # -7.74086
@
%
Most likelihood functions accept additional arguments; these are
documented on their online help pages. The only additional argument
accepted for birth-death model is to disable conditioning on survival
(by default, the likelihood is conditional on two lineages surviving
to the present, following \citep{Nee-1994-305}.
<<>>=
lik(c(.1, .03), condition.surv=FALSE) # -10.74823
@
\begin{figure}
\centering
<<bd-tree,fig=TRUE,results=hide>>=
plot(phy, no.margin=TRUE, show.tip.label=FALSE)
@
\caption{A birth-death tree, with speciation rate $\lambda=0.1$
and extinction rate $\mu=0.03$, simulated until it has 100 species
(the actual simulation runs until the speciation event that would
create the 101st species, but does not add this species).}
\label{fig:bd-tree}
\end{figure}
To do a ML model fit, pass the likelihood function and a starting
point guess to the \code{find.mle} function:
<<>>=
fit <- find.mle(lik, c(.1, .03), method="subplex")
@
%
The final argument here selects the method ``\code{subplex}'' for the
ML search; other methods are available. (The default for birth-death
models is ``\code{nlm}'', which may produce warnings about failure to
converge for the example here.)
To extract the coefficients from the fitted object, use the
\code{coef} function:
<<>>=
coef(fit)
@
and to extract the log-likelihood value at the ML point, use the
\code{logLik} function
<<>>=
logLik(fit)
@
which extracts the coefficients with some additional information, or
extract the \code{lnLik} element from the list directly:
<<>>=
fit$lnLik
@
Does this model fit much better than a model without extinction (a
Yule, or pure birth, model)? You can constrain parameters of
likelihood functions using the \code{constrain} function. To specify
that the extinction rate, $\mu$ should be zero:
<<>>=
lik.yule <- constrain(lik, mu ~ 0)
@
%
Argument names here must match those given by \code{argnames}. Run
the ML search the same way as above, specifying a single starting
parameter (I've used the speciation rate from the full model here):
<<>>=
fit.yule <- find.mle(lik.yule, coef(fit)[1], method="subplex")
@
To perform a likelihood ratio test, use the \code{anova}
function\footnote{This is an unfortunate convention in R: many
packages use this function as a general model comparison function,
and I've taken their lead here -- in a future version, I may add a
\code{lrt} function, which should be clearer. No analysis of
variance is performed.}
The model with the nonzero extinction estimate is preferred, with
$\chi^2_1=\Sexpr{chi(fit, fit.yule)}$
<<>>=
anova(fit, yule=fit.yule)
@
Alternatively, we can use Markov chain Monte Carlo (MCMC) to perform a
Bayesian analysis. Here, I will use a uniform prior on the interval
$[0,\infty)$ for both parameters by not specifying any prior. The
\code{w} parameter is the tuning parameter. Here, it affects how many
function evaluations will be needed per sample, but will not generally
affect the rate of mixing (see the online help for \code{mcmc} for
more information).
<<bd-run-mcmc,cache=TRUE>>=
set.seed(1)
samples <- mcmc(lik, fit$par, nsteps=10000, w=.1, print.every=0)
@
The posterior distribution of these parameters, and the code to
generate it, is in figure \ref{fig:bd-mcmc-distr}.
