Analytical methods: Maximum Likelihood

Model-based methods

The maximum likelihood aproach shares a great deal with parsimony, but it uses a different optimality criterion.

In maximum likelihood, the tree that is most likely to have given rise to the observed data is considered best.

To determine this likelihood, it is necessary to use a probabilistic model of sequence (i.e., character-state) evolution.

Parsimony assumes that homoplasy (character-state reversal) is rare, but does not make use of explicit probabilistic models. With certain data types, particularly those with a limited number of character states, the assumption that homoplasy is rare can be shown to be violated.

Maximum Likelihood

What is Likelihood?

Probability refers to the chance of drawing things from a defined population

But many abstract concepts, like evolutionary trees, don't draw from a defined and finite population.

Thus probability per se can't apply.

Likelihood is proportional to probability, but the ratio of proportion is arbitrary.

L(H|R) -- the likelihood of a hypothesis "H" given a set of data "R" proportional to a conditional probability P(H|R).

The model relates the hypothesis (tree) to the data (sequences)

Calculating the likelihood of a tree

Given a tree with branch lengths

For each character, assign the appropriate character state to the tips of the tree

Sum the probabilities of all possible ancestral states

This gives a site likelihood, i.e., the likelihood for that character, or site within the sequence

This is the likelihood of observing those character states given the tree

To calculate these site likelihoods, it is necessary to know the probability of character-state change over time. We will discuss models of character-state evolution shortly.

This sum is a likelihood because all possible terminal character states are not considered; the calculations only reflect those that are present in the dataset.

Computing the site likelihoods efficiently relies on calculating conditional likelihoods for sub-trees

Start at a node where all of the descendant nodes are tips on the tree. Ignoring ambiguity and missing data for the time being, the tips will all have a known character-state, and consequently will have a likelihood of 1 for that character-state.

For that node, calculate the likelihood of each of the four character-states.

Assuming the branches are of finite length, even nodes with descendents that have the same character state will have a non-zero likelihood for each of the other possible character-states.

Recursively calculate the likelihood of the character-states for each of the nodes moving down the tree.

The tree likelihood is the product of all of the site likelihoods

This is the likelihood of the entire dataset given the tree

To make calculation easier, most implementations convert site likelihoods to logarithms, and then sum the log likelihoods.

This likelihood is very small; this reflects the fact that the observed sequence alignment is only one of many that could have resulted, given the tree.

Models of character-state change

Evolution of a two-state system

Assume character state changes are independent

The past states held by a given site (character) do not change the future states it can hold

This is a Markov model

The probability of character state change is a function of time and rate of change. This value (time x rate) is the branch length.

DNA sequence evolution

Jukes-Cantor

Four possible character states

All substitutions are equally likely

All nucleotides occur with equal frequency

Kimura two-parameter

Transition-transversion ratio

1. 	A	C	G	T
   A		Transversion	Transition	Transversion
   C	Transversion		Transversion	Transition
   G	Transition	Transversion		Transversion
   T	Transversion	Transition	Transversion

In the evolution of real sequences transitions are typically observed more often than transversions.

Example of a substitution probability matrix consistent with the K2P model.

	A	C	G	T
A	0.6	0.1	0.2	0.1
C	0.1	0.6	0.1	0.2
G	0.2	0.1	0.6	0.1
T	0.1	0.2	0.1	0.6

These values represent the probability of the corresponding event occurring within a unit of time, t.

The values in the diagonals are selected such that each row adds up to one. Each row has to add up to one because the substitution matrix takes into account all possible events within the model.

 

Felsenstein, Chapters 9 & 16

Hillis, Moritz & Mable, pp. 426-446

Felsenstein, J. 1978. Cases in which parsimony or compatibility methods will be positively misleading. Systematic Zoology 27: 401-410.

Li, W.-H. 1997. Molecular Evolution. Sinauer Associates, Inc., Sunderland, MA. Pp. 116-119.