Tree Shape

Ten to twenty years ago, or so, there was a great deal of literature published on tree shape, tree balance and tree asymmetry. The general questions of interest were the pattern of diversification, based on the argument that the shape of the tree would relate to particular patterns of speciation and extinction.

Since that time, tree shape has been less and less discussed while the use of likelihood estimation of diversification rate parameters has become popular. If (hypothetically) we can obtain estimates of p and q (the rates of branching and termination) and test for differences , is there anything else left for tree shape analysis to tell us?

Reconstructing The Var-Covar Matrix of Evolutionary Rates

Since the concepts of G-matrices and P-matrices were first introduced, there has been considerable interest in examining how these matrices change with evolution and how they may affect the pattern of trait evolution. One of the most focused on questions is how the correlation between traits might influence the direction of evolution, an idea espoused by Schluter (1996?) as evolution along 'lines of least resistance'. Hunt, 2008, used fossil data from deep-sea ostracods and a morphology-based phylogeny to test this hypothesis (and found it to be true).

Obviously, being able to test such a hypothesis using extant taxa would be very beneficial, which I see as the goal of Revell and Collar's 2009 paper. However, they find that there is a strict limit in both computational power and the number of parameters, as each new trait examined rapidly increases the number of parameters (due to the covariance parameters between traits). Their own dataset limits their analysis to examining only two traits.

So, two questions:
1) What other methods might be employed to get the rate matrix?
2) What if reconstructing the var-covar matrix is simply too costly to do in most of our analyses? Could this detrimentally affect our analyses by not taking into account the evolutionary correlation between traits?

evolutionary correlation between traits

After reading the Collar et al. 2009 paper I am still confused as to what they mean by evolutionary correlation. For example, they introduce the topic by giving examples of how evolutionary correlation can arise by drift or can constrain evolution, and how adaptive radiations can result from the decoupling of phenotypes, thus providing a greater occupation of morphospace.

Then they talk about Centrarchid fishes. The authors hypothesize that because Micropterus specialize on evasive prey they are likely to have a different rate matrix from other Centrarchids. They don't say whether it will be more correlated or less, although it seems to me that they are implying it will be more strongly correlated.

Enter the results. In their two-matrix model the evolutionary correlation in R1 (other fish) = 0.35, whereas the correlation in R2 (micropterus) = 0.801. This implies to me that the two traits are more strongly correlated in micropterus. However, in their discussion, the authors describe this result as...
"the strong, positive association between evolution of gape width and buccal length in Micropterus suggests that changes in buccal cavity size are occurring without modification of shape. In contrast, the low evolutionary correlation between the characters in the other centrarchid lineages reveals that evolution has resulted in modifications to both buccal cavity size and shape."

if change in one trait is occurring without change to the other, as in Micropterus, I feel like this indicates that the traits are less correlated. Maybe its simply the way they are phrasing this which is confusing.

Revell & Collar and eigenvectors

Revell & Collar (2009) show results from "tests of shared eigenstructure between evolutionary rate matrices" in table 4. I can understand why they would want to test models of equality, proportionality, and no shared structure between rate matrices. However, I may have missed their explanation of this in the paper but I do not have a grasp of the biological significance of forcing the rate matrices to share eigenvectors. Is this a constraint that one makes sense in biology? I understand why they wanted to show the data but does placing any restrictions on the two matrices make sense if you have already decided that they should be different?

ASR and Character State Frequency

Last week in class we talked about how the frequency of character states can impact the ancestral state reconstruction (ASR) when likelihood is used (while MP is immune to this problem). This impact was nicely demonstrated in the Buschbom & Barker (2006) paper we read.

If I understand things correctly, Pagel's likelihood methods to detect correlated evolution will also be affected by the frequency of character states. So my first question is: is this actually the case or am I misunderstanding things?

If it IS affected, what can be done to get around this problem? Buschbom & Barker (2006) added 'dummy' taxa to account for the missing taxa (they knew their relative phylogenetic position, and character state[s]). This seems like a good solution, but this isn't always possible if you are working with a large group (say a group of 10,000 species). Are there any alternatives? Can taxa instead be weighted somehow (in a way such that taxon 1 represents 50 species [all with the same character state], while taxon 2 represents 10 species [all with the same character state])?

