MULTISTATE - Mark Pagel

MULTISTATE - Mark Pagel
Introductory User’s Manual for
MULTISTATE
(copyright M. Pagel)
Mark Pagel
School of Animal and Microbial Sciences
University of Reading
Reading RG6 6AJ
UK
email: [email protected]
(www.ams.reading.ac.uk/zoology/pagel/)
MULTISTATE
Multistate is a computer program for the analysis of discrete characters on phylogenetic
trees. The program implements a continuous time Markov model and was initially
described in Pagel (1994). Several papers since then describe other features of my
related Discrete model (Pagel, 1997, 1999a,b).
This manual provides a very short introduction to using the Multistate method. Some
familiarity with my Discrete method will be helpful (a manual is available from for this
method: www.ams.reading.ac.uk/zoology/pagel/). The compressed folder contains
executable files to run on PC or Mac OSX computers. This version of Multistate allows
one to analyze variables with between 2 and 6 states, using an interactive command. A
control file and a test data file for a 4-state variable are also included.
Users who may wish to analyze a sample of trees rather than just one can use Multistate
in conjunction with our Bayesian-Multistate scripts also available on this website.
These scripts are intended to be used as a way of accounting for phylogenetic uncertainty
in comparative studies (see Lutzoni, Pagel, and Reeb, 2001; Pagel and Lutzoni, 2002).
The Bayesian-Multistate scripts along with Multistate make it possible to test
comparative hypotheses in each of the trees in a sample of trees and then combine those
tests across trees.
The approach that the Bayesian-Multistate scripts adopt documents the between-tree
variability in comparative parameters, that is the variation that arises from using different
phylogenetic trees to test the same hypothesis. For example, if Multistate were used to
estimate the ancestral state at a node, the Bayesian-Multistate scripts document how that
estimate varies from tree to tree. Another component of variability is that attributable to
the inherent uncertainty about the parameters of the model of trait evolution within any
given tree. Multistate finds the maximum likelihood (ML) values of these parameters,
but it is also possible to ask how the likelihood of the model of trait evolution varies
when the parameters are fixed at other than the ML values. The Bayesian-Multistate
scripts are not designed to investigate this within-tree variability. To do so, it is
necessary to vary the parameters of the model of trait evolution by hand from within
Multistate.
This manual describes how to use Multistate and some of its hypothesis testing
capabilities. It is intended to provide enough of an introduction to use the program,
although it does not attempt to describe all of the things for which one might use the
program.
Some of the main uses of the program are:
•
•
•
find ancestral states (see especially Pagel, 1999b),
test for rates of evolution,
detect directional trait evolution,
•
and, investigate the tempo and mode of trait evolution.
INTRODUCTION
Multistate implements a continuous-time Markov model in a maximum likelihood
framework. This makes it possible to analyse and test hypotheses about trait evolution
without the need ever to reconstruct ancestral states (although ancestral states can be
estimated). Instead, the parameters of trait evolution are estimated having summed the
likelihood over all possible states at each node of the tree (see Pagel, 1994 for further
explanation of the likelihood approach).
An advantage of the likelihood approach over other methods is that uncertainty in the
ancestral state reconstructions is automatically taken into account in all likelihood
calculations. By comparison, for example, parsimony methods first infer the ancestral
states and often treat them in later calculations as if they are known without error. This
lends a false degree of certainty to calculations and biases p-values. This bias is most
pronounced when traits evolve more than once on a tree, that is when trait evolution is
relatively rapid. Under such circumstances parsimony methods are known to
underestimate the amount of change, especially in long branches of the tree (Pagel, 1999a
gives an example).
Multistate proceeds by estimating the instantaneous rates of transitions among all
possible pairs of states. It then uses these to calculate the probabilities of change in a
branch of a given length. For a trait that can take only two values (e.g., 0,1), two rates
must be estimated, one for transition from 0!1, and the other for transitions from 1! 0.
These parameters are sufficient to characterise the evolution of traits in isolation from
one another. Six parameters are required for a trait with 3-states, and in general, n(n-1)
parameters are required for a character with n states. For three states, one estimates the
transitions from 0 to 1, 0 to 2, 1 to 0, 1 to 2, 2 to 0, and 2 to 1 (states are always denoted
0,1,…n-1). These transition rates can be shown to be sufficient to calculate the
probability of any kind of change in any branch of the tree, and they can be used to chart
the most probable course of evolution from the ancestral state to the contemporary
derived state.
Running Multistate
This version of the program runs on PC’s under the Windows operating system or on
Macintosh OSX systems. Users of pre-OSX Macintosh computers can run Multistate by
installing Connectix Virtual PC on their computers.