\begin{figure}
\centering
<<bd-mcmc-distr,fig=TRUE>>=
samples$r <- samples$lambda - samples$mu
col <- c("#eaab00", "#004165", "#618e02")
profiles.plot(samples[c("lambda", "mu", "r")], col.line=col, las=1,
opacity=.75, legend.pos="topright")
abline(v=c(.1, .03, .07), col=col, lty=2)
@
\caption{Posterior probability distributions for the parameters of the
constant rate birth death model. True values are indicated by the
solid vertical lines. The bars at the bottom of the distributions
and the shaded areas correspond to the 95\% credibility
intervals. These include the two parameter $\lambda$ and $\mu$,
though the true diversification rate $r$ lies above the 95\%
credibility interval for that parameter.}
\label{fig:bd-mcmc-distr}
\end{figure}
Analyses can also use trees where only a fraction of species are
present in the phylogeny. To demonstrate this, let's drop 25 of the
100 species from the original tree at random:
<<>>=
set.seed(1)
phy.sub <- drop.tip(phy, sample(100, 25))
@
When constructing the likelihood function, pass an argument
\code{sampling.f} in, with a value on $(0,1]$ representing the
fraction of species that are descended from the root node that are
included in the phylogeny (here, $75/100$). Then, run a ML analysis
with \code{find.mle} as before:
<<>>=
lik.sub <- make.bd(phy.sub, sampling.f=75/100)
fit.sub <- find.mle(lik.sub, c(.1, .03), method="subplex")
coef(fit.sub)
@
With fewer included species, test to see whether the full model is
still preferred over the Yule model:
<<>>=
lik.sub.yule <- constrain(lik.sub, mu ~ 0)
fit.sub.yule <- find.mle(lik.sub.yule, coef(fit.sub)[1],
method="subplex")
anova(fit.sub, yule=fit.sub.yule)
@
In this small tree, the support for the model with extinction is
only
$\chi^2_1=\Sexpr{chi(fit.sub, fit.sub.yule)}$
(compared with
$\chi^2_1=\Sexpr{chi(fit, fit.yule)}$ when we had all
species), which is no longer significant at the 5\% level.
\clearpage % Force figure to appear.
\section{Markov models of discrete character evolution}
As with the birth death models above, diversitree supports both
simulating discrete characters and estimating rates of character
evolution. The implemented methods are a little idiosyncratic, as they
are primarily here for completeness (representing special cases of the
joint diversification-character evolution models), but are also useful
in their own right.
First, we will simulate a binary trait on a birth death tree. The
tree above with $100$ tips is a bit unwieldy to visualise, so we'll
make a smaller $50$ tip tree:
<<>>=
set.seed(2)
phy <- tree.bd(c(.1, .03), max.taxa=50)
@
% Dammit - this is broken apparently...
Then, on this tree, simulate a binary character where the rate of
transition from state $0$ to state $1$ is $0.1$, and the reverse rate
is $0.2$. We'll start the tree in state $0$ (specified by \code{x0}).
The argument \code{model="mk2"} specifies that we will use a Mk2
model:
<<>>=
set.seed(1)
states <- sim.character(phy, c(.1, .2), x0=0, model="mk2")
@
The simulation remembers the history at nodes too, which is displayed
in figure \ref{fig:mk2-tree} (eventually full history will be
recorded).
\begin{figure}
\centering
<<mk2-tree,fig=TRUE>>=
plot(phy, show.tip.label=FALSE, no.margin=TRUE)
col <- c("#004165", "#eaab00")
tiplabels(col=col[states+1], pch=19, adj=1)
nodelabels(col=col[attr(states, "node.state")+1], pch=19)
@
\caption{Character history for simulated trait and tree. Blue is
state $0$, yellow is state $1$.}
\label{fig:mk2-tree}
\end{figure}
Next, build a likelihood function with the \code{make.mk2} function,
and run a ML analysis with \code{find.mle}, using an initial parameter
guess of $(0.1, 0.1)$:
<<>>=
lik.mk2 <- make.mk2(phy, states)
fit.mk2 <- find.mle(lik.mk2, c(.1, .1), method="subplex")
coef(fit.mk2)
@
In the fit, the $q_{10}$ parameter is higher than $q_{01}$, the
difference being a little larger than in the true model.
See if this difference is statistically justified by running a model
where the two $q$ values are constrained to be equal:
<<>>=
lik.mk1 <- constrain(lik.mk2, q10 ~ q01)
fit.mk1 <- find.mle(lik.mk1, .1, method="subplex")
anova(fit.mk2, mk1=fit.mk1)
@
This is significant ($\chi^2_1=\Sexpr{chi(fit.mk2, fit.mk1)}$) so we
can conclude that asymmetric model fits better.
