An introduction to maximum likelihood

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An introduction to maximum likelihood
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What does parsimony assume?

• Traditional view Just character independence -
  with enough characters you should converge to the
  true phylogeny
• Felsenstein (1978) used a simple example to show
  that parsimony assumes more

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Four taxon case

• Two states 0 1
• Changes in each direction equally probable
• Probability of change of states on a branch P
  or Q
• Which data patterns favor the true tree?

A B C D 0 0 1 1 1 1 0 0
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How can the 1100 pattern arise?

• Change on branches AB (PQ) and either
• No change on the other three (1-Q)2(1-P)
• Change on the other three PQ2
• PQ(1-Q)2(1-P) Q2P

0
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What is the probability of those outcomes?

• Prob1100 PQ(1-Q)2(1-P) Q 2 P
• Prob0011 (1-P)(1-Q)Q(1-Q)(1-P)(1-Q)QP

0
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Consider the probability of data favoring the
tree (A,C)(B,D)

• Prob1010 P(1-Q)Q2(1-P) (1-Q)2P
• Prob0101 (1-P)QQ(1-Q)PQ(1-Q)(1-P)

0
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Probability of consistency

• Parsimony will be consistent if
• Prob1010 Prob0101 Prob1100 Prob0011
• If we assume Q is less than 0.5, consistency
  requires that P2 Q(1-Q)

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Consistency is not guaranteed
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Inconsistency

• When the model is inconsistent the tree gets
  worse as you add more data
• Long branch attraction (LBA)

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Possible responses

• It only applies to four taxa and two states
• Still applies to 4-state data
• Gets worse with more taxa
• Consistency is not so important
• Real data are not in the Felsenstein zone

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Maximum likelihood

• A general approach to estimating parameters in
  statistics
• Has many desirable statistical properties
• Felsenstein suggested it could be applied to
  phylogenetic inference and that it should avoid
  LBA

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The maximum likelihood criterion

• The best estimate of a parameter is the value
  that would be most likely to generate the
  observed data

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Application to phylogeny

• Assume a model of evolution
• Find the tree that would be most likely to give
  the observed data given the model
• Branch lengths are taken into account
• Uses all data (variant and invariant)

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An example (from Swofford et al. 1996)

• What can we say about the placement of another
  taxon with state C?

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An example (from Swofford et al. 1996)

• Parsimony the new taxon could attach in several
  places

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An example (from Swofford et al. 1996)

• ML - One place is favored
• State at ? most likely A

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An outline of the ML approachConsider one
character, i
(It is useful to arbitrarily root the tree)
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Sum across all possible histories for i
There are 4(n-2) arrangements for n taxa
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For each tree we calculate the likelihood of
getting the observed states L(i)
A
G
G
G
t2
t3
t4
t5
A
t1
A
L(i) PA x PA-A(t1)x PA-G(t1)x PA-G(t1)x
PA-A(t1)x PA-G(t1)
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Multiply across all sites (assume independence)
L will be very smalllnL will be a large negative
number
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Tree searching

• Search for the set of branch-lengths that
  maximize L ( lower -lnL score)
• Record that score
• Search for tree topologies with the best score

Time consuming
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Critical issues glossed over

• Where do we get Pn - the probability of state n
  at the arbitrary root node?
• Equiprobable (25)
• Empirical (frequency in the entire matrix)
• Estimated (optimized by ML on each tree)
• Where do we get Pi-j(t) - the probability of
  going from state i to state j in time t?

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Typical Simplifying Assumptions

• Stationarity
• Reversibility
• Site independence
• Markovian process (no memory)

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The simplest model of molecular evolution
Jukes-Cantor
Instantaneous rate matrix (Q-matrix)
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The simplest model of molecular evolution
Jukes-Cantor
Instantaneous rate matrix (Q-matrix)
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Calculating probabilities of change

• To convert the Q matrix into a matrix giving the
  probability of starting at state i and ending in
  state j, t time units later uses the formula

P(t) eQt
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The simplest model of molecular evolution
Jukes-Cantor
Substitution probability matrix (P-matrix)
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More complicated (realistic) models for DNA

• Allow deviation from equiprobable base
  frequencies
• HKY85 F81GTR
• Allow two substitution types (ti and tv)
• K2P HKY85
• Allow for six substitution types
• GTR

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Relationship among models
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Relationship between MP and ML

• One argument - MP is inherently nonparametric ?
  No direct comparison possible
• MP is an ML model that makes particular
  assumptions

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The Goldman (1990) model(see Lewis 1998 for more)

• We force all branch lengths to be equal
• The Likelihood for a character only considers the
  set of ancestral states that maximizes the
  likelihood

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Why use MP

• The model is clearly less realistic, but
• We can do more thorough searches and data
  exploration (computational efficiency)
• Robust results will usually still be supported

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Why use ML

• The model (assumptions) are explicit
• We can statistically compare alternative models
• We can conduct parametric statistical tests
  (under the assumption that we have used the
  correct model)
• But, even the most complex model is still
  unrealistically simple