Homoplasy. Selection of models of molecular evolution. Evolutionary correction. Saturation

Transcription

1 Homoplasy Selection of models of molecular evolution David Posada Homoplasy indicates identity not produced by descent from a common ancestor. Graduate class in Phylogenetics, Campus Agrário de Vairão, Vila do Conde, Portugal, 6 April 2006 David Posada University of Vigo, Spain, Saturation Evolutionary correction Models of substitution describe how sequences evolve through time and help correct for the saturation effect. Multiple hits lead to saturation at some point in time. David Posada University of Vigo, Spain, David Posada University of Vigo, Spain,

2 Models of nucleotide substitution Proportion of invariable sites If some sites do not change it is possible that fast evolving sequences show less divergence than slower sequences. b a c f d e Q A: rate = 0.005, p-inv = 0.2 B: rate = 0.02, p-inv = 0.5 P(t = e-qt David Posada University of Vigo, Spain, Rate variation among sites David Posada University of Vigo, Spain, Models of evolution A discrete gamma distribution with four categories is commonly used David Posada University of Vigo, Spain, David Posada University of Vigo, Spain,

3 Models are abstractions Models are important Models have a clear influence in Models asume independence of sites, stationarity, equilibrium frecuencies... All models are wrong, but indeed,...some are useful. - Box, 1976 Parameter estimation (ti/tv,!,p i,.. Topology estimation Phylogenetic confidence Hypothesis testing David Posada University of Vigo, Spain, David Posada University of Vigo, Spain, Effect of model in topology estimation: HIV pol Statistical tests of models Model adequacy is an absolute measure of the fit of a model to a particular data set. Model choice is a relative procedure to select the models that results in the best fit to a particular data set. We measure fit using the likelihood. A NJ K80 B NJ - GTR +! There are differences in the position of the subtype A. David Posada University of Vigo, Spain, David Posada University of Vigo, Spain,

4 Likelihood In phylogenetics see 18,19 we define the likelihood (L as (proportional to the probability of the data (D given a model of evolution (M, a vector of K model parameters! = (! 1,! 2,...,! K, a tree topology (", and a vector of S branch lengths,! = (! 1,! 2,...,! S : Statistical methods for model selection Hierarchical LRTs BIC Bayes factors AIC L = P(D M,!,",# David Posada University of Vigo, Spain, David Posada University of Vigo, Spain, Parsimony principle Likelihood ratio tests (LRT All model selection methods are somehow based on the principle of parsimony. LRT = 2 (! 1!! 0! 1 is the log of the maximized likelihood under the alternative hypothesis (complex model! 0 is the log of the maximized likelihood under the null hypothesis (simple model If models are nested, the LRT is distributed as a " 2 (or a mixed " 2, with q degrees of freedom, where q is the difference in number of free parameters between the two models 22 David Posada University of Vigo, Spain, David Posada University of Vigo, Spain,

5 Hierarchical LRTs David Posada University of Vigo, Spain,

6 Problems with hlrts Bayes factors 1 Maybe there is not an optimal model 2 The optimal model depends on the significance level 3 Local optima Bayes factors 23 are the Bayesian analogue of the LRT 24. B ij = P(D M i P(D M j Evidence for M 2 is considered very strong if B ij >150, strong if 12<B ij <150, positive if 3<B ij <12, barely worth mentioning if 1<B ij <3, and negative (supports M j if B ij <1 25. Used to infer the occurrence of recombination 26, to compare different phylogenetic hypothesis and for model selection 30 David Posada University of Vigo, Spain, David Posada University of Vigo, Spain, Posterior probabilities For multiple models P(M i D =! P(D M i P(M i R r =1 P(D M r P(M r Bayesian Information Criterion (BIC Simple approximation to the log marginal likelihood of a model BIC =!2! + K logn We generally use MCMC to approximate model likelihoods: harmonic mean of the likelihood. Given equal priors for all competing models, choosing the model with the smallest BIC is equivalent to selecting the model with the maximum posterior probability David Posada University of Vigo, Spain, David Posada University of Vigo, Spain,

7 Model selection uncertainty Model averaging We can establish a "95% credible set of models by summing the posterior probabilities from largest to smallest until the sum is just # 0.95 (Occam s window. We can make inferences based on the entire set of candidate models. the overall posterior mean of the shape of the gamma distribution (! would be: G E(! D = "! ˆ P(M D i i, i =1 David Posada University of Vigo, Spain, David Posada University of Vigo, Spain, Parameter importance The relative importance for the shape of the gamma distribution across all candidate models is: Kullback-Leibler distance ( = f (x log I f, g " f (x % ( $ ' dx # g(x! & R w + (! = " P(D M i P(M i I! (M i, i =1 where " I! (M i = 1 if! is in model M i # $ 0 otherwise I(f,g is the information lost when using g to approximate f David Posada University of Vigo, Spain, David Posada University of Vigo, Spain,