Data Input Format
Multistate requires a bifurcating and rooted phylogeny and data on species. It uses its
own input format (‘pag-format’) to describe the phylogenetic tree and the data. The
attraction of the tree format is its transparency, although it is not as compact as Phylip
(Newick) or Nexus formats. We make available software for converting from Nexus to
pag formats (www.ams.reading.ac.uk/zoology/pagel/).
The input format is simple. Consider the tree below of four species, and three internal
nodes, 1, 2, and 3 where node 3 is also the root. Trees must be rooted.
s1
s2
s3
s4
|
|
|
|
|t1
|t2
|
|
|__________|
|
|
|1
| t3
|
|t5
|
|t4
|_____________|
|
|2
|
|t6
|
|_____________|
|3
The ‘pag’ input format for this tree is:
41
# example phylogenetic tree. Comments can precede tree if
# preceded by "#" as in this line.
s1, n1, t1, data
s2,n1, t2, data
s3, n2, t3, data
s4,n3, t4,data
n1, n2, t5
n2, n3, t6
The first line in the file gives the number of species and number of traits. The “data” are
the comparative data measured across species. Multistate takes one trait (but see
Appendix for a special case exception to this). Colons should not be used in the data file.
Read the first line of the input tree as "species 1 goes to node "1" over ‘time’ or length t1
with data 1; the second line as species 2 goes to node 1 over time or length t2, with data
1", and so on. Data points must be real numbers. Missing data are not allowed. Species
with missing data must be removed from the tree.
Beginning with the fifth line of the file, the connections among the internal nodes are
described, "node 1 goes to node 2 over time or length 4, and node 2 goes to node 3 over
time or length t6". Nodes do not have data, and the branch lengths can be any real
number. Branch lengths can be any units but units of time and genetic distance
(operational time) are especially useful. If no branch length information is available, one
option is to assign them all an arbitrary length of 1.0 (although it should be borne in mind
that doing so implies that more total evolution has taken place between the root and the
tips of the tree for those species with more ancestors).
Only tips (species) have data, and the tree must be bifurcating and rooted. If you have
so-called polytomies in your tree, resolve them to bifurcations, or if this is not possible,
remove species until a bifurcating node remains. If the species that are removed all have
the same values on the traits, the analyses will not be affected in any substantial way.
Trees must be rooted, although the root itself is not described but inferred by the program
from the input format.
The species names can be any alphanumeric character but should start with a letter. They
must be one word. Internal nodes can be integers or alphanumeric characters (Note that
this input format is more flexible than that for my related program Continuous that
analyses quantitative comparative data. Users anticipating using both methods may wish
to have their input formats conform to that required for Continuous – see its manual).
It may often be easiest to number the species from left to right beginning with species s1
to species sn. Then label the nodes as n+1, n+2 and so on until you reach the root. Every
node in a bifurcating tree must have two and only two descendants. The root does not "go
to" any other node as is not described any further. Items are separated by commas.
At the moment, Multistate does not give very much useful information when it tries to
read in a treefile that is wrong in some respect. The most common errors are failing to
separate items by commas, or inserting more than one comma, having too few or too
many descendants of a node, specifying the wrong descendant, or incorrectly specifying
the number of species or variables. The best way to de-bug a treefile that is not working
is to print it out and compare it line by line to a picture of the phylogeny.
Running MULTISTATE cont’d
Once you have an input file, to launch Multistate on Windows, double click on the
executable and a window will appear asking for the name of the control file. On
Macintosh OSX, launch Multistate from the Terminal command-line. Operation is then
the same for both systems. Type ‘control’ (or whatever you have named it – an example
control file is given below). If you have specified the input file name in the control file,
it will read it in. Otherwise you will be prompted from the window for the name of the
input file. The input file must be a text file and in the same folder as the executable and
the control file. Once you’ve read in the input file name the program will ask for the
name of the results file. This is the file to which the contents of the active window will
be written, including all of the output from Multistate (the output also is printed on the
screen. The output frequently scrolls up off the screen so you may wish to set up
scrolling windows within Windows).
The program will then await your next command. Typing in ‘test’ fits the model to the
data, and returns a likelihood.
Restricting parameters
Parameter values of the transition rates can be restricted using the ‘restrict’ command
within the window (alternatively, restrictions can be placed in the control file and thus
can be saved from one session to the next). For a four state model, the parameter values
that can be restricted are (states are numbered 0-3):
q01
q10
q20
q30
q02
q12
q21
q31
q03
q13
q23
q32
The qii (e.g., q00, q11, q22, q33) are functions of the others in their row and cannot be
changed. In any given row, at least one of the transitions must be allowed to vary.