It was just alarming to see how incomplete sampling can have such a drastic effect on ASR, and it seems few studies, especially at broad phylogenetic scales have complete sampling.

Pagel 1994

I found this paper very helpful for its lucid explanation of how a Markov model is used to test for character correlation. It also answered a question I had last week about how a likelihood ratio test is used – in this case to compare the likelihood of the model of dependent change, L (D), with the model of independence, L (I). How does an investigator know if the LR statistic is chi-squared distributed, that is, that the L (D) can be expressed as a special case of the L (I)? Also, how does including unresolved nodes in the analysis “reduce the power of the test to discriminate between the models of independence and dependence” (pg 42)? Lastly, how are choices regarding branch lengths typically made?

Maddison 1990

This paper highlights the fundmental difficulty of testing for character correlation on tree topologies with ambiguous ancestral state reconstructions, and helps me to appreciate the methods of estimating levels of uncertainty that have been developed since this paper was published. I thought the impact of coding characters as binary vs. multistate was an interesting part of the model discussion and underscored the importance and subjectivity of character coding.

An Old But Valid Complaint

So, here's a methodological issue that impacts on both discrete and continuous character reconstruction, one not raised by the papers we read. When an active directional trend exists in the true character evolution, ancestral state reconstruction will often be highly misleading (see references below). With some knowledge of taxa closer to the base node, we can control for this (such as fossil taxa). But obviously we don't have such taxa for many groups and for many of the characters we wish to understand better.

So, in those later cases, could we test for whether a trend may or may not be present in our character history of interest? How pervasive do we believe character trends to be, and should we be concerned if we cannot actually control for them?

Food for thought.

Refs:
Finarelli, J. A., and J. J. Flynn. 2006. Ancestral State Reconstruction of Body Size in the Caniformia (Carnivora, Mammalia): The Effects of Incorporating Data from the Fossil Record. Systematic Biology 55(2):301-313.

Oakley, T. H., and C. W. Cunningham. 2000. Independent Contrasts Succeed Where Ancestor Reconstruction Fails in a Known Bacteriophage Phylogeny. Evolution 54(2):397-405.

Webster, A. J., and A. Purvis. 2002. Testing the accuracy of methods for reconstructing ancestral states of continuous characters. Proceedings of the Royal Society of London. Series B: Biological Sciences 269(1487):143-149.

Huelsenbeck 2003 : Simulated Character Histories

When I read Huelsenbeck 2003, I was unsure why they were using the simulation-based approach to determining probable character histories. It seemed that running lots of simulated character histories and throwing out everything that did not have the observed character states on the terminal taxa would be computationally intensive. I knew there were ways of calculating the likelihood of ancestral character states, and I wondered why they were not doing that instead. Is it quantification of how frequently that state is present in consistant character histories is more readily interpretable than likelihood ratios in terms of how much confidence should be placed in that value? How does this compare to methods of generating confidence intervals?

Huelsenbeck 2003

Huelsenbeck et al. (2003) write that "the parsimony method has been the only method available for mapping characters onto a phylogenetic tree" (Discussion, p. 147). They then go on to cite many of the Schluter, Pagel etc. papers that have used Markov-chain methods in a maximum-likelihood context to reconstruct ancestral states and examine character correlation. At the time of publication (2003), were ML methods not already being used for mapping morphological characters? It seems to me that now, folks favor standard ML methods for character mapping over parsimony (that much is clear), but I have seen Huelsenbeck's et al. stochastic Bayesian approach less frequently. So, I am confused as to where this paper fits into the 'history' of development of phylogenetic analysis. The authors push it as a novel method for mapping morphological characters, but retrospectively it appears to me that the novel contribution of this paper is the ability to map changes along the branches of the tree, rather than the use of Markov models to examine morphological or other character evolution on a tree per se.

Nearest-Neighbor Statistic vs. (Pagel's) Lambda

On p. 474, Buschbom & Barker (2006) discuss the use of the nearest-neighbor statistic to test for the random phylogenetic distribution of sorediate and non-sorediate taxa in Porpidia (s.l.). Could (Pagel's) lambda have been used for this as well? Would one expect it to yield similar results?

How many MCs?