\subsection{Drawing samples with MCMC}
It is possible to run an MCMC analysis. However, care should be taken
to choose priors carefully, as while $q_{10}/(q_{01} + q_{10})$ is
usually well characterised by the data, the overall rate $(q_{01} +
q_{10})$ is poorly defined. For small trees like this, essentially
infinite values of character evolution are consistent with the data,
with the tip states just drawn from the stationary distribution of the
process.
There are two supplied prior functions, but any function that takes a
vector of parameters and returns the log prior probability may be
used. First, consider an exponential prior with rate $10$, which
gives a mean of $1/10$, and assume the same prior distribution for
both parameters.
<<>>=
prior.exp <- make.prior.exponential(10)
@
To run the MCMC, we need to specify a starting point (again, I have
used $(.1, .1)$, but the ML point might be preferable). I have
discarded the first 500 samples (10\%), which is probably overkill for
this model, as the autocorrelation between samples is extremely small.
<<mk2-mcmc,cache=TRUE>>=
set.seed(1)
samples <- mcmc(lik.mk2, c(.1, .1), nsteps=5000, prior=prior.exp,
w=.1, print.every=0)
samples <- subset(samples, i > 500)
@
\begin{figure}
\centering
<<mk2-mcmc-distr,fig=TRUE>>=
col <- c("#004165", "#eaab00")
profiles.plot(samples[c("q01", "q10")], col.line=col, las=1,
legend.pos="topright")
abline(v=c(.1, .2), col=col)
@
\caption{Posterior probability distributions for the parameters of the
Mk2 model. True values are indicated by the solid vertical lines.}
\label{fig:mk2-mcmc}
\end{figure}
The marginal distributions of these parameters are shown in figure
\ref{fig:mk2-mcmc}, and overlap substantially. However, that is
because the distributions are correlated (increasing $q_{01}$ fits
best if $q_{10}$ is also increased). The $q_{01}$ parameter is
estimated to be greater than the $q_{10}$ parameter only a small
fraction of the time:
<<>>=
mean(samples$q01 > samples$q10)
@
% TODO: Meristic and multitrait.
\clearpage
\section{Binary traits and diversification: BiSSE}
\label{sec:bisse}
The BiSSE (Binary State Speciation and Extinction) model combines
the features of the constant-rates birth-death model with the
two-state Markov model. Again, start with a simulated tree. The
parameters here are in the order $\lambda_0$, $\lambda_1$, $\mu_0$,
$\mu_1$, $q_{01}$, $q_{10}$, so the parameters below correspond to an
asymmetry in the speciation rate where state $1$ speciates at twice
the rate as state $0$. All other parameters are equal between states.
<<>>=
pars <- c(0.1, 0.2, 0.03, 0.03, 0.01, 0.01)
set.seed(4)
phy <- tree.bisse(pars, max.t=30, x0=0)
states <- phy$tip.state
head(states)
@
This gives a 52 species tree, shown with its true history in figure
\ref{fig:bisse-tree}. The character states are now stored in the
\code{states} vector. This vector is named, so that each element can
be easily associated with a tip in the tree.
\begin{figure}
\centering
<<bisse-tree,fig=TRUE,results=hide>>=
par(mar=rep(0, 4))
col <- c("#004165", "#eaab00")
plot(history.from.sim.discrete(phy, 0:1), phy, col=col)
@
\caption{A BiSSE tree, with parameters $\lambda_0=0.1$,
$\lambda_1=0.2$, $\mu_0=\mu_1=0.03$, and $q_{01}=q_{10}=0.01$.
Blue is state $0$, yellow is state $1$.}
\label{fig:bisse-tree}
\end{figure}
The \code{make.bisse} takes as its first two arguments a tree and set
of character states (these are the only mandatory arguments):
<<>>=
lik <- make.bisse(phy, states)
lik(pars) # -159.71
@
To perform an ML search, we need a starting point. The
\code{starting.point.bisse} function produces a basic heuristic guess
of a sensible starting point, based on the character-independent
birth-death fit. There are no guarantees that this is at all close to
the ML point, or that the ML point can be reached from this point
while climbing uphill only (which most optimisers assume).
