PowerPoint PresentationSlide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Phylogenetic Signal (Blomberg, Garland, and Ives 2003)Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48Slide 49Slide 50Slide 51Slide 52Slide 53Slide 54Slide 55Slide 56Slide 57Phylogenetic comparative methodsComparative studies (nuisance)Evolutionary studies (objective)Community ecology (lack of alternatives)Current growth of phylogenetic comparative methodsNew statistical methodsAvailability of phylogeniesCultureOne of many possible types of problemsy=b0+b1x+ε€ y = b0+ εor as a special caseThis model structure can be used for a variety of types of problemsy=b0+b1x+εAssumptions:y takes continuous valuesx 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 knowny=b0+b1x+εStatistical methods(P)IC = GLSPhylogenetic independent contrastsGeneralized Least Squares(these are methods, not models)Other methods for other statistical modelsML, REML, EGLS, GLM, GLMM, GEE, “Bayesian” methodsy=b0+b1x+ε are random variables with expectation 0 and finite (co)variances that are knownPhylogeny provides a hypothesis for these covariancesClose Relatives Tend to Resemble Each OtherABCDEFGHABCDEFGHI0 1 2 3 4-101234XYABCDEFGHABCDEFGHI0 1 2 3 4-101234XY What does this represent?How is it constructed?Is it known for certain?ABCDEFGHABCDEFGHI0 1 2 3 4-101234XY Assume that this represents time and is known without errorTranslate into the pattern of covariances in among speciesVHypothetical trait for a single species under Brownian motion evolutionTrait valueTimepossible course of evolutionTrait valueTimeanother possible course of evolutionTrait valueTimeanother possible course of evolutionBrownian motion evolution gives the hypothetical variance of a traitTrait valueTimeVarianceBrownian motion evolutionTrait valueTimeVarianceBrownian motion evolution of a hypothetical trait during speciationVariance between species = TimeTotal variance = Total timeVariance between species = TimeCovariance = Shared timeTotal variance = Total timeVariance between species = Time€ ⇒ VBrownian motionCovariance matrix giving phylogenetic covariances among speciesdiagonal elements give the total variance for species ioff-diagonal elements give covariances between species i and species j€ vi i€ vij€ VABCDEFGHABCDEFGHI0 1 2 3 4-101234XYI 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 evolutionOrnstein-Uhlenbeck evolutionStabilizing selection with strength given by dTimeselectionVariance between species < TimeVariance between species < TimeTotal variance << Total timeOrnstein-Uhlenbeck evolutionTimeVarianceStabilizing selection means information is “lost” through timePhylogenetic correlations between species decreasePhylogenetic Signal(Blomberg, Garland, and Ives 2003)€ ⇒ V(d)€ V(d) = measures the strength of signalOU process€ V(d) = y=b0+b1x+εAssumptions:y takes continuous valuesx can be a random variable or a set of known numbersy is linearly related to x are random variables with expectation 0 and finite (co)variances that are knownIf d must be estimated, cannot be analyzed using PIC or GLSIf 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)€ ⇒ V(d)€ V(d) = measures the strength of signalOU processy=b0+b1x+εStatistical methods(P)IC = GLSPhylogenetic independent contrastsGeneralized Least Squares(these are methods, not models)Other methods for other statistical modelsML, REML, EGLS, GLM, GLMM, GEE, “Bayesian” methodsPICy1y2y3y41234€ Δyij= β1Δxij+ ν'i+ν'jεij€ ν'i= νi+ν'kν'lν'k+ν'ly1y2y3y41234€ y4=y1ν1+ y2ν21 ν1+1 ν2=y1ν1+y2ν2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ν1ν2ν1+ ν2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟€ Δy12= y1− y2€ Δy34= y3− y4€ ν'4= ν4+ν1ν2ν1+ ν2PIC€ Δyijν'i+ν'j= β1Δxijν'i+ν'j+ εijRegression through the origin€ Δyij= β1Δxij+ ν'i+ν'jεijPIC€ Δyijν'i+ν'j= β1Δxijν'i+ν'j+ εij€ Δyijν'i+ν'j= β1Δ˜ x iju'i+u'j+ εijYou could also use different branch lengths for x:Branch lengths of yBranch lengths of xPIC€ Δyijν'i+ν'j= β1Δxijν'i+ν'j+ εijWhen could this be justified?You could also use different branch lengths for x:€ Δyijν'i+ν'j= β1Δ˜ x iju'i+u'j+ εijWhen could this be justified?€ Δyij= β1Δxij+ ν'i+ν'jεijNever (?)€ Δyijν'i+ν'j= β1Δ˜ x iju'i+u'j+ εijy=b0+b1x+εStatistical methods(P)IC = GLSPhylogenetic independent contrastsGeneralized Least Squares(these are methods, not models)Other methods for other statistical modelsML, REML, EGLS, GLM, GLMM, GEE, “Bayesian” methodsElements of V are given by shared branch lengths under the assumption of “Brownian motion” evolutionE εε'[ ] =σ2V ≠σ2Iy=b0+b1x+εy= y1,y2,...,yn[ ]'X = 1,x[ ]b= b0,b1[ ]'ˆ b = X'V−1X( )−1X'V−1y( )ˆ σ 2= y−Xˆ b ( )'V−1y−Xˆ b ( )n−2( )Generalized Least Squares, GLSOrdinary least squares€ ˆ b = X'X( )−1X'y( )ˆ σ 2= y − Xˆ b ( )'y − Xˆ b ( )n − 2( )V = IDVD'=Iz=DyU =DXRelated to ordinary least squaresy=Xb+εDy=DXb+Dεz=Ub+α€ z = Ub + αE αα'[ ]=E Dε Dε( )'[ ]=DE εε'[ ]D'=Dσ2VD' =σ2I€ z = Ub + αValues of € z = Dyare linear combinations of yi€ E αα'[ ]= σ2IABCDEFGHABCDEFGHI0 1 2 3 4-101234XYGLS LSparametertrue valueestimate 95% confidenceintervalestimate95% confidenceintervalb00 2.28 [-0.82, 5.38] -1.10 [-3.69, 1.49]b10 -0.43 [-1.45, 0.60] 1.45 [0.28, 2.62]σ22 3.35 1.39E{Yh} 2.84 [ -0.35 , 6.03] 3.84 [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
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