Pagel et al. (2004) disregard the utility of MCMCMC in phylogenetics because their unpublished experience is that it is not as effective as having completely separate MCMC chains. I have no experience with either method but I am curious if there is a general consensus that MCMCMC is less useful than just MCMC. Does anyone have experience or opinions in this area?

More from David Collar

For those of you still tantalized by the multi-rate BM models from previous weeks, Collar et al. has another paper out (today!), this time applying a similar methodology to dragon lizards. I haven't read it entirely yet, but a number of our issues with the paper, such as considering alternative phylogenies from a posterior Bayesian sample, AICc weights and a much simpler and understandable data transformation are all taken into account.

OU models are not considered at all to my shock, and there is a short argument about how BM models have been previously shown to adequetly fit adaptive evolution and that multi-rate BM models are the most realistic.

Collar, D. C., J. A. Schulte, B. C. O'Meara, and J. B. Losos. 2010. Habitat use affects morphological diversification in dragon lizards. Journal of Evolutionary Biology 23(5):1033-1049.

ABSTRACT

Habitat use may lead to variation in diversity among evolutionary lineages because habitats differ in the variety of ways they allow for species to make a living. Here, we show that structural habitats contribute to differential diversification of limb and body form in dragon lizards (Agamidae). Based on phylogenetic analysis and ancestral state reconstructions for 90 species, we find that multiple lineages have independently adopted each of four habitat use types: rock-dwelling, terrestriality, semi-arboreality and arboreality. Given these reconstructions, we fit models of evolution to species' morphological trait values and find that rock-dwelling and arboreality limit diversification relative to terrestriality and semi-arboreality. Models preferred by Akaike information criterion infer slower rates of size and shape evolution in lineages inferred to occupy rocks and trees, and model-averaged rate estimates are slowest for these habitat types. These results suggest that ground-dwelling facilitates ecomorphological differentiation and that use of trees or rocks impedes diversification.

Huelsenbeck et al. 2003 & Maddison & Maddison 2003

It seems the stochastic mapping procedure provides the means to estimate phylogenetic uncertainty in character reconstruction that parsimony, based on the reading of the MacClade chapter 4, does not provide (examining character reconstructions on numerous parsimonious trees can become impractical especially when dealing with polytomies). Huelsenbeck et al. make the point that parsimony works best at rates of evolution that are low and branch lengths are short, but could not the same be said for the stochastic method of mapping? MacClade does provide the means to weight characters using various transformations. To what extent does this improve the performance of parsimony at higher evolutionary rates? For example, how would the analysis of heterostyly/self incompatibility done with stochastic mapping compare with a parsimony analysis using the dollo transformation on self incompatibility (given that self incompatibility is thought to evolve slowly, but lost quickly)? Unfortunately, the Pontederiaceae data was not included in figure 19 of the Huelsenbeck paper. The outgroup analysis in MacClade seems straightforward, however it was unclear to me from the other readings how outgroup data was integrated into the alternative character evolution models.

Pagel 1999

Pg 613: What does it mean to measure branch lengths in units of the "opportunity for selection"?

comment

I have been having trouble getting my comments to post--they seem to just disappear into outer space. So this is in response to my original thread, Adaptive Radiations and OU:

Dave,
I agree, paramaterization would be tough. One thing would be to necessitate in the model that each taxon must differ from its sister by a certain quantitative amount, at the tips and through time. The lineages would be playing bumper cars through time. The amount of required difference between taxa could vary among models, and one could choose among the best one and compare to BM in which taxa are allowed to overlap completely. This assumes that sisters overlap in geography. I have no clue how hard this would be to implement--it all sounds hard to me!

Adaptation important to PCMs?

Is "correcting for patterns of maladaptation in related species" tantamount to ruling out the influence of non-adaptive traits on the observed phylogenetic structure? I may be misunderstanding, but this seems to be excluding the possibility that a trait evolved NOT through direct selection. i.e., A trait could also show up in a population because it is linked to a gene that IS under selection, but not for the trait in question (pleiotropy) -- right?

I guess I'm confused about why the adaptiveness / maladaptiveness of a trait is important to the use of comparative methods. A given trait could go from adaptive to nonadaptive in an afternoon given an appropriately sized environmental change. I thought the main importance of PCMs is to account for evolutionary relatedness in species comparisons, not necessarily the role of adaptation..?