<<>>=
p <- starting.point.bisse(phy)
p
@
Start an ML search from this point (this may take some time)
<<bisse-ml,cache=TRUE>>=
fit <- find.mle(lik, p)
@
As above, the \code{fit.mle} object has an element \code{lnLik} with
the log-likelihood value
<<>>=
fit$lnLik
@
and coefficients may be extracted with \code{coef} (rounded for clarity):
<<>>=
round(coef(fit), 3)
@
Let's test the hypothesis that the speciation rates are different for
the different states. We can use \code{constrain} to enforce equal
speciation rates to be equal across character states:
<<>>=
lik.l <- constrain(lik, lambda1 ~ lambda0)
@
and then start the ML search again:
<<bisse-ml-constrained,cache=TRUE>>=
fit.l <- find.mle(lik.l, p[argnames(lik.l)])
fit.l$lnLik # -158.74
@
%
(the statement ``\code{p[argnames(lik.l)]}'' drops the $\lambda_1$
element from the starting parameter vector). This fit has quite
different parameters to the full model (compare $\mu_0$)
<<>>=
round(rbind(full=coef(fit), equal.l=coef(fit.l, TRUE)), 3)
@
(the \code{TRUE} argument forces \code{coef} to return values for
constrained parameters). However, the difference in fits is not
statistically supported, with
$\chi^2_1=\Sexpr{chi(fit, fit.l)}$
<<>>=
anova(fit, equal.l=fit.l)
@
\subsection{Analysis with MCMC}
Because we are fitting six parameters to a tree with only 52 species,
priors will be needed so that the posterior distribution is proper. I
will use an exponential prior with rate $1/(2r)$, where $r$ is the
character independent diversification rate:
<<>>=
prior <- make.prior.exponential(1 / (2 * (p[1] - p[3])))
@
The MCMC sampler in diversitree uses slice sampling
\citep{Neal-2003-705} for parameter updates. The ``step size''
(argument \code{w}) does not need to be carefully tuned as it does not
affect the rate of mixing -- just the number of function evaluations
per update. Ideally it will be on the same order as the width of the
``high probability region''. An easy way of setting this is to run a
short chain (say, 100 steps) and use the range of observed samples as
a measure of this.
<<cache=TRUE>>=
set.seed(1)
tmp <- mcmc(lik, fit$par, nsteps=100, prior=prior,
lower=0, w=rep(1, 6), print.every=0)
w <- diff(sapply(tmp[2:7], range))
w
@
Run the chain for 10,000 steps (this will take a while)
<<bisse-mcmc,cache=TRUE>>=
samples <- mcmc(lik, fit$par, nsteps=10000, w=w, lower=0, prior=prior,
print.every=0)
@
The marginal distributions for the two speciation rates are shown in
figure \ref{fig:bisse-mcmc}, which shows the 95\% credibility
intervals for $\lambda_0$ completely overlapping those for
$\lambda_1$.
\begin{figure}
\centering
<<bisse-mcmc-fig,fig=TRUE>>=
col <- c("#004165", "#eaab00")
profiles.plot(samples[c("lambda0", "lambda1")], col.line=col, las=1,
xlab="Speciation rate", legend="topright")
abline(v=c(.1, .2), col=col)
@
\caption{Posterior probability distributions for $\lambda_0$ and
$\lambda_1$ for a BiSSE model. True values are indicated by the
solid vertical lines. The bars at the bottom of the distributions
and the shaded areas correspond to the 95\% credibility intervals.}
\label{fig:bisse-mcmc}
\end{figure}
\subsection{Incomplete taxonomic sampling}
Not all phylogenies are complete, but the basic BiSSE calculations
assume that they are. If given tree contains a random sample of all
extant species, the calculations can be corrected. To demonstrate
this, we will generate a larger tree (150 taxa), and drop 50 taxa from
it. Here, I am using the same parameters as earlier.
