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 y b0 b1x or as a special case y b0 This model structure can be used for a variety of types of problems y b0 b1x 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 y b0 b1x 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 y b0 b1x are random variables with expectation 0 and finite co variances that are known Phylogeny provides a hypothesis for these covariances Close Relatives Tend to Resemble Each Other A 4 B G H 3 C D 2 F E A Y E C 1 F 0 D G B 1 0 1 2 H X I 3 4 A What does this represent 4 B G H 3 C D F E How is it constructed 2 A Y E C 1 F 0 Is it known for certain D G B 1 0 1 2 H X I 3 4 Assume that this represents time and is known without error A 4 B G D 2 Y E H 3 C E A Translate into the pattern of covariances in among species C 1 F F 0 G D B 1 0 H 1 2 V X I 3 4 Trait value Hypothetical trait for a single species under Brownian motion evolution possible course of evolution Time Trait value another possible course of evolution Time Trait value another possible course of evolution Time Trait value Brownian motion evolution gives the hypothetical variance of a trait Variance Time Trait value Brownian motion evolution Variance Time Brownian motion evolution of a hypothetical trait during speciation Variance between species Time Total variance Total time Variance between species Time Total variance Total time Covariance Shared time Variance between species Time A 4 B G Brownian motion V H 3 C D 2 F E A Y E C 1 F 0 D G B 1 0 1 2 3 4 H X I V Covariance matrix giving phylogenetic covariances among species v i i diagonal elements give the total variance for species i v ij off diagonal elements give covariances between species i and species j 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 Ornstein Uhlenbeck evolution Stabilizing selection with strength given by d selection Time Variance between species Time Total variance Total time Variance between species Time Ornstein Uhlenbeck evolution Time Variance Stabilizing selection means information is lost through time Phylogenetic correlations between species decrease Phylogenetic Signal Blomberg Garland and Ives 2003 OU process V d V d measures the strength of signal V d y b0 b1x 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 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 Phylogenetic Signal Blomberg Garland and Ives 2003 OU process V d V d measures the strength of signal y b0 b1x 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 PIC y ij 1 x ij i j ij k l i i k l 4 1 y4 2 3 y1 y2 y3 4 1 y4 2 3 y1 y2 y3 y12 y1 y 2 y1 1 y 2 2 y1 y 2 1 2 y4 1 1 1 2 1 2 1 2 y 34 y 3 y 4 1 2 4 4 1 2 PIC y ij 1 x ij i j ij y ij i j 1 x ij i j ij Regression through the origin PIC y ij i j 1 x ij i j ij You could also use different branch lengths for x y ij i j 1 x ij u i u j ij Branch lengths of y Branch lengths of x PIC y ij i j 1 x ij i j ij You could also use different branch lengths for x y ij i j 1 x ij u i u j When could this be justified ij When could this be justified y ij i j 1 x ij u i u j ij y ij 1 x ij i j ij Never y b0 b1x 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 y b0 b1x 2 2 E V I Elements of V are given by shared branch lengths under the assumption of Brownian motion evolution Generalized Least Squares GLS y y1 y2 yn X 1 x b b0 b1 1 1 1 b X V X X V y 1 n 2 y Xb V y Xb 2 Ordinary least squares b X X 1 X y y Xb y Xb n 2 2 V I Related to ordinary least squares DVD I z Dy U DX y Xb Dy DXb D z Ub z Ub E E D D DE D 2 2 D VD I z Ub E I 2 Values of z Dy are linear combinations of yi A 4 B G H 3 C D 2 F E A Y E C 1 F 0 D G B 1 0 1 2 H X I 3 4 GLS parameter true value estimate 95 confidence LS estimate interval 95 confidence interval b0 0 2 28 0 82 5 38 1 10 3 69 1 49 b1 0 0 43 1 45 0 60 1 45 0 28 2 62 2 2 3 35 1 39 2 84 0 35 6 03 3 84 E Yh 0 35 7 33 If IC and GLS can yield identical results and the authors refer to IC as a special case of GLS models p 3032 in what situation s would GLS be a more appropriate method In other words why not just use IC Divergence time for desert and montane ringtail populations assumed to be 10 000 years QuickTime and a TIFF LZW decompressor are needed to see this picture Predicting values for ancestral and new species …
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