scholarly journals A variable-rate quantitative trait evolution model using penalized-likelihood

PeerJ ◽  
2021 ◽  
Vol 9 ◽  
pp. e11997
Author(s):  
Liam J. Revell
Keyword(s):  
Brownian Motion ◽  
Quantitative Trait ◽  
Penalized Likelihood ◽  
Comparative Methods ◽  
Character Trait ◽  
Likelihood Method ◽  
Rate Variation ◽  
Phylogenetic Comparative Methods ◽  
Trait Evolution ◽  
Rate Of Evolution

In recent years it has become increasingly popular to use phylogenetic comparative methods to investigate heterogeneity in the rate or process of quantitative trait evolution across the branches or clades of a phylogenetic tree. Here, I present a new method for modeling variability in the rate of evolution of a continuously-valued character trait on a reconstructed phylogeny. The underlying model of evolution is stochastic diffusion (Brownian motion), but in which the instantaneous diffusion rate (σ2) also evolves by Brownian motion on a logarithmic scale. Unfortunately, it’s not possible to simultaneously estimate the rates of evolution along each edge of the tree and the rate of evolution of σ2 itself using Maximum Likelihood. As such, I propose a penalized-likelihood method in which the penalty term is equal to the log-transformed probability density of the rates under a Brownian model, multiplied by a ‘smoothing’ coefficient, λ, selected by the user. λ determines the magnitude of penalty that’s applied to rate variation between edges. Lower values of λ penalize rate variation relatively little; whereas larger λ values result in minimal rate variation among edges of the tree in the fitted model, eventually converging on a single value of σ2 for all of the branches of the tree. In addition to presenting this model here, I have also implemented it as part of my phytools R package in the function multirateBM. Using different values of the penalty coefficient, λ, I fit the model to simulated data with: Brownian rate variation among edges (the model assumption); uncorrelated rate variation; rate changes that occur in discrete places on the tree; and no rate variation at all among the branches of the phylogeny. I then compare the estimated values of σ2 to their known true values. In addition, I use the method to analyze a simple empirical dataset of body mass evolution in mammals. Finally, I discuss the relationship between the method of this article and other models from the phylogenetic comparative methods and finance literature, as well as some applications and limitations of the approach.

scholarly journals A variable-rate quantitative trait evolution model using penalized-likelihood

2021 ◽  
Author(s):  
Liam J. Revell
Keyword(s):  
Brownian Motion ◽  
Quantitative Trait ◽  
Penalized Likelihood ◽  
Simulated Data ◽  
Character Trait ◽  
Likelihood Method ◽  
Rate Variation ◽  
Trait Evolution ◽  
Penalty Term ◽  
Rate Of Evolution

In recent years it's become increasingly popular to use phylogenetic comparative methods to investigate heterogeneity in the rate or process of quantitative trait evolution across the branches or clades of a phylogenetic tree. Here, I present a new method for modeling variability in the rate of evolution of a continuously-valued character trait on a reconstructed phylogeny. The underlying model of evolution is stochastic diffusion (Brownian motion), but in which the instantaneous diffusion rate (σ2) also evolves by Brownian motion on a log-scale. Unfortunately, it's not possible to simultaneously estimate the rates of evolution along each edge of the tree and the rate of evolution of σ2 itself using Maximum Likelihood. As such, I propose a penalized-likelihood method in which the penalty term is equal to the log-transformed probability density of the rates under a Brownian model, multiplied by a 'smoothing' coefficient, λ, selected by the user. λ determines the magnitude of penalty that's applied to rate variation between edges. Lower values of λ penalize rate variation relatively little; whereas larger λ values result in minimal rate variation among edges of the tree in our fitted model, eventually converging on a single value of σ2 for all of the branches of the tree. In addition to presenting this model here, I've also implemented it as part of my phytools R package in the function multirateBM. Using different values of the penalty coefficient, λ, I fit the model to simulated data with: Brownian rate variation among edges (the model assumption); uncorrelated rate variation; rate changes that occur in discrete places on the tree; and no rate variation at all among the branches of the phylogeny. I then compare the estimated values of σ2 to their known true values. In addition, I use the method to analyze a simple empirical dataset of body mass evolution in mammals. Finally, I discuss some applications and limitations of the approach.

scholarly journals Beyond Brownian motion and the Ornstein-Uhlenbeck process: Stochastic diffusion models for the evolution of quantitative characters