<<>>=
pars <- c(0.1, 0.2, 0.03, 0.03, 0.01, 0.01)
set.seed(4)
phy <- tree.bisse(pars, max.taxa=150, x0=0)
states <- phy$tip.state
phy.s <- drop.tip(phy, setdiff(seq_len(150), sample(150, 50)))
states.s <- states[phy.s$tip.label]
@
Calculate what the sampling fraction is for this tree. You can either
assume that the sampling fraction is independent of the character state:
<<>>=
sampling.f <- 50 / 150
@
or you can assume that it varies with character state
<<>>=
sampling.f <- as.numeric(table(states.s) / table(states))
sampling.f
@
Pass this in to \code{make.bisse} and construct a new likelihood
function that accounts for the sampling:
<<>>=
lik.s <- make.bisse(phy.s, phy.s$tip.state, sampling.f=sampling.f)
@
This can then be optimised, as before:
<<bisse-s-ml,cache=TRUE>>=
p <- starting.point.bisse(phy)
fit.s <- find.mle(lik.s, p)
fit.s[1:2]
@
\subsection{Terminally unresolved trees}
Another way that phylogenies might be incompletely resolved is that
higher level relationships may be known (say, genera), but little or
nothing is known about species relationships within these groups.
This results in trees where some ``taxa'' represent a number of
species -- ``terminal clades''. There are a couple of different ways
that unresolved clade information may be specified. To demonstrate
this, I will use an example of sexual dimorphism in shorebirds; this
is the same example as in \citep{FitzJohn-2009-595}. The phylogeny is
a supertree constructed by \citet{Thomas-2004-28}, and the data on
sexual size dimorphism are derived from \citet{Lislevand-2007-1605}.
The required files can be downloaded from
\url{http://www.zoology.ubc.ca/prog/diversitree/files/}
Read in the phylogenetic tree
<<>>=
tree <- read.nexus("data/Thomas-tree.nex")
tree <- ladderize(tree)
@
The tree contains many polytomies; the original tree, and a simplified
tree with polytomies converted into clades are shown in figure
\ref{fig:shorebird-tree}.
\setkeys{Gin}{width=1.0\textwidth}
\begin{figure}[p]
\centering
<<shorebirds-tree,fig=TRUE,width=8,height=5,results=hide>>=
par(mar=rep(.5, 4), mfrow=c(1, 2))
plot(tree, show.tip.label=FALSE)
tree.clade <- clades.from.polytomies(tree)
plt <- plot(tree.clade, show.tip.label=FALSE, clade.fill="gray")
@
\caption{The shorebird supertree \citep{Thomas-2004-28}. On the left,
the original tree with all polytomies and 350 species. On the
right, the polytomies have been collapsed, to leave 135 tips, with
tips representing from 1 to 47 species.}
\label{fig:shorebird-tree}
\end{figure}
\setkeys{Gin}{width=0.8\textwidth}
The character states are stored as the size of the difference of mass
between sexes, divided by the mean across sexes
\citep[see][]{FitzJohn-2009-595}.
<<>>=
d <- read.csv("data/Lislevand-states.csv", as.is=TRUE)
states <- d$dimorph
names(states) <- d$species
states <- states[tree$tip.label]
names(states) <- tree$tip.label
head(states)
@
These will need converting to a binary character for use, for example
-- to convert this into a binary character where an absolute relative
difference of 10\% would be considered ``dimorphic'':
<<>>=
head((abs(states) > .1) + 0)
@
The simplest way of working with this tree is to use the
\code{clades.from.polytomies} function. This collapses all daughters
of any polytomy into a clade. (Be careful - if you have a polytomy at
the base of your tree, the entire tree will collapse into a single
clade!) This can be visualised with \code{plot} functions as normal
-- see \code{?plot.clade.tree} for more information. This tree can
then be passed into \code{make.bisse}, along with a plain vector of
state names. The catch is that every taxon still needs state
information, not just those at the tips. Our state vector
\code{states} includes states for all 350 species, so we are OK to use
this.