Most of the above is directed toward Hansen et al. 2008

Incorporating topological and branch length uncertainty into OU model comparison

If I understand Collar et al. (2009) correctly, they took a single tree from Near et al. (2005), and performed 500 character state simulations on this tree in SIMMAP. They then took these 500 reconstructed character histories and calculated AICc scores for the OU/BM models for each of these 500 character histories. I think this is great, but feel the next step should be to not just use a single tree (perhaps the best tree from their BEAST analysis?), but instead take all sampled post-burnin trees from a BEAST analysis. This will allow for uncertainty in topology and branch length estimates to be incorporated in the OU/BM model selection procedure. Are people doing this already? If not, Any ideas how difficult this would be to code and if an analysis such as this is computationally feasible?

Regression slopes and allometry

Hansen et al. 2008 left me feeling somewhat unsure about the possibility of estimating the optimal regression slope from comparative data. Although their method partially accounts for maladaptation, their simulations suggest that slopes will still often be underestimated (though they think this may be partly an effect of the topology of the primate tree they tested it on). I think that in many applications the value of the slope will not matter, but in others such as hypotheses about allometry it could be important. Is the timescale of adaptation fast enough that this isn't a problem? Are there better ways of calculating allometric slopes?

Adaptive radiations and OU?

Reading Butler and King (2004) led me to think more about the use of these models in testing for character displacement and/or character evolution in adaptive radiations. As I understand it (or think I understand it), the OU model allows optima to be designated a priori; different species on a tree can have different specified optima (corresponding to observed trait values), and different models can be tested, each with a different assumption about evolution towards a terminal character state in each lineage. Stabilizing selection towards this optimum is thought to be the main force drive character evolution. This is what they did for the Anolis example, by constraining terminal taxa to be either small or large in body size and comparing models in which ancestral taxa (internal nodes) were either intermediate or unspecified (Fig 5). However, I was wondering about the possibility of more explicitly including competition among the taxa in a model. That is, instead of designating optima a priori for each taxon, create a model in which the evolution of a character along a lineage depends not only on the incremental change through time along that lineage (BM), but depends also on the evolving state in neighboring lineages. In other words, each lineage is initially free to evolve towards any optimum, but through time is constrained by the directions the sister lineages take, via competition and/or the filling up of niche space. I guess this could be thought of as density-dependent character evolution, where stabilizing selection is a lesser force to interspecific competition and stochasticity in the model determines which lineages reach different trajectories first and consequently drive the trajectories of other lineages in the tree. I have no idea of this has been done before, how complicated it would be, or whether the OU models do actually allow for this type of analysis and I just didn’t get it. Thoughts?

Behavioral evolution and phylogenies

While reading Martins and Hansen (1997) I was reminded of something that was starting to concern me from the Blomberg et al. (2003) paper. The malleability of behaviors makes me question the validity of studying their evolution across phylogenies. Can we really say that a given species shows a certain behavior when individuals undoubtedly show different behaviors at different times, regardless of their evolutionary history? So are we actually just measuring a predisposition to perform a behavior? I am particularly concerned about this because I am very interested in the evolution of certain ant behaviors. Any thoughts?

Model weights and stochastic mapping

I'm writing in reference to the Collar et al. 2009 paper assigned for this week. In this paper they analyze rates of morphological evolution with regard to different feeding strategies, and test adaptive OU models (which should constrain evolution towards an optimum) versus brownian evolution with a single or multiple rates.

One thing that I questioned was their reporting of model support. Instead of using model weights, they only report -ln L and AIC scores. I think this is might be misleading in some cases. For example, they argue that the best model for evolution of PC4 was a brownian model with two rates, a slow rate for highly piscivorous lineages, and a faster rate for others. However, this model had only a moderately higher aic weight than the other when I calculated it from the reported AIC scores (w(BM2b)=0.38, w(BM1)=0.22). As you can see, this isn't strikingly high support. By looking at fig. 3, it appears that there is a lower rate in the highly piscivorous clade, and I'm not questioning the result, just stating that I think they could report the support of the different models in a more clear way.

Alternatively, they also reported model support with respect to the proportion of trees in which the preferred model received greatest support. In this case the BM2b model was favored in 95% of the stochastic reconstructions. I thought that this was cool, does anyone have experience using this SIMMAP program?