2016 ◽  
Author(s):  
Simon Phillip Blomberg
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Comparative Methods ◽  
Diffusion Models ◽  
Mathematical Framework ◽  
Phylogenetic Comparative Methods ◽  
Uhlenbeck Process ◽  
Trait Evolution ◽  
Ornstein Uhlenbeck Process ◽  
Non Gaussian

AbstractGaussian processes such as Brownian motion and the Ornstein-Uhlenbeck process have been popular models for the evolution of quantitative traits and are widely used in phylogenetic comparative methods. However, they have drawbacks which limit their utility. Here I describe new, non-Gaussian stochastic differential equation (diffusion) models of quantitative trait evolution. I present general methods for deriving new diffusion models, and discuss possible schemes for fitting non-Gaussian evolutionary models to trait data. The theory of stochastic processes provides a mathematical framework for understanding the properties of current, new and future phylogenetic comparative methods. Attention to the mathematical details of models of trait evolution and diversification may help avoid some pitfalls when using stochastic processes to model macroevolution.

scholarly journals On the effect of asymmetrical trait inheritance on models of trait evolution

2019 ◽  
Author(s):  
Pablo Duchen ◽  
Michael L. Alfaro ◽  
Jonathan Rolland ◽  
Nicolas Salamin ◽  
Daniele Silvestro
Keyword(s):  
Quantitative Trait ◽  
Comparative Methods ◽  
Simulation Framework ◽  
Trait Evolution ◽  
Individual Level ◽  
Parameter Estimations ◽  
Wide Range ◽  
Biological Interpretation ◽  
Individual Based Simulation ◽  
The Individual

AbstractCurrent phylogenetic comparative methods modeling quantitative trait evolution generally assume that, during speciation, phenotypes are inherited identically between the two daughter species. This, however, neglects the fact that species consist of a set of individuals, each bearing its own trait value. Indeed, because descendent populations after speciation are samples of a parent population, we can expect their mean phenotypes to randomly differ from one another potentially generating a “jump” of mean phenotypes due to asymmetrical trait inheritance at cladogenesis. Here, we aim to clarify the effect of asymmetrical trait inheritance at speciation on macroevolutionary analyses, focusing on model testing and parameter estimation using some of the most common models of quantitative trait evolution. We developed an individual-based simulation framework in which the evolution of species phenotypes is determined by trait changes at the individual level accumulating across generations and cladogenesis occurs then by separation of subsets of the individuals into new lineages. Through simulations, we assess the magnitude of phenotypic jumps at cladogenesis under different modes of trait inheritance at speciation. We show that even small jumps can strongly alter both the results of model selection and parameter estimations, potentially affecting the biological interpretation of the estimated mode of evolution of a trait. Our results call for caution when interpreting analyses of trait evolution, while highlighting the importance of testing a wide range of alternative models. In the light of our findings, we propose that future methodological advances in comparative methods should more explicitly model the intra-specific variability around species mean phenotypes and how it is inherited at speciation.

scholarly journals Fast and Robust Inference of Phylogenetic Ornstein-Uhlenbeck Models Using Parallel Likelihood Calculation

2017 ◽  
Author(s):  
Venelin Mitov ◽  
Tanja Stadler
Keyword(s):  
Brownian Motion ◽  
Bayesian Inference ◽  
Mixed Model ◽  
False Positive Rate ◽  
Comparative Methods ◽  
Neutral Evolution ◽  
Stabilizing Selection ◽  
Full Potential ◽  
Mammal Species ◽  
Trait Evolution

AbstractPhylogenetic comparative methods have been used to model trait evolution, to test selection versus neutral hypotheses, to estimate optimal trait-values, and to quantify the rate of adaptation towards these optima. Several authors have proposed algorithms calculating the likelihood for trait evolution models, such as the Ornstein-Uhlenbeck (OU) process, in time proportional to the number of tips in the tree. Combined with gradient-based optimization, these algorithms enable maximum likelihood (ML) inference within seconds, even for trees exceeding 10,000 tips. Despite its useful statistical properties, ML has been criticised for being a point estimator prone to getting stuck in local optima. As an elegant alternative, Bayesian inference explores the entire information in the data and compares it to prior knowledge but, usually, runs in much longer time, even for small trees. Here, we propose an approach to use the full potential of ML and Bayesian inference, while keeping the runtime within minutes. Our approach combines (i) a new algorithm for parallel likelihood calculation; (ii) a previously published method for adaptive Metropolis sampling. In principle, the strategy of (i) and (ii) can be applied to any likelihood calculation on a tree which proceeds in a pruning-like fashion leading to enormous speed improvements. As a showcase, we implement the phylogenetic Ornstein-Uhlenbeck mixed model (POUMM) in the form of an easy-to-use and highly configurable R-package. In addition to the above-mentioned usage of comparative methods, the POUMM allows to estimate non-heritable variance and phylogenetic heritability. Using simulations and empirical data from 487 mammal species, we show that the POUMM is far more reliable in terms of unbiased estimates and false positive rate for stabilizing selection, compared to its alternative - the non-mixed Ornstein-Uhlenbeck model, which assumes a fully heritable and perfectly measurable trait. Further, our analysis reveals that the phylogenetic mixed model (PMM), which assumes neutral evolution (Brownian motion) can be a very unstable estimator of phylogenetic heritability, even if the Brownian motion assumption is only weakly violated. Our results prove the need for a simultaneous account for selection and non-heritable variance in phylogenetic evolutionary models and challenge stabilizing selection hypotheses stated in numerous macro-evolutionary studies.

