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Phylogenetic comparative methods Comparative studies (nuisance) Evolutionary studies (objective)

Phylogenetic comparative methods Comparative studies (nuisance) Evolutionary studies (objective) Community ecology (lack of alternatives). Current growth of phylogenetic comparative methods New statistical methods Availability of phylogenies Culture. One of many possible types of problems.

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Phylogenetic comparative methods Comparative studies (nuisance) Evolutionary studies (objective)

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  1. Phylogenetic comparative methods Comparative studies (nuisance) Evolutionary studies (objective) Community ecology (lack of alternatives)

  2. Current growth of phylogenetic comparative methods New statistical methods Availability of phylogenies Culture

  3. One of many possible types of problems or as a special case This model structure can be used for a variety of types of problems

  4. Assumptions: • y takes continuous values • x can be a random variable or a set of known values (continuous or not) • y is linearly related to x •  are random variables with expectation 0 and finite (co)variances that are known

  5. Statistical methods • (P)IC = GLS • Phylogenetic independent contrasts • Generalized Least Squares • (these are methods, not models) • Other methods for other statistical models • ML, REML, EGLS, GLM, GLMM, GEE, “Bayesian” methods

  6. are random variables with expectation 0 and finite (co)variances that are known Phylogeny provides a hypothesis for these covariances

  7. Close Relatives Tend to Resemble Each Other

  8. What does this represent? How is it constructed? Is it known for certain?

  9. Assume that this represents time and is known without error Translate into the pattern of covariances in  among species V

  10. Hypothetical trait for a single species under Brownian motion evolution Trait value possible course of evolution Time

  11. another possible course of evolution Trait value Time

  12. another possible course of evolution Trait value Time

  13. Brownian motion evolution gives the hypothetical variance of a trait Trait value Variance Time

  14. Brownian motion evolution Trait value Variance Time

  15. Brownian motion evolution of a hypothetical trait during speciation

  16. Variance between species = Time

  17. Total variance = Total time Variance between species = Time

  18. Total variance = Total time Covariance = Shared time Variance between species = Time

  19. Brownian motion Covariance matrix giving phylogenetic covariances among species diagonal elements give the total variance for species i off-diagonal elements give covariances between species i and species j

  20. I am confused by the authors use of "branch lengths" on page 3023. I'm not sure if"different types of branch lengths" mean different phylogenetic analyses or something else I'm not aware of. Digression - non-Brownian models of evolution

  21. Ornstein-Uhlenbeck evolution Stabilizing selection with strength given by d

  22. Variance between species < Time

  23. Total variance << Total time Variance between species < Time

  24. Ornstein-Uhlenbeck evolution Time Variance Stabilizing selection means information is “lost” through time Phylogenetic correlations between species decrease

  25. Phylogenetic Signal(Blomberg, Garland, and Ives 2003) OU process measures the strength of signal

  26. Assumptions: • y takes continuous values • x can be a random variable or a set of known numbers • y is linearly related to x • are random variables with expectation 0 and finite (co)variances that are known If d must be estimated, cannot be analyzed using PIC or GLS

  27. If we are dealing with a recent, rapid radiation, (supported clade but with short branches) will the lack of branch length data render any PIC not very informative biologically, because we would expect non-significant probabilities, based solely on the branch lengths alone? page 3022, second paragraph.

  28. Phylogenetic Signal(Blomberg, Garland, and Ives 2003) OU process measures the strength of signal

  29. Statistical methods • (P)IC = GLS • Phylogenetic independent contrasts • Generalized Least Squares • (these are methods, not models) • Other methods for other statistical models • ML, REML, EGLS, GLM, GLMM, GEE, “Bayesian” methods

  30. PIC 1 y1 4 y4 2 y2 3 y3

  31. 1 y1 4 y4 2 y2 3 y3

  32. PIC Regression through the origin

  33. PIC You could also use different branch lengths for x:

  34. Branch lengths of y Branch lengths of x

  35. PIC You could also use different branch lengths for x: When could this be justified?

  36. When could this be justified? Never (?)

  37. Statistical methods • (P)IC = GLS • Phylogenetic independent contrasts • Generalized Least Squares • (these are methods, not models) • Other methods for other statistical models • ML, REML, EGLS, GLM, GLMM, GEE, “Bayesian” methods

  38. Elements of V are given by shared branch lengths under the assumption of “Brownian motion” evolution

  39. Generalized Least Squares, GLS

  40. Ordinary least squares V = I

  41. Related to ordinary least squares

  42. Values of are linear combinations of yi

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