Introduction to Biosystematics - Zool 575

Transcription

1 Introduction to Biosystematics Lecture Introduction to maximum likelihood - synopsis of how it works - likelihood of a single sequence - likelihood across a single branch - likelihood as branch length changes - likelihood of 1 site on a tree - likelihood of a tree 2. Mechanistic comparison with Parsimony Derek S. Sikes University of algary Zool 575 Statistical Phylogenetics - rew from mathematical, computer, evolutionary, numerical studies, not as much from systematics 1. Parsimony (avalli-sforza & Edwards, 196) tree with minimum changes preferred 2. (avalli-sforza & Edwards, 1964) tree that maximizes probability of the data is preferred Made available for DN based phylogenetics by Felsenstein in the early 1980s Optimality riteria - iven 2+ trees Maximum Parsimony he tree hypothesizing the fewest number of character state changes is the best he tree maximizing the probability of the observed data is best Parsimony & Parsimony & ladistics - Strict cladists typically use only parsimony methods & justify this choice on philosophical grounds eg it provides the least falsified hypothesis - Parsimony has also been interpreted as a fast approximation to maximum likelihood avalli-sforza and Edwards (1967:555), stated that parsimony s success is probably due to the closeness of the solution it gives to the projection of the maximum likelihood tree and parsimony certainly cannot be justified on the grounds that evolution proceeds according to some minimum principle. (ML) Slow incorporation into systematics 1. omputationally complex - limited early uses to small molecular datasets (<12 OUs) 2. Molecular only (until 2001). Slow development of good software 4. Slow development of objective means to choose process models Now we have more sophisticated software, faster computers, more realistic models, and means to choose among models ML is becoming a widely used method Joseph Felsenstein and David Swofford instrumental in developing software to use ML (ML) statistical model-fitting exercise Which model (process + tree [topology+branch lengths]) is most likely to have generated the observed data? Maximizing the probability of the data given a model L=Pr(D H) Select process model using I etc. select tree & model parameters using ML 1

2 (ML) Note: we could select our process model using ML instead of the I - just pick the model with the highest likelihood Why do we use the I? (or similar criterion like the LR [Likelihood Ratio est] or the BI [Bayesian Information riterion]) Recall the I rewards models for high likelihoods (ML) oal: o estimate the probability that we would observe a particular dataset, given a phylogenetic tree and some notion of how the evolutionary process worked over time ( ) Probability of given - is not the probability that the model or tree (hypothesis) is true - is the likelihood of seeing the data if we assume the model and tree are true hypothesis 1: coin is fair ( P heads = 0.5) hypothesis 2: coin is biased (P heads = 0.21) -lnl (hypo 1) = lnl (hypo 2) = But what about other values for (P heads)? (say 0.?) eg. oin flipping flips and 21 are heads hypothesis 1: coin is fair ( P heads = 0.5) hypothesis 2: coin is biased (P heads = 0.21) -lnl (hypo 1) = lnl (hypo 2) = lnl <- Estimator (MLE) for P heads (0.21) <- P heads ML says nothing about the likelihood of the hypothesis being true it simply allows us to select among many hypotheses based on which makes the data most likely eg. Loud sounds coming from your attic (data) Hypothesis 1: gremlins are bowling upstairs Hypothesis 2: gremlins are sleeping upstairs hypothesis 1 makes the data more likely than hypothesis 2 - but itself is highly improbable! - he data are constant - it is the models (hypotheses) that we are varying - he likelihood is usually very small so we take the natural logarithm (log-likelihood), often abbreviated lnl (or lnl) which is typically negative Eg. lnl = lnl = which is better? he larger value some programs report -lnl so the value is positive, or simply ignore the negative 2