<<>>=
states.15 <- (abs(states) > 0.15) + 0
tree.clade <- clades.from.polytomies(tree)
lik <- make.bisse(tree.clade, states.15)
@
\setkeys{Gin}{width=1.0\textwidth}
\begin{figure}[p]
\centering
<<shorebirds-tree-with-traits,fig=TRUE,width=6.5,height=6.75,results=hide,echo=FALSE>>=
## Diversitree imports:
group.label.tip <- diversitree:::group.label.tip
filled.arcs <- diversitree:::filled.arcs
add.alpha <- diversitree:::add.alpha
## Determine family for each tip:
families <- read.csv("data/shorebird-families.csv", as.is=TRUE)
family <- families$family[match(tree.clade$tip.label, families$spp)]
## Summarise trait over clades, and make vector where 1 is grey, 0 is
## black, and values in between are darker or lighter.
nn <- sapply(tree.clade$clades, function(x)
tabulate(states.15[x]+1, 2))
p <- states.15[tree.clade$tip.label]
p[colnames(nn)] <- nn[2,] / colSums(nn)
cols <- add.alpha("black", (1 - p)*2/3 + 1/3)
names(cols) <- tree.clade$tip.label
w <- max(branching.times(tree)) / 30
plt <- plot(tree.clade, show.tip.label=FALSE, no.margin=TRUE,
transform=sqrt, clade.fill="gray", label.offset=33,
type="fan")
group.label.tip(plt, family, "black", "black", 1, 2*w, 3*w, cex=.8)
theta <- plt$xy$theta[seq_along(tree.clade$tip.label)]
dt <- (plt$n.taxa/2 - .1) / plt$n.spp * 2 * pi
filled.arcs(theta - dt, theta + dt, max(plt$xy$x) + w/2, w, cols)
@
\caption{Phylogenetic tree of the 350 species of
shorebirds (Charadriiformes) and measures of sexual dimorphism,
based on \citet{Thomas-2004-28}. Gray triangles indicate
unresolved clades, with the height of the triangle being
proportional to the square root of the number of species.
Character states at the 15\% threshold level are indicated at the
tips; gray indicates sexually dimorphic, black indicates sexually
monomorphic, and white indicates no data. For unresolved clades,
the degree of shading indicates the proportion of species in each
state. For clarity, only family or subfamily names are shown.}
\label{fig:shorebird-tree-with-traits}
\end{figure}
\setkeys{Gin}{width=0.8\textwidth}
A sensible starting point can still be computed with
\code{starting.point.bisse}; this takes into account the clade
information automatically.
<<>>=
p <- starting.point.bisse(tree.clade)
@
You can then perform a ML search or MCMC analysis as usual. However,
unresolved clades slow down the analysis considerably, and this will
take several minutes.
<<shorebirds-ml,cache=TRUE>>=
fit.full <- find.mle(lik, p)
@
In the full model, speciation rates for dimorphic species (state $1$)
are greater than those for monomorphic species, but character
transition rates away from dimorphism ($q_{10}$) are greater than the
reverse transition ($q_{10}$):
<<>>=
round(coef(fit.full), 3)
@
To test whether these differences are significant, we can use a
likelihood ratio test. First, construct the reduced models,
constraining $\lambda_1 \sim \lambda_0$ or $q_{10} \sim q_{01}$:
<<>>=
lik.l <- constrain(lik, lambda1 ~ lambda0)
lik.q <- constrain(lik, q10 ~ q01)
@
Then, rerun the
<<shorebirds-ml-constrain,cache=TRUE>>=
fit.l <- find.mle(lik.l, p[argnames(lik.l)])
fit.q <- find.mle(lik.q, p[argnames(lik.q)])
@
These constrained models are significantly worse fits than the full
model ($p \approx 0.04$ for both).
<<>>=
anova(fit.full, equal.l=fit.l, equal.q=fit.q)
@
The analysis can also be run with MCMC. Here, I am using an
exponential prior with rate $1 / (2r)$, as earlier.
