Maximum likelihood(cont.)

Midterm

• Relatedness – is not sharing a common ancestor but sharing a relatively recent CA

• Homoplasy includes convergence and/or reversal

• PTP looks at tree length; g1 looks at skew in tree length

• Both introgression and LGT result in a dominant and minor history

Midterm

• ILD test looks at the sum of the lengths (or likelihood) of the optimal trees from each partition

• Topology tests evaluate whether the data support one tree better than another. It can be used to evaluate support for a clade or to assess discordance, but that is an application

Midterm

• Parsimony criterion: Tree that can explain the data with the lowest number of character state changes (weighted by the inferred evidential value of each character state transition)

• Assumptions: Single; independence; branch lengths short and fairly even

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Incomplete lineage sorting

The cause is the retention of a

polymorphism – does not depend on gene

duplication

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A-B coalescen

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Relationship among models9

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Site-to-site rate heterogeneity• Two main methods:

– Some proportion of invariant sites

– Rates distributed as a discrete approximation to a gamma distribution

• Both use one parameter – I = proportion of invariant sites– α = shape parameter

Nested models

• Simpler models have fewer parameters than more complex models

• Two models are “nested” if all parameters in the simpler model are also in the more complex model– Nested: GTR-HKY; HKY-JC; GTR-JC; – Not nested: F81-K2P; JC+I-HKY

Which of these pairs are nested?

• HKY-K2P

• GTR+Γ-GTR

• HKY+I-HKY+Γ

• HKY+I-HKY

In the case of nested models

• Log-likelihood under the simpler model = Ls

• Log-likelihood under the complex model = Lc

• It will always be the case that Lc ≥ Ls

• But how much better can we explain the data under the more complex model?

Log-Likelihood ratios

• The likelihood ratio is 2(Lc-Ls)

• For nested models the expected LR is distributed as a Χ2 with as many degrees of freedom as the number of extra parameters in the more complex model (kc-ks)

Hierarchical LR tests

• If the LR is significant under a chi-square then favor the more complex model

• If the LR is not significant then stick with the simpler model

Akaike Information Criterion

• Another approach to choosing among models

• Can be used even among non-nested models

• Pick the model with the lowest AIC:– k = number of parameters in model– AIC = -2 ln L + 2k

Relationship between MP and ML

• One argument - MP is inherently nonparametric No direct comparison possible

• MP is an ML model that makes particular assumptions

Parsimony-like likelihood model(see Lewis 1998 for more)

• Estimate branch-length independently for each character (a VERY complex model)

• Only sum over maximum likelihood ancestral states

Why use MP

• The model is less realistic, but:– We can do more thorough searches and data

exploration (computational efficiency)– Robust results will usually still be supported

Why use ML

• The model is explicit

• We can statistically compare alternative models of molecular evolution

• We can conduct parametric statistical tests

Likelihood based topology test

• Kishino-Hasegawa test

• Likelihood ratio test of zero length branches