Parameters can be restricted to a constant or to each other.
‘restrict q10 = 0.0’ forces that parameter to be zero in the model;
‘restrict q10 = q12’ forces these two parameters to be equal to each other in the model
By comparing the likelihoods of restricted and unrestricted models one can test whether a
restriction has an effect (see below and also Discrete manual or Pagel, 1994).
Be sure to unrestrict parameters when making subsequent tests, by using the ‘clear’
command.
Common sets of restrictions
A 3-state model estimates 6 parameters, a 4-state model estimates 12 (shown above), 5states requires 20 parameters, and 6 states requires 30 (in general n2 – n). A good rule of
thumb is to limit the number of parameters so that there are ten data points per parameter.
Thus, one might want at least 300 species to estimate a 6-parameter model.
However, by using restrictions one can retain the flexibility of many states but reduce the
number of parameters. A very useful model to compare to the ‘full model’ (the one that
estimates all n2 – n parameters) is a simple ‘forward/backward’ model. This model is
created by restricting all of the forward parameters to be equal to each other, and the
same for all the backward ones. For example, for the 4-state model the following
restrictions set all of the forward transitions to each other:
‘restrict q01 = q02’
‘restrict q01 = q03’
‘restrict q01 = q12’
‘restrict q01 = q13’
‘restrict q01 = q23’
The same can be done for the backward transitions by setting all of them to say q10.
You may wish to make up special control files that contain a set of restrictions that you
will commonly use. This saves typing them all in again at the beginning of each new
session. An alternative to the forward/backward model is one in which you restrict the
parameters in each row to be equal to each other. This would be equivalent to saying that
each state has its own rate of moving either forward or backward. For a four-state model
this set of restrictions would create a 4-parameter model.
Commands available in the window
quit : terminates interactive session
help : displays this message
status : print information on current restrictions
scaling [0|1]
: set/unset branch length scaling
clear [<parameter>] : removes restrictions on parameters
restrict <parameter> = <value | parameter> : restrict a parameter
test : calculate likelihood
fossil [<state> <node>]: sets fossil at given node to given state
Scaling
The ‘scaling’ command sets or unsets the branch length scaling (see Pagel, 1994 or
Discrete manual). It is often good to fit this parameter at least once as it is often zero or
very near to zero for discrete data. Then in subsequent runs, the scaling parameter can be
restricted to its maximum likelihood value. To restrict it, type in ‘restrict scale = x.xxx,
where x.xx is the real number value to which you want to restrict it. This often greatly
improves the fit of the data to the model (compare the likelihoods with scaling turned on
and off).
Reconstructing Ancestral States: the Fossils command
The ‘fossil’ command is used to reconstruct ancestral states. It fixes the interior nodes of
the tree to given values. The synatx is fossil state node, where ‘state’ is only those values
that are legitimate in the model (so, 0,1,2,3 in a 4-state model), and node is the node
name corresponding to the node in the pag file. Having fixed a node, run the ‘test’
command to get the likelihood given the fixed node.
To determine the support for various states at a node, successively fix the node to its n
possible states and compare the likelihoods. The proportional support for a given value
at a node can be calculated from the following. Define ‘numerator’ as Exp[log-likelihood
for state i], where Exp is simply the base of the natural logarithm, e. That is, one raises e
to the power of the log-likelihood for state i. Most hand held calculators will do this.
Then define ‘denominator’ as the sum of Exp[log-likelihood for state i] for all n states.
That is, raise e to the power of the log-likelihood separately for each of the states and add
up the n results.
Finally, divide numerator by denominator to get the proportion of the likelihood
associated with state i. In general the support for one state over another at a given node is
treated as significant if the difference between their log-likelihoods is more than the value
2 (see Pagel, 1999).
Remember to ‘un-fix’ the fossil state at a node before going on to other calculations
(although it is acceptable to fix more than one node at a time). To do this type in “fossil
127 node”. For example, “fossil 127 N12” would un-fossil node N12.
As an alternative, if you wish to reconstruct a binary trait, see the ‘testall’ command
described in the Appendix (below). This will ‘fossil’ nodes and calculate probabilities
automatically using “global” and “local” approaches (Pagel, 1999b), for all nodes.
Testing Likelihoods
To compare two log-likelihoods simply subtract the smaller from the larger. The
absolute value of their difference is distributed as a chi-squared variate with degrees of
freedom equal to the difference in the number of parameters in the two likelihoods.