scholarly journals Is horizontal transmission really a problem for phylogenetic comparative methods? A simulation study using continuous cultural traits

2010 ◽  
Vol 365 (1559) ◽  
pp. 3903-3912 ◽  
Author(s):  
Thomas E. Currie ◽  
Simon J. Greenhill ◽  
Ruth Mace
Keyword(s):  
Cultural Evolution ◽  
Horizontal Transmission ◽  
Comparative Methods ◽  
Phylogenetic Comparative Methods ◽  
Ancestral State ◽  
Regression Analyses ◽  
Trait Evolution ◽  
Phylogenetic Methods ◽  
Cultural Traits ◽  
Future Work

Phylogenetic comparative methods (PCMs) provide a potentially powerful toolkit for testing hypotheses about cultural evolution. Here, we build on previous simulation work to assess the effect horizontal transmission between cultures has on the ability of both phylogenetic and non-phylogenetic methods to make inferences about trait evolution. We found that the mode of horizontal transmission of traits has important consequences for both methods. Where traits were horizontally transmitted separately , PCMs accurately reported when trait evolution was not correlated even at the highest levels of horizontal transmission. By contrast, linear regression analyses often incorrectly concluded that traits were correlated. Where simulated trait evolution was not correlated and traits were horizontally transmitted as a pair , both methods inferred increased levels of positive correlation with increasing horizontal transmission. Where simulated trait evolution was correlated, increasing rates of separate horizontal transmission led to decreasing levels of inferred correlation for both methods, but increasing rates of paired horizontal transmission did not. Furthermore, the PCM was also able to make accurate inferences about the ancestral state of traits. These results suggest that under certain conditions, PCMs can be robust to the effects of horizontal transmission. We discuss ways that future work can investigate the mode and tempo of horizontal transmission of cultural traits.

scholarly journals On the Effect of Asymmetrical Trait Inheritance on Models of Trait Evolution

2020 ◽  
Author(s):  
Pablo Duchen ◽  
Michael L Alfaro ◽  
Jonathan Rolland ◽  
Nicolas Salamin ◽  
Daniele Silvestro
Keyword(s):  
Quantitative Trait ◽  
Intraspecific Variability ◽  
Comparative Methods ◽  
Trait Evolution ◽  
Individual Level ◽  
Parameter Estimations ◽  
Wide Range ◽  
Biological Interpretation ◽  
Individual Based Simulation ◽  
The Individual

Abstract Current phylogenetic comparative methods modeling quantitative trait evolution generally assume that, during speciation, phenotypes are inherited identically between the two daughter species. This, however, neglects the fact that species consist of a set of individuals, each bearing its own trait value. Indeed, because descendent populations after speciation are samples of a parent population, we can expect their mean phenotypes to randomly differ from one another potentially generating a “jump” of mean phenotypes due to asymmetrical trait inheritance at cladogenesis. Here, we aim to clarify the effect of asymmetrical trait inheritance at speciation on macroevolutionary analyses, focusing on model testing and parameter estimation using some of the most common models of quantitative trait evolution. We developed an individual-based simulation framework in which the evolution of phenotypes is determined by trait changes at the individual level accumulating across generations, and cladogenesis occurs then by separation of subsets of the individuals into new lineages. Through simulations, we assess the magnitude of phenotypic jumps at cladogenesis under different modes of trait inheritance at speciation. We show that even small jumps can strongly alter both the results of model selection and parameter estimations, potentially affecting the biological interpretation of the estimated mode of evolution of a trait. Our results call for caution when interpreting analyses of trait evolution, while highlighting the importance of testing a wide range of alternative models. In the light of our findings, we propose that future methodological advances in comparative methods should more explicitly model the intraspecific variability around species mean phenotypes and how it is inherited at speciation.