8 Akaike s information criterion (AIC The AIC estimates the expected K-L information. The AIC penalized the likelihood by the number of parameters (K AIC =!2! + 2K We prefer the model with smallest AIC. For small samples ($ n/k < 40: AIC c = AIC + 2K(K +1 n! K!1 Akaike differences AIC (% AIC difference (% i are rescaled AICs reescalated with the minimum AIC (% i =0 As a rule of thumb 31 :! i = AIC i " min AIC % i & 1-2 receive substantial support and are considered when making inferences 4 & % i & 7 have considerably less support % i > 10 receive no support David Posada University of Vigo, Spain, David Posada University of Vigo, Spain, Akaike weights (w i Akaike 32 suggested that the exp(!1/2" i approximate the relative likelihood of the models given the data L(M i D. Akaike weights (w i for R models: w i = # exp(!1/2" i R r =1 exp(!1/2" r Model selection uncertainty wity the AIC The Akaike weights are very useful for assessing model-selection uncertainty. We can establish a 95% confidence set of models for the best K-L model by summing the Akaike weights from largest to smallest until the sum is just # We can interpret them as the probability of a model is the best approximation of the true model, given the data. David Posada University of Vigo, Spain, David Posada University of Vigo, Spain,

9 Model averaging We can also make inferences based on the entire set of candidate models using the AIC Estimating phylogenies with 24 models! ˆ = " R i=1 w i I(M i! i w + (!, where R w + (! = " w i I! (M i, i=1 " I! (M i = 1 if! is in model M i # $ 0 otherwise. Also, w + (! estimates the relative importance of any parameter. David Posada University of Vigo, Spain, David Posada University of Vigo, Spain, Akaike weights Model AIC ' ( Cumulative ( GTR GTR+* SYM SYM+* HKY HKY+* K K80+* GTR+* SYM+* HKY+* F F81+* K80+* JC JC+* F81+* JC+* JC F K HKY SYM GTR Consensus tree for 24 models using Akaike weights David Posada University of Vigo, Spain, David Posada University of Vigo, Spain,

10 Parameter importance and model-averaged estimates Parameter Importance Model-averaged estimates fa fc fg ft TiTv rac rag rat rcg rct pinv(i alpha(g pinv(i+ig alpha(g+ig AIC and Bayes AIC approximates reality Bayesian approaches try to identify the most probable model If, as is usually the case, none of the models is correct, what sense can we make of prior or posterior odds? Bayesian priors What is exactly the sample size n? *for 56 models David Posada University of Vigo, Spain, David Posada University of Vigo, Spain, Model selection strategies Modeltest Good properties for model selection methods hlrt Bayesian AIC Applies easily to nonnested models No Yes Yes Allows for the simultaneous comparison of multiple models It does not depend on a subjective significance level No Yes Yes No Yes Yes Incorporates topology uncertainty No Yes* No Easy to compute Yes No* Yes Implements hlrt, AIC for 56 models. It will implement the BIC and tools for model selection uncertainty and model averaging. Google Any OS Assesses model selection uncertainty No Yes Yes So far, it needs PAUP* Allows model averaging No Yes Yes Provides the possibility of specifying prior information for models No Yes* Yes Web-server Provides the possibility of specifying prior information for model parameters Designed to approximate, rather than to identify, truth No Yes* No No No Yes *Not the BIC David Posada University of Vigo, Spain, David Posada University of Vigo, Spain,

11 Take home 1. Models are abstractions to learn about processes. 2. We can statistically select the best-fit model for our data. I recommend AIC or Bayes. 3. It is easy to implement using PAUP* and Modeltest. Model selection is a useful tool for research, but it is not a substitute for careful thinking and common sense reasoning - Browne, David Posada University of Vigo, Spain, David Posada University of Vigo, Spain, References 1. Felsenstein, J. Cases in which parsimony or compatibility methods will be positively misleading. Systematic Zoology 27, ( Huelsenbeck, J.P. & Hillis, D.M. Success of phylogenetic methods in the four-taxon case. Systematic Biology 42, ( Penny, D., Lockhart, P.J., Steel, M.A. & Hendy, M.D. The role of models in reconstructing evolutionary trees. in Models in Phylogenetic Reconstruction, Vol. 52 (eds. Scotland, R.W., Siebert, D.J. & Williams, D.M (Clarendon Press, Oxford, Bruno, W.J. & Halpern, A.L. 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