<<>>=
prior <- make.prior.exponential(1 / (2*p[1]))
@
The MCMC itself takes a very long time to run (approximately
4--5~s/sample)
<<shorebirds-mcmc,eval=FALSE>>=
set.seed(1)
samples <- mcmc(lik, coef(fit.full), 10000, w=.3, prior=prior,
print.every=0)
@
<<shorebirds-mcmc-run,results=hide,echo=FALSE>>=
.save <- "cache/shorebirds-mcmc-samples.Rdata"
if ( file.exists(.save) ) {
load(.save)
} else {
<<shorebirds-mcmc>>
save(samples, file=.save)
}
@
\begin{figure}[p]
\centering
<<shorebird-mcmc-plot,fig=TRUE>>=
par(mar=c(4.1, 4.1, .5, .5))
col <- c("#004165", "#618e02")
## col <- c("#004165", "#eaab00")
profiles.plot(samples[c("lambda0", "lambda1")], col, las=1,
legend.pos="topright", xlab="Speciation rate estimate")
@
\caption{Posterior probability distributions for speciation rates
for non-dimorphoc species (blue) and dimorphic species (yellow),
estimated using MCMC.}
\label{fig:shorebird-mcmc-plot}
\end{figure}
Setting up the likelihood function above assumed that the tree being
used contained polytomies, and this is the source of the unresolved
clades. However, it is probably more common to have an ``exemplar''
tree, where the unresolved species were never included in the first
place. The tree \code{Thomas-tree-exemplar.nex} (which was derived
from the tree above) does not contain any reference the species that
are contained within the unresolved clades.
<<>>=
tree.ex <- read.nexus("data/Thomas-tree-exemplar.nex")
states.ex <- states.15[tree.ex$tip.label]
names(states.ex) <- tree.ex$tip.label
@
We need to define a \code{data.frame} with information about the
unresolved clades. The file \code{Thomas-unresolved.csv} contains the
information in the correct format:
<<>>=
unresolved <- read.csv("data/Thomas-unresolved.csv", as.is=TRUE)
head(unresolved)
@
All the columns here are required:
\begin{itemize}
\item \code{tip.label}: the tip label within the tree
\item \code{Nc}: the total number of species that the tip represents
\item \code{n0}: the number of species known to be in state $0$
\item \code{n1}: the number of species known to be in state $1$
\end{itemize}
(additional columns are fine and will be silently ignored). Note that
\code{Nc} can be greater than \code{n0 + n1}: this allows for species
with unknown state.
This \code{unresolved} object is passed into the \code{make.bisse}
function:
<<>>=
lik.ex <- make.bisse(tree.ex, states.ex, unresolved=unresolved)
@
This likelihood function should be identical to the one created above.
<<>>=
lik.ex(coef(fit.full))
lik(coef(fit.full))
@
\section{Multiple state characters and diversification: MuSSE}
MuSSE (Multiple State Speciation and Extinction) generalises the
BiSSE model to allow characters with more than two states.
Following \citet{Pagel-1994-37}, this can also allow for multiple
characters, each of which might be binary, by recoding the states.
To illustrate, we'll start with a simple simulated example with a
three-level state. The tree is simulated where character evolution is
only possible among neighbouring states (i.e., $1\to 3$ and $3\to 1$
transitions are disallowed). All other transitions are equal, and
both speciation and extinction rates increase as the character number
increases. For a three state case, the parameter vector is in the order
$(\lambda_1, \lambda_2, \lambda_3, \mu_1, \mu_2, \mu_3, q_{12}, q_{13},
q_{21}, q_{23}, q_{31}, q_{32})$. This order can be seen here (sorry
-- clunky at the moment)
<<>>=
diversitree:::default.argnames.musse(3)
@
(the order of the $q$ parameters is row-wise through the transition
rate matrix, skipping diagonal elements).
Simulate a 30 species tree, with the tree starting in state $1$.
<<>>=
pars <- c(.1, .15, .2, # lambda 1, 2, 3
.03, .045, .06, # mu 1, 2, 3
.05, 0, # q12, q13
.05, .05, # q21, q23
0, .05) # q31, q32
set.seed(2)
phy <- tree.musse(pars, 30, x0=1)
@
The tree and its real character history are shown in figure
\begin{figure}
\centering
<<fig=TRUE,results=hide>>=
col <- c("#eaab00", "#004165", "#618e02")
h <- history.from.sim.discrete(phy, 1:3)
plot(h, phy, cex=1, col=col, no.margin=TRUE, font=1)
@
\caption{Simulated MuSSE tree. Yellow is state 1, blue is state 2,
and green is state 3}
\label{fig:musse-tree}
\end{figure}
The states are numbered $1,2,3$, rather than $0,1$ in BiSSE.
