Taxonomic

Taxonomic Phylogenetic

Modern Comparative Methods

Phylogenetic


Taxonomic

Nested ANOVA

Analysisof Covariance

[Source: Garland 2001, pp. 107-132 in [Source: Garland 2001, pp. 107-132 in Hypoxia: From Genes to the BedsideHypoxia: From Genes to the Bedside, Academic Press, NY], Academic Press, NY]

Phylogenetic


Taxonomic

Nested ANOVA

Analysisof Covariance

[Source: Dunham & Miles 1985, Am Nat. 126: 231-257][Source: Dunham & Miles 1985, Am Nat. 126: 231-257]

Phylogenetic


Taxonomic

Independent Contrasts

Generalized Linear Models

General Linear ModelGeneral Linear Model

[Source: Martins & Hansen 1997, [Source: Martins & Hansen 1997, Am. Nat.Am. Nat. 149: 646-667] 149: 646-667]

y = a + βx1 + βx2…βxn + ε

Meanstructure

Errorstructure

Cheverud et al. (1985)

Grafen(1989)

Components of ErrorComponents of Error

• Error in measurement of Error in measurement of X’sX’s and and YY

• Error due to stochastic evolutionary processes Error due to stochastic evolutionary processes

• Error due to imperfect phylogenetic Error due to imperfect phylogenetic information (topology or branch lengths)information (topology or branch lengths)

[Source: Martins & Hansen 1997, [Source: Martins & Hansen 1997, Am. Nat.Am. Nat. 149: 646-667] 149: 646-667]

additive

not additive

Phylogenetic


Taxonomic

Independent Contrasts

Generalized Linear Models

Independent ContrastsIndependent Contrasts

• A set of N-1 phylogenetically independent data A set of N-1 phylogenetically independent data pointspoints

• Can be used in any univariate or multivariate Can be used in any univariate or multivariate analysesanalyses

• Yield identical statistical power as do analyses of Yield identical statistical power as do analyses of raw dataraw data

[Source: Felsenstein 1985, [Source: Felsenstein 1985, Amer. Nat.Amer. Nat. 125: 1-15] 125: 1-15]

The value of XA and XB are not independent

But XA-XB is clearly independent of XC-XD. These two values are independent contrasts.

Independent contrasts for trait Xi, where i is the specific taxon.

BA C DXA XB XC XD


The value of Xi for an ancestor (e.g., E) is the weighted mean of Xi for the descendants.

Independent contrasts for trait Xi, where i is the specific taxon.

BA C DXA XB XC XD

E FXE = (1/vA)XA + (1/vB)XB

1/vA + 1/vB


vBvA

What do you need to compute What do you need to compute independent contrasts?independent contrasts?

• A fully resolved cladogram A fully resolved cladogram

• Branch lengthsBranch lengths

• Data for two or more Data for two or more traitstraits

[Source: Garland et al. 1992, [Source: Garland et al. 1992, Syst. Biol.Syst. Biol. 41: 18-32] 41: 18-32]

Standardizing ContrastsStandardizing Contrasts

• Taxa that have been separated by longer periods of time should have larger contrasts.

• Must standardize contrasts before using them in statistical analyses (mean = 0, variance = 1).

• Because the expected mean of contrasts is zero, all

you need to do is divide the contrasts by their standard deviation (= square root of the sum of branch lengths)


Assessing Standardization

Adequate standardization is assessed by regressing standardized contrasts onto the standard deviation of a contrast (square root of the sum of branch lengths)

A correlation (+ or -) implies inadequate standardization.


Use of Contrasts in Statistical AnalysesUse of Contrasts in Statistical Analyses

• Contrasts can be used in any subsequent statistical analyses as you would use raw data.

• However, the following restrictions hold:

1) All standard assumptions of parametric statistics (e.g., homoscedasticity, normality of residuals) also apply to the analysis of contrasts.

2)2) Statistical models should not include an intercept because the intercept Statistical models should not include an intercept because the intercept of a relationship between two sets of contrasts is theoretically equal to of a relationship between two sets of contrasts is theoretically equal to zero.zero.


Can contrasts be computed for Can contrasts be computed for environmental variables?environmental variables?

• Some have argued that the use of contrasts for environmental variables is inappropriate (e.g., see Martins 2000).

