Hostname: page-component-546b4f848f-w58md Total loading time: 0 Render date: 2023-06-02T06:20:59.427Z Has data issue: false Feature Flags: { "useRatesEcommerce": true } hasContentIssue false

Phylogenetic confidence intervals for the optimal trait value

Published online by Cambridge University Press:  30 March 2016

Krzysztof Bartoszek*
Affiliation:
Uppsala University
Serik Sagitov*
Affiliation:
Chalmers University of Technology and the University of Gothenburg
*
Postal address: Department of Mathematics, Uppsala University, 751 06 Uppsala, Sweden. Email address: bartoszekkj@gmail.com
∗∗Postal address: Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, 412 96 Göteborg, Sweden.
Rights & Permissions[Opens in a new window]

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a stochastic evolutionary model for a phenotype developing amongst n related species with unknown phylogeny. The unknown tree is modelled by a Yule process conditioned on n contemporary nodes. The trait value is assumed to evolve along lineages as an Ornstein-Uhlenbeck process. As a result, the trait values of the n species form a sample with dependent observations. We establish three limit theorems for the sample mean corresponding to three domains for the adaptation rate. In the case of fast adaptation, we show that for large n the normalized sample mean is approximately normally distributed. Using these limit theorems, we develop novel confidence interval formulae for the optimal trait value.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 2015 