3 Requires explicit process model of evolution Strength: assumptions are spelled out unlike Parsimony - which also requires a model but the model is implicit & only recently understood statistically (next lecture) - he hypotheses (the model) includes: 1. he process model (eg J69, R+) 2. he topology (tree). he branch lengths of (2n-) branches for n taxa - Must calculate MLEs for parameters of the process model (eg the base frequencies) and each of the branch lengths for a given topology - Recall the topologies are supplied by 1) a starting tree that is then 2) subjected to branch swapping (same method to get trees as Parsimony) - For every topology there are huge numbers of possible branch lengths per branch to evaluate - his is a LO of calculations! - single sequence Let s consider a single short sequence We don t need the rate matrix since nothing is changing! We only need a base composition model If we consider a sequence of length 2: Sequence: ga he probability of observing this sequence is the product of the probabilities of observing each character e.g! g = 0.4;! a =0.15 (for instance) Likelihood(D M) = 0.4 x 0.15 = 0.06 (likelihood of the data given the model) - single sequence Why we use log-likelihoods: ake a sequence with 894 sites, (: 272, :297, :95, :20) Using a model that allows an unequal base composition of! = [0.0425, 0.221, , ] he Likelihood would be = x x x Which is a INY number.. - single sequence Why we use log-likelihoods: So we convert to log-likelihoods, which we can then sum rather than multiply: lnl = 272ln(0.0425) + 297ln(0.221) + 95ln( ) + 20ln( ) lnl = Note, if we used the J69 model with its base composition of 25 % for each base, the lnl would be 894ln(0.25) = his is 6 log-likelihood units worse! - single branch Example process model Substitution probabilities (rate matrix) P = note rows sums to 1* Base composition:! = [0.1, 0.4, 0.2, 0.] *he rows of this matrix sum to 1 - meaning that for every nucleotide, we have covered all the possibilities of what might happen to it. he columns do not sum to anything in particular

4 Example model So this model says the probability of an changing to a is nd the probability of it staying the same is nd that makes up 10% of the bases in the data etc. - single branch With this we can calculate the likelihood of going from one sequence to another - single branch simple likelihood calculation lignment ccat ccgt he likelihood going from the first to the second sequence across a short branch would be site = (! c P c"c ) (! c P c"c ) (! a P a"g ) (! t P t"t ) (values are supplied by our model) = (0.4x0.98)(0.4x0.98)(0.1x)(0.x0.979) Likelihood (D M) = variable branch length But this is for a short branch - variable branch length Different branch lengths yield different likelihoods: Probability of change increases as branches get longer We know at least 1 change has happened so the branch length isn t 0.0 (substitutions per site) We also know that all the sites haven t changed so the branch length isn t 1.0 or higher We can use ML to estimate the branch length that makes the data most likely - variable branch length s the branch length increases the probability of a base changing increases (and the probability of it staying the same decreases - of course, they can t both increase!) For a branch times longer than our initial branch: - trees o determine the likelihood score of a tree: - Must have a topology - Find the MLEs for the parameters of the model, minimally this involves finding the MLEs for the 2n- branches of the tree P = he likelihood of each site is calculated taking into account all possible ancestral states (these are unknowns) - sum all possible likelihoods for each site 0.08 his yields a likelihood = (which is better than our short branch likelihood of ) he likelihood of the tree is the sum of all the individual site log-likelihoods (which we ve already done ) 4

5 - branch lengths his is key (1) ML estimates corrected branch lengths (corrected for unobserved changes) nd (2) uses these branch lengths to adjust the probabilities of change Branch lengths are estimated in units of expected number of changes per character he probability of change is a product of mutation rate (µ) and time (t) Branches can be long due to large t, or higher µ, or both (hard to tease apart t from µ) if all branches have same µ then the data fit a molecular clock (ultrametric) 1. he most parsimonious topology is often, but not always, the same as the maximum likelihood topology 2. Parsimony does not correct the data for unobserved changes and so parsimony branch lengths are typically underestimates of actual branch lengths - Parsimony branch lengths = the estimated number of changes that occurred & are mapped onto the tree after it has been found (not used during tree searching - only total tree length is used to select the best tree). Because Parsimony ignores branch length information during searches it is - much faster than ML - unable to use this information to help it find the most probable tree (some strict cladists argue that they do not use parsimony to find the most probable tree, they claim there is no connection between minimal tree length and probability of being correct, truth is unknowable ) - especially unable to use branch lengths (~ rates of change) to detect areas of the tree that are likely to experience higher rates of homoplasy than other regions 4. Parsimony minimizes homoplasy - ML will sometimes prefer a tree with more than the minimum amount of homoplasy (if it makes the data more likely given the model) - gain, it is branch lengths which indicate where homoplasy may be common in the tree - longer branches are more likely to be involved in homoplastic events than shorter ones [Note: Parsimony is sometimes called Maximum Parsimony and abbreviated MP in contrast to ML] ML Parsimony would never prefer the correct, but longer tree on the left whereas ML would (more on this in next lecture) - Parsimony also ignores autapomorphies D B autapomorphies B 1 2 D MP 2 convergence events 1 convergence event hus we expect MP to yield the same tree as ML when branch lengths are similar across the tree (when there is little branch-length heterogeneity) ontrary to some statements to the opposite, it doesn t matter how much change has occurred (some have said that MP will only work if there has been little change - low rates of evolution) his is wrong, MP can do fine with high rates of change as long as there is little branch length heterogeneity - and conversely, even with INY amounts of change MP can fail if branches are of significantly different lengths 5