Martins & Hansen 1997; Butler & King 2004

Both papers provide explanations of comparative methods that I found very helpful. In Butler and King (p. 684) there is a discussion on how to improve a BM model by transforming the phylogeny. Apparently, the downside to this process is that transforming branch lengths will distort the phylogeny. Has anyone tried this approach? Specifically, how is the phylogeny distorted?

Also, I wonder how often the Hansen model is used and what the impact of non-clocklike phylogenetic behavior will have on the analysis. Because the Hansen model "requires phylogenetic branch lengths to be on a common timescale" (p. 686) how applicable is this model to the study of angiosperms (given phylogenies generally do not utilize mtDNA and are often non-clocklike)?

Hansen 1997

I have finished reading the Stabilizing Selection and the Comparative Analysis of Adaptation paper. My impression (as a beginner to comparative methods) is that this paper connects nicely with the Hunt 2007 paper that emphasizes the prevalence of the evolutionary stasis model - compared to those of directional change and random walks.
On page 1343 of the Hansen paper the author states: "the problem of inferring causation from correlation remains." How is this topic addressed using comparative methods? It seems the Hansen paper benefits from the utilization of the well known fossil record of horses, and knowledge of the historical selective regime and primary optimum.

Selectivity and Phylogenetic Independent Contrasts

Sometimes, we want to analyze selectivity during some rapid event, say recent climate change, in relation to a dataset of multiple taxa. We would be looking at whether these taxa have experienced some sort of selective event or not as a result: they might have undergone globally extinction, locally extinction, range shifts, decreased growth rate, etc. In our little example, we have reason to believe it is tied to some phenotypic variable (for example, body size). To measure the correlation between the selectivity and the phenotypic trait should the dataset be analyzed using independent contrasts or not? Is it necessary? Is it valid?

(Tangent: Is there such a thing as a use of independent contrasts that is not necessary but is valid? It seems all situations where independent contrasts could be applied are also situations in which they should be applied.)

I think that it is not a valid approach, because if we are interested in the instantaneous selection acting on the trait itself, then the actual determining factor is the actual phenotypic traits held at that time. But what do you guys think?

As I pointed out in class, this question is important for those interested in the fossil record because its the same exact situation when we want to understand selectivity during a mass extinction event. Analyses like Dave Jablonski's analysis of geographic range and extinction risk in K-T molluscs was performed on the raw variables. I've done similar analyses on morphological correlates of extinction risk for graptolites in the Late Ordovician, without taking into phylogenetic structure (not yet published). I'm not troubled by this, because I think the point of these analyses is to understand the causal relationship between the actual phenotypic trait and the occurrence of extinction, not the possible relationship between a potential unmeasured evolving trait that decides extinction risk and the phenotypic traits.

Matt Friedman thinks differently, and he gives a fairly good argument against my position in his paper, cited below. Note that PIC and raw values gave two very different results for his analysis: body size was a significant correlate until he corrected for phylogeny.

Friedman, M. 2009. Ecomorphological selectivity among marine teleost fishes during the end-Cretaceous extinction. Proceedings of the National Academy of Sciences 106(13):5218-5223.

http://www.pnas.org/content/106/13/5218.full

So, what do you guys think?

IC formulas

There was a question in class yesterday about how you calculate each of the information criteria. The formulas are standard and in Burnham and Anderson 2002, but I'll post them here so you can find them easily:

AIC = -2*lnL(model) + 2*K
AICc = -2*lnL(model) + 2*K*(n/(n-K-1))
BIC = -2*lnL(model) + K*ln(n)

and the weights are calculated using:

w_i = exp(-0.5 * Δi) / (Σ exp(-0.5 * Δr))

where Δi is the difference in information criterion value between model i and the best-fit model in the set you are considering(lowest information criterion), and Δr is the difference between each of models in the set you are considering and the best-fit model in that set. The denominator will be the same for all models in your set, so that all Σw_i = 1.0.

What's the point of the PCM blog?

This blog is intended primarily for students of CEB35300, a graduate seminar in phylogenetic comparative methods at University of Chicago. Students are required each week to post at least twice: one question and one response, two questions, or two responses. This is also an excellent venue to post any insights regarding methods, analysis approaches, code, books and web resources, etc. that would be of interest to the rest of the class.