References

[1] Adamczak, R. and Milos, P. (2014). U-statistics of Ornstein-Uhlenbeck branching particle system. J. Theoret. Prob. 27, 10711111.CrossRefGoogle Scholar
[2] Adamczak, R. and Milos, P. (2015). CLT for Ornstein-Uhlenbeck branching particle system. Electron. J. Prob. 20, 35 pp.CrossRefGoogle Scholar
[3] Aldous, D. and Popovic, L. (2005). A critical branching process model for biodiversity. Adv. Appl. Prob. 37, 10941115.CrossRefGoogle Scholar
[4] Ané, C. (2008). Analysis of comparative data with hierarchical autocorrelation. Ann. Appl. Statist. 2, 10781102.CrossRefGoogle Scholar
[5] Ané, C., Ho, L. S. T. and Roch, S. (2014). Phase transition on the convergence rate of parameter estimation under an Ornstein-Uhlenbeck diffusion on a tree. Preprint. Available at http://arxiv.org/abs/1406.1568.Google Scholar
[6] Bartoszek, K. (2014). Quantifying the effects of anagenetic and cladogenetic evolution. Math. Biosci. 254, 4257.CrossRefGoogle ScholarPubMed
[7] Bartoszek, K. et al. (2012). A phylogenetic comparative method for studying multivariate adaptation. J. Theoret. Biol. 314, 204215.CrossRefGoogle ScholarPubMed
[8] Boettiger, C., Coop, G. and Ralph, P. (2012). Is your phylogeny informative? Measuring the power of comparative methods. Evolution 66, 22402251.CrossRefGoogle ScholarPubMed
[9] Bohrnstedt, G. W. and Goldberger, A. S. (1969). On the exact covariance of products of random variables. J. Amer. Statist. Assoc. 64, 14391442.CrossRefGoogle Scholar
[10] Bokma, F. (2010). Time, species, and separating their effects on trait variance in clades. Syst. Biol. 59, 602607.CrossRefGoogle ScholarPubMed
[11] Butler, M. A. and King, A. A. (2004). Phylogenetic comparative analysis: a modeling approach for adaptive evolution. Amer. Naturalist 164, 683695.CrossRefGoogle ScholarPubMed
[12] Crawford, F. W. and Suchard, M. A. (2013). Diversity, disparity, and evolutionary rate estimation for unresolved Yule trees. Syst. Biol. 62, 439455.CrossRefGoogle ScholarPubMed
[13] Edwards, A. W. F. (1970). Estimation of the branch points of a branching diffusion process. J. R. Statist. Soc. B 32, 155174.Google Scholar
[14] Felsenstein, J. (1985). Phylogenies and the comparative method. Amer. Naturalist 125, 115.CrossRefGoogle Scholar
[15] Garland, T. Jr. and Ives, A. R. (2000). Using the past to predict the present: confidence intervals for regression equations in phylogenetic comparative methods. Amer. Naturalist 155, 346364.CrossRefGoogle ScholarPubMed
[16] Garland, T. Jr., Midford, P. E. and Ives, A. R. (1999). An introduction to phylogenetically based statistical methods, with a new method for confidence intervals on ancestral values. Amer. Zool. 39, 374388.CrossRefGoogle Scholar
[17] Gascuel, O. and Steel, M. (2014). Predicting the ancestral character changes in a tree is typically easier than predicting the root state. Syst. Biol. 63, 421435.CrossRefGoogle Scholar
[18] Gernhard, T. (2008). New analytic results for speciation times in neutral models. Bull. Math. Biol. 70, 10821097.CrossRefGoogle ScholarPubMed
[19] Gernhard, T. (2008). The conditioned reconstructed process. J. Theoret. Biol. 253, 769778.CrossRefGoogle ScholarPubMed
[20] Hansen, T. F. (1997). Stabilizing selection and the comparative analysis of adaptation. Evolution 51, 13411351.CrossRefGoogle ScholarPubMed
[21] Hansen, T. F., Pienaar, J. and Orzack, S. H. (2008). A comparative method for studying adaptation to a randomly evolving environment. Evolution 62, 19651977.Google ScholarPubMed
[22] Ho, L. S. T. and Ane, C. (2013). Asymptotic theory with hierarchical autocorrelation: Ornstein-Uhlenbeck tree models. Ann. Statist. 41, 957981.CrossRefGoogle Scholar
[23] Ho, L. S. T. and Ané, C. (2014). A linear-time algorithm for Gaussian and non-Gaussian trait evolution models. Syst. Biol. 63, 397408.Google ScholarPubMed
[24] Ho, L. S. T. and Ané, C. (2014). Intrinsic inference difficulties for trait evolution with Ornstein-Uhlenbeck models. Meth. Ecol. Evol. 5, 11331146.CrossRefGoogle Scholar
[25] Huelsenbeck, J. P. and Rannala, B. (2003). Detecting correlation between characters in a comparative analysis with uncertain phylogeny. Evolution 57, 12371247.CrossRefGoogle Scholar
[26] Huelsenbeck, J. P., Rannala, B. and Masly, J. P. (2000). Accommodating phylogenetic uncertainty in evolutionary studies. Science 288, 23492350.CrossRefGoogle ScholarPubMed
[27] Ives, A. R., Midford, P. E. and Garland, T. Jr. (2007). Within-species variation and measurement error in phylogenetic comparative methods. Syst. Biol. 56, 252270.CrossRefGoogle ScholarPubMed
[28] Martins, E. P. and Hansen, T. F. (1997). Phylogenies and the comparative method: a general approach to incorporating phylogenetic information into the analysis of interspecific data. Amer. Naturalist 149, 646667.CrossRefGoogle Scholar
[29] Mooers, A. et al. (2012). Branch lengths on birth-death trees and the expected loss of phylogenetic diversity. Syst. Biol. 61, 195203.CrossRefGoogle ScholarPubMed
[30] Mossel, E. and Steel, M. (2014). Majority rule has transition ratio 4 on Yule trees under a 2-state symmetric model. J. Theoret. Biol. 360, 315318.CrossRefGoogle Scholar
[31] Mulder, W. H. and Crawford, F. W. (2015). On the distribution of interspecies correlation for Markov models of character evolution on Yule trees. J. Theoret. Biol. 364, 275283.CrossRefGoogle ScholarPubMed
[32] R Development Core Team (2013). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing Vienna, Austria. Available at http://www.r-project.org.Google Scholar
[33] Rohlf, F. J. (2001). Comparative methods for the analysis of continuous variables: geometric interpretations. Evolution 55, 21432160.CrossRefGoogle ScholarPubMed
[34] Rohlf, F. J. (2006). A comment on phylogenetic correction. Evolution 60, 15091515.CrossRefGoogle ScholarPubMed
[35] Sagitov, S. and Bartoszek, K. (2012). Interspecies correlation for neutrally evolving traits. J. Theoret. Biol. 309, 1119.CrossRefGoogle ScholarPubMed
[36] Slater, G. J. et al. (2012). Fitting models of continuous trait evolution to incompletely sampled comparative data using approximate Bayesian computation. Evolution 66, 752762.CrossRefGoogle ScholarPubMed
[37] Stadler, T. (2008). Lineages-through-time plots of neutral models for speciation. Math. Biosci. 216, 163171.CrossRefGoogle Scholar
[38] Stadler, T. (2009). On incomplete sampling under birth-death models and connections to the sampling-based coalescent. J. Theoret. Biol. 261, 5868.CrossRefGoogle ScholarPubMed
[39] Stadler, T. (2011). Simulating trees with a fixed number of extant species. Syst. Biol. 60, 676684.CrossRefGoogle ScholarPubMed
[40] Stadler, T. and Steel, M. (2012). Distribution of branch lengths and phylogenetic diversity under homogeneous speciation models. J. Theoret. Biol. 297, 3340.CrossRefGoogle ScholarPubMed
[41] Steel, M. and Mckenzie, A. (2001). Properties of phylogenetic trees generated by Yule-type speciation models. Math. Biosci. 170, 91112.CrossRefGoogle ScholarPubMed
[42] Stone, E. A. (2011). Why the phylogenetic regression appears robust to tree misspecification. Syst. Biol. 60, 245260.CrossRefGoogle ScholarPubMed
[43] Symonds, M. R. E. (2002). The effects of topological inaccuracy in evolutionary trees on the phylogenetic comparative method of independent contrasts. Syst. Biol. 51, 541553.CrossRefGoogle ScholarPubMed
[44] Yule, G. U. (1925). A mathematical theory of evolution, based on the conclusions of Dr. J. C. Willis, F. R. S. Philos. Trans. R. Soc. London B 213, 2187.CrossRefGoogle Scholar