<<>>=
states <- phy$tip.state
table(states)
@
Making a likelihood function is basically identical to BiSSE. The
third argument needs to be the number of states. In a future version
this will probably be \code{max(states)}, but there are some pitfalls
about this that I am still worried about.
<<>>=
lik <- make.musse(phy, states, 3)
@
The argument names here are in the same order as for the simulation.
Just adding one more state (compared with BiSSE) has moved us up to
10 parameters.
<<>>=
argnames(lik)
@
Rather than start with the full model, and constrain things, here I
will start with a very simple model and expand. This model has all
$\lambda_i$, $\mu_i$, and $q_i$ the same (except for $q_{13}$ and
$q_{31}$, which are still zero).
<<>>=
lik.base <- constrain(lik, lambda2 ~ lambda1, lambda3 ~ lambda1,
mu2 ~ mu1, mu3 ~ mu1,
q13 ~ 0, q21 ~ q12, q23 ~ q12, q31 ~ 0, q32 ~ q12)
argnames(lik.base)
@
Find the ML point for this model
<<musse-ml-base,cache=TRUE>>=
p <- starting.point.musse(phy, 3)
fit.base <- find.mle(lik.base, p[argnames(lik.base)])
@
Now, allow the speciation rates to vary
<<musse-ml-lambda,cache=TRUE>>=
lik.lambda <- constrain(lik, mu2 ~ mu1, mu3 ~ mu1,
q13 ~ 0, q21 ~ q12, q23 ~ q12, q31 ~ 0, q32 ~ q12)
fit.lambda <- find.mle(lik.lambda, p[argnames(lik.lambda)])
@
There is very little improvement here (this is a small tree)
<<>>=
anova(fit.base, free.lambda=fit.lambda)
@
% TODO: Include a multitrait case here -- possibly just the case from
% the diversitree paper.
\section{Quantitative traits and diversification: QuaSSE}
The QuaSSE method is by far the slowest method in diversitree.
This is because to compute the likelihood, we have to solve a system
of partial differential equations, which is substantially more
complicated than the system of ordinary differential equations or
simple algebraic expressions in the other methods. The basic
interface is very similar to the other methods, but there are more
options to control the behaviour of the integrator.
We'll start with a simulated tree. The tree simulation differs
slightly from the other methods, because there is no longer a
canonical argument list (speciation and extinction rates are arbitrary
functions of the character state).
%
Here is a set of functions; speciation rate is a sigmoidal function
which ranges from $0.1$ to $0.2$ with an inflection point at $x=0$,
extinction is constant at rate $0.03$, and the model of character
evolution is Brownian motion with diffusion parameter $0.025$.
<<>>=
lambda <- function(x) sigmoid.x(x, 0.1, 0.2, 0, 2.5)
mu <- function(x) constant.x(x, 0.03)
char <- make.brownian.with.drift(0, 0.025)
@
Simulate the tree:
<<cache=TRUE>>=
set.seed(1)
phy <- tree.quasse(c(lambda, mu, char), max.taxa=15, x0=0,
single.lineage=FALSE)
@
We need to specify the standard deviation for the states; here I will
just assume that all taxa have a state standard deviation of $1/200$
<<>>=
states <- phy$tip.state
states.sd <- 1/200
@
Then, build the likelihood as usual. The difference compared with
other models is that we have to specify the speciation and extinction
functions (here, \code{sigmoid.x} and \code{constant.x},
respectively). There are a number of other provided functions (see
\code{?constant.x} for a list), but any function that takes \code{x}
as the first argument may be used.
<<>>=
lik <- make.quasse(phy, states, states.sd, sigmoid.x, constant.x)