• Garland et al. (1992) noted that contrasts could be computed for a continuous environmental variable if environmental states are passed from ancestor to descendant.

[Sources: Martins 2000, [Sources: Martins 2000, Trends Ecol. Evol.Trends Ecol. Evol. 15: 296-299; 15: 296-299;Garland et al. 1992, Garland et al. 1992, Syst. Biol.Syst. Biol. 41: 18-32] 41: 18-32]

[Source: Angilletta et al. 2004, [Source: Angilletta et al. 2004, Am. Nat.Am. Nat. 164: in press] 164: in press]

Can contrasts be used for Can contrasts be used for intraspecific comparisons?intraspecific comparisons?

• When gene flow among populations is limited, phylogenetic comparative methods can be applied in studies of microevolution.

• Alternatively, Felsenstein (2002) proposed a method of computing contrasts for intraspecific data that accounts for gene flow among populations.

[Source: Niewiarowski et al. 2004, [Source: Niewiarowski et al. 2004, EvolutionEvolution 58: 619-633] 58: 619-633]

[Source: Cheverud et al. 1985, [Source: Cheverud et al. 1985, EvolutionEvolution 39: 1335-1351] 39: 1335-1351][Source: Cheverud et al. 1985, [Source: Cheverud et al. 1985, EvolutionEvolution 39: 1335-1351] 39: 1335-1351]

VT = total variance

VP = phylogenetic variance

VS = specific variance

cov(P,S) = covariance between phylogenetic

and specific values

Phylogenetic AutocorrelationPhylogenetic Autocorrelation

VT = VP + VS + 2[cov(P,S)]

[Source: Cheverud et al. 1985, [Source: Cheverud et al. 1985, EvolutionEvolution 39: 1335-1351] 39: 1335-1351][Source: Cheverud et al. 1985, [Source: Cheverud et al. 1985, EvolutionEvolution 39: 1335-1351] 39: 1335-1351]

y = vector of standardized trait values (n x 1)p = phylogenetic autocorrelation coefficient W = matrix of phylogenetic connectivity (n x n)e = vector of residual trait values

y = pWy + e

Phylogenetic AutocorrelationPhylogenetic Autocorrelation

Phy

loge

netic

sim

ilarit

y

P

The autocorrelation coefficient is standardized like a typical correlation coefficient: -1 ≤ p ≤ 1

Interpreting the Autocorrelation CoefficientInterpreting the Autocorrelation Coefficient

p = 0 → no phylogenetic effect

0 < p ≤ 1 → phylogenetic similarity

-1 ≤ p < 0 → phylogenetic dissimilarity

[Source: Cheverud et al. 1985, [Source: Cheverud et al. 1985, EvolutionEvolution 39: 1335-1351] 39: 1335-1351]

[Source: Cheverud et al. 1985, [Source: Cheverud et al. 1985, EvolutionEvolution 39: 1335-1351] 39: 1335-1351]

Examples of Autocorrelation CoefficientsExamples of Autocorrelation Coefficients


Do different methods yield similar results?Do different methods yield similar results?


Do different methods yield similar results?Do different methods yield similar results?

Are phylogenetic methods necessary?Are phylogenetic methods necessary?

• Some biologists have argued that phylogenetic Some biologists have argued that phylogenetic comparative methods are too conservativecomparative methods are too conservative

• If the phylogenetic information is imperfect, a If the phylogenetic information is imperfect, a comparative method could be misleading. comparative method could be misleading.

• One recommendation is to investigate the extent One recommendation is to investigate the extent of phylogenetic signal prior to choosing a of phylogenetic signal prior to choosing a comparative method. comparative method.

[Source: Blomberg et al. 2003, [Source: Blomberg et al. 2003, EvolutionEvolution 57: 717-745] 57: 717-745]

Phylogenetic Signal (K)

If a trait evolves according to a model of Brownian motion, then K = 1.

[Source: Garland 2001, pp. 107-132 in [Source: Garland 2001, pp. 107-132 in Hypoxia: From Genes to the BedsideHypoxia: From Genes to the Bedside, Academic Press, NY], Academic Press, NY][See also Garland et al. 1999, [See also Garland et al. 1999, Am. ZoAm. Zool. 39: 374-388]

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Taxonomic