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+ε

€

y = b0 + ε

or as a special case

This model structure can be used for a variety of types of problems

y=b0 +b1x+εAssumptions:

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+εStatistical methods

(P)IC = GLS

Phylogenetic independent contrastsGeneralized Least Squares

(these are methods, not models)

Other methods for other statistical models

ML, REML, EGLS, GLM, GLMM, GEE, “Bayesian” methods

y=b0 +b1x+ε

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

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What does this represent?

How is it constructed?

Is it known for certain?

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Assume that this represents time and

is known without error

Translate into the pattern of covariances

in among species

V

Hypothetical trait for a single species under Brownian motion evolution

Tra

it va

lue

Time

possible course of evolution

Tra

it va

lue

Time

another possible course of evolution

Tra

it va

lue

Time

another possible course of evolution

Brownian motion evolution gives the hypothetical variance of a trait

Tra

it va

lue

Time

Variance

Brownian motion evolutionT

rait

valu

e

Time

Variance

Brownian motion evolution of a hypothetical trait during speciation

Variance between species = Time

Total variance = Total time

Variance between species = Time

Covariance = Shared time

Total variance = Total time

Variance between species = Time

€

⇒ VBrownia

n 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

€

v i i

€

v ij

€

V

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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

Time

selection

Variance between species < Time

Variance between species < Time

Total variance << Total 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)

€

⇒ V(d)

€

V(d) =

measures the strength of signal

OU process

€

V(d) =

y=b0 +b1x+εAssumptions:

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)

€

⇒ V(d)

€

V(d) =

measures the strength of signal

OU process

y=b0 +b1x+εStatistical methods

(P)IC = GLS

Phylogenetic independent contrastsGeneralized Least Squares

(these are methods, not models)

Other methods for other statistical models

ML, REML, EGLS, GLM, GLMM, GEE, “Bayesian” methods

PIC

y1

y2

y3

y4

1

2

3

4

€

Δy ij = β1Δx ij + ν 'i +ν ' jε ij

€

'i = ν i +ν 'k ν 'l

ν 'k +ν 'l

y1

y2

y3

y4

1

2

3

4

€

y4 =y1 ν 1 + y2 ν 2

1 ν 1 +1 ν 2

=y1

ν 1

+y2

ν 2

⎛

⎝ ⎜

⎞

⎠ ⎟

ν 1ν 2

ν 1 + ν 2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

Δy12 = y1 − y2

€

Δy34 = y3 − y4

€

'4 = ν 4 +ν 1ν 2

ν 1 + ν 2

PIC

€

Δy ij

ν 'i +ν ' j

= β1

Δx ij

ν 'i +ν ' j

+ ε ij

Regression through the origin

€

Δy ij = β1Δx ij + ν 'i +ν ' jε ij

PIC

€

Δy ij

ν 'i +ν ' j

= β1

Δx ij

ν 'i +ν ' j

+ ε ij

€

Δy ij

ν 'i +ν ' j

= β1

Δ˜ x iju'i +u' j

+ ε ij

You could also use different branch lengths for x:

Branch lengths of y

Branch lengths of x

PIC

€

Δy ij

ν 'i +ν ' j

= β1

Δx ij

ν 'i +ν ' j

+ ε ij

When could this be justified?

You could also use different branch lengths for x:

€

Δy ij

ν 'i +ν ' j

= β1

Δ˜ x iju'i +u' j

+ ε ij

When could this be justified?

€

Δy ij = β1Δx ij + ν 'i +ν ' jε ij

Never (?)

€

Δy ij

ν 'i +ν ' j

= β1

Δ˜ x iju'i +u' j

+ ε ij

y=b0 +b1x+εStatistical methods

(P)IC = GLS

Phylogenetic independent contrastsGeneralized Least Squares

(these are methods, not models)

Other methods for other statistical models

ML, REML, EGLS, GLM, GLMM, GEE, “Bayesian” methods

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

E εε'[ ] =σ 2V≠σ 2I

y=b0 +b1x+ε

y= y1,y2,...,yn[ ]'

X= 1,x[ ]

b= b0,b1[ ]'

ˆ b = X'V−1X( )−1

X'V−1y( )

ˆ σ 2 = y−Xˆ b ( )'V−1 y−Xˆ b ( ) n−2( )

Generalized Least Squares, GLS

Ordinary least squares

€

ˆ b = X'X( )−1

X'y( )

ˆ σ 2 = y − X ˆ b ( )'

y − X ˆ b ( ) n − 2( )

V = I

DVD'=I

z=Dy

U =DX

Related to ordinary least squares

y=Xb+ε

Dy=DXb+Dε

z=Ub+α

€

z = Ub + α

E αα'[ ]=E Dε Dε( )'[ ]

=DE εε'[ ]D'

=Dσ 2VD'=σ 2I

€

z = Ub + α

Values of

€

z = Dy

are linear combinations of yi€

E αα '[ ] = σ 2I

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GLS LS

parameter true value estimate 95% confidence

interval

estimate 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]

σ2 2 3.35 1.39

{E 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 just use IC?

Divergence time for desert and montane ringtail populations assumed to be 10,000 years

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Predicting values for ancestral and new species

€

Δy ij = β1Δx ij + ν 'i +ν ' jε ij

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Is the prediction of the estimate of y for species I more or less precise than what you would expect from a standard regression analysis?

When dealing with multiple, incongruent gene trees, we can perform multiple PIC's on each tree, and find a correlation or not. How do we know which is the "right" answer?

The three main phylogenetically based statistical methods described in the reading (IC, GLS, and Monte Carlo simulations) rely on correct information about tree topology and branch lengths. If we are unsure of the correctness of these basic assumptions, what is the best way to analyze our data?

I'm unclear how data can be statistically significant when transformed, but not significant otherwise. This seems like cheating/lying.

The paper discussed researchers' decisions about branch lengths, especially in terms of transformations (OU, ACDC). Do researchers use ultrametric trees for these analyses?