Thus, to test the full 12 parameter likelihood of the 4-state model to the likelihood
obtained from the forward/backward model, compare the difference in their loglikelihoods to a chi-squared distribution with 10 degrees of freedom. If the
forward/backward model is not significantly worse that the full model this provides an
argument for using it as a simpler representation of the data. Use a liberal alpha value for
this test, of perhaps 0.15 to 0.25, because if the two models differ you want to have
power to detect that.
To test whether the scaling parameter improves the likelihood, compare the likelihoods
with and without it in the model, and test against a chi-squared with 1 df.
Other tests follow this same logic, except for the tests of ancestral state reconstructions
(fossils). Here, the model is the same for all states at the node, and one uses the rule of
thumb that if the log-likelihoods differ by two or more, then one state is ‘significantly’
better supported at that site than the other.
*****************************
Example “control” file. It is best to leave untouched the ‘interactive’ and ‘partial’
commands. Commands with a # in front are turned off, removing the # swithces it on. A
set of restrict commands are shown, each one setting the value of a parameter to zero.
#datafile = test4
interactive = 2
#restrict
#restrict
#restrict
#restrict
#restrict
#restrict
q02= 0.0
q03= 0.0
q20= 0.0
q21= 0.0
q12=0.0
q13=0.0
#restrict q23=
#restrict q30=
#restrict q31=
#restrict q32=
#converge =
#number =
#step =
#rate =
partial= 1
0.0
0.0
0.0
0.0
************************************
References.
Please consider citing one or more of these as appropriate when you publish work using
Multistate.
Pagel, M. Detecting correlated evolution on phylogenies: a general method for the
comparative analysis of discrete characters. Proceedings of the Royal Society (B)
255 37-45 (1994).
Pagel, M. Inferring evolutionary processes from phylogenies. Zoologica Scripta 26, 331348 (1997).
Pagel, M. Inferring the historical patterns of biological evolution. Nature, 401, 877-884
(1999a)
Pagel, M. The maximum likelihood approach to reconstructing ancestral character states
of discrete characters on phylogenies. Systematic Biology, 48, 612-622 (1999b).
Lutzoni, F. ,Pagel, M., and Reeb, V. 2001. Major fungal lineages derived from lichensymbiotic ancestors. Nature, 411, 937-940.
Pagel, M. and Lutzoni, F. 2002. Accounting for phylogenetic uncertainty in comparative
studies of evolution and adaptation. Pp 148-161 in Biological Evolution and
Statistical Physics (M. Lässig, and A. Valleriani, eds). Berlin: Springer-Verlag.
Notes.
Multistate was adapted from the computer program Discrete, by Richard Forster, with
subsequent modifications by Peter Fredericks, Dr. Andrew Meade and Dr. Daniel Barker
Appendix: the testall command
Multistate’s ‘testall’ command causes Multistate to reconstruct the ancestral state of
every node in the tree. It is currently only implemented for traits that take two states, 0
and 1. Use of ‘testall’ with the Bayesian-Multistate scripts is not recommended.
The testall command must be issued from within Multistate (i.e., at the Multistate
prompt), and has syntax:
testall outputfilename
testall ignores any user-defined fossils but does respect user-defined restrictions on qij
and scale.
Intermediate calculations are written to the screen, and the probabilities of state 0 for
each trait and node are written to the file outputfilename. If the file already exists, it will
be overwritten. Since only two states are allowed, the probability of state 1 is simply
1 – (probability of state 0).
outputfilename is a tab-delimited text file. The first row contains column headings. The
first column gives the trait number, counting from 0. Subsequent columns give the
probability of state 0 at each node. For each trait, probabilities for the ‘global’ approach
are given, followed by probabilities for the ‘local’ approach (Pagel, 1999b). For terminal
nodes, probabilities are output as 0 or 1 (if the state is 1 or 0, respectively).
In addition to probabilities, outputfilename contains a summary of user-defined
restrictions in force during the testall command. These are given below the block of
probabilities. ‘none’ indicates no user-defined restrictions.
Note the use (or non-use) of scaling is not currently recorded by ‘testall’, and should be
recorded by the user.
One may continue to use Mulstistate interactively after a testall command, but any userdefined fossils are cleared by testall. They will have to be set again by the user.
Continued use of the program after testall has only been lightly tested.
If there is more than one trait, traits must be supplied as a comma-separated list. An
example data file with four species and three traits is:
43
s1, 1, 0.5, 1, 0, 0
s2, 1, 0.5, 0, 0, 0
s3, 2, 0.5, 1, 1, 1
s4, 3, 1.0, 0, 0, 1
1, 2, 1.0
2, 3, 1.0
‘test’, described in the main part of the manual, will only use the first trait, ignoring the
rest. ‘testall’ will use all traits.
Data files of incorrect format may have unpredictable effects.
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