scholarly journals Trait Evolution with Jumps: Illusionary Normality

2017 ◽  
Author(s):  
Krzysztof Bartoszek
Keyword(s):  
Stochastic Process ◽  
Weak Convergence ◽  
Comparative Methods ◽  
The Other ◽  
Phylogenetic Comparative Methods ◽  
Evolutionary Models ◽  
Trait Evolution ◽  
Other Hand ◽  
Extinction Component

ABSTRACTPhylogenetic comparative methods for real-valued traits usually make use of stochastic process whose trajectories are continuous. This is despite biological intuition that evolution is rather punctuated than gradual. On the other hand, there has been a number of recent proposals of evolutionary models with jump components. However, as we are only beginning to understand the behaviour of branching Ornstein–Uhlenbeck (OU) processes the asymptotics of branching OU processes with jumps is an even greater unknown. In this work we build up on a previous study concerning OU with jumps evolution on a pure birth tree. We introduce an extinction component and explore via simulations, its effects on the weak convergence of such a process. We furthermore, also use this work to illustrate the simulation and graphic generation possibilities of the mvSLOUCH package.

scholarly journals Realistic scenarios of missing taxa in phylogenetic comparative methods and their effects on model selection and parameter estimation

PeerJ ◽  
2019 ◽  
Vol 7 ◽  
pp. e7917 ◽  
Author(s):  
Rafael S. Marcondes
Keyword(s):  
Parameter Estimation ◽  
Model Selection ◽  
Phylogenetic Signal ◽  
Comparative Methods ◽  
Phylogenetic Comparative Methods ◽  
Trait Evolution ◽  
Continuous Trait ◽  
Starting Point ◽  
Almost All ◽  
Future Work

Model-based analyses of continuous trait evolution enable rich evolutionary insight. These analyses require a phylogenetic tree and a vector of trait values for the tree’s terminal taxa, but rarely do a tree and dataset include all taxa within a clade. Because the probability that a taxon is included in a dataset depends on ecological traits that have phylogenetic signal, missing taxa in real datasets should be expected to be phylogenetically clumped or correlated to the modelled trait. I examined whether those types of missing taxa represent a problem for model selection and parameter estimation. I simulated univariate traits under a suite of Brownian Motion and Ornstein-Uhlenbeck models, and assessed the performance of model selection and parameter estimation under absent, random, clumped or correlated missing taxa. I found that those analyses perform well under almost all scenarios, including situations with very sparsely sampled phylogenies. The only notable biases I detected were in parameter estimation under a very high percentage (90%) of correlated missing taxa. My results offer a degree of reassurance for studies of continuous trait evolution with missing taxa, but the problem of missing taxa in phylogenetic comparative methods still demands much further investigation. The framework I have described here might provide a starting point for future work.

scholarly journals A Penalized Likelihood Method for Mapping Epistatic Quantitative Trait Loci With One-Dimensional Genome Searches

Genetics ◽  
2002 ◽  
Vol 162 (2) ◽  
pp. 951-960 ◽  
Author(s):  
Martin P Boer ◽  
Cajo J F ter Braak ◽  
Ritsert C Jansen
Keyword(s):  
Quantitative Trait Loci ◽  
Quantitative Trait ◽  
Statistical Tests ◽  
Penalized Likelihood ◽  
Likelihood Method ◽  
Likelihood Methods ◽  
One Dimensional ◽  
Statistical Framework ◽  
Multidimensional Search ◽  
Trait Loci

AbstractEpistasis is a common and important phenomenon, as indicated by results from a number of recent experiments. Unfortunately, the discovery of epistatic quantitative trait loci (QTL) is difficult since one must search for multiple QTL simultaneously in two or more dimensions. Such a multidimensional search necessitates many statistical tests, and a high statistical threshold must be adopted to avoid false positives. Furthermore, the large number of (interaction) parameters in comparison with the number of observations results in a serious danger of overfitting and overinterpretation of the data. In this article we present a new statistical framework for mapping epistasis in inbred line crosses. It is based on reducing the high dimensionality of the problem in two ways. First, epistatic QTL are mapped in a one-dimensional genome scan for high interactions between QTL and the genetic background. Second, the dimension of the search is bounded by penalized likelihood methods. We use simulated backcross data to illustrate the new approach.

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