6 So it is not the total amount of change in the tree that matters - it is how evenly it is distributed (recall that amount of change is related to length of branches) aveat - as you will read (Siddall 1998; Swofford et al 2001) the failure of MP with high branch-length heterogeneity depends on where the longer branches are in the tree - more on this in the next lecture It s all about the branch lengths Recall - Statistical descriptions of MP indicate that this method does not assume a common mechanism, ie does not assume a common set of branch lengths for all characters - this means that MP does not assume a shared probability of change for different states, ie each character can have its own probability of change for every branch - this is a strong assumption - the question is Do the data fit this model? - Instead of one rate matrix to estimate from the dataset MP has to estimate a separate rate matrix for every character by branch combination! - If every unique substitution probability (minus 1) is a parameter, then MP requires the estimation of a HUE number of parameters - his is like assigning a separate J69 model for each character in the dataset - and thus using only a single character to estimate the parameters needed (more on this later) Parsimony Branch Lengths Parsimony will help find the shortest tree but even on this shortest tree there are often many different ways the character evolution can be mapped on the tree 2 common, but arbitrary, ways to map characters using parsimony: RN - accelerated transformation - when alternative mappings are possible this causes characters to change closer to the root and later reverse (prefers reversals) DELRN - delayed transformation - when alternative mappings are possible this causes characters to change further from the root (prefers parallelisms) Parsimony Branch Lengths haracter mapping is directly related to the estimated branch length Note: RN and DELRN have nothing to do with finding the shortest tree - they only relate to how the character evolution is mapped on the tree lso, these are extremes across a continuum of options Unlike ML there is no objective means of choosing which of many character mappings to prefer (since branch lengths are ignored by parsimony) More on this later - in lecture on character evolution erms - from lecture & readings Felsenstein maximum parsimony maximum likelihood L=Pr(D H) Model components Process model MLE log-likelihood branch lengths site log-likelihoods expected number of changes per character mutation rate time branch length heterogeneity RN DELRN 6

7 Study questions Parsimony can be considered a fast approximation to Maximum Likelihood - describe what Minimum Evolution (ME) is considered relative to parsimony and what NJ is considered relative to ME, and finally, how all these methods, NJ, ME, MP, ML are related to one another. Why was ML slow to be incorporated into common use by systematics and what has changed to make it so commonly used today? When we say L=Pr(D H) what does this mean? what are the subcomponents of H? If we use ML to obtain the MLEs of parameters of the model why don!t we just use ML to decide which model to use? Why do we use tools like the I instead of just the log-likelihood score? Study questions When we say given the model, what does given mean? If the likelihood is not the probability of the hypothesis being true, what is it? Why do we work with log-likelihoods instead of likelihoods? Provide the formula (but not the answer) to calculate the loglikelihood of a single sequence of 289 base pairs using the J69 model. When we consider longer branches what happens to the values in the substitution (rate) matrix? ontrast the diagonals with the off-diagonals. In words describe what we need to determine the log-likelihood of a tree. Study questions When would you expect parsimony to prefer a different tree than ML? nd why? What is more important to the performance of parsimony - the total amount of evolutionary change or the degree of branch length heterogeneity? Why? What are the statistical implications of the fact that MP does not assume all characters share a single set of branch lengths (ie all characters do not share a single rate matrix)? [hint: Would the I ever prefer MP over a common mechanism ML model?] 7