Lecture Outline

• On Correlated Observations and Errors
• Multivariate Phylogenetic Comparative Methods
  1. Phylogenetic Least Squares (ANOVA/Regression/Covariation)
  2. Phylogenetic Signal
  3. Phylogenetic Ordination
  4. Comparing Evolutionary Models

Motivation: Correlated Observations

• Imagine the following scenario:
  • We assemble a large database of data for many mammal species to ask:
• Is home range size predicted by body size?

• The observations are not independent (species are related by a phylogeny)!

Comparative Biology Tradition

• To study adaptation, biologists look comparatively across taxa, and examine trait correlations

• Species values commonly utilized

• But this has the same problem: the observations are not independent!

• To resolve this, Phylogenetic Comparative Methods are required

Phylogenetic Comparative Methods (PCMs)

• PCMs condition the data on the phylogeny under an evolutionary model (i.e., account for the phylogeny during the analysis)

• Empirical Goal: Evaluate evolutionary hypotheses while accounting for (phylogenetic) non-independence

• This requires a phylogeny, and an evolutionary model of how trait variation is expected to accumulate

• But to understand PCMs more deeply, let's first go back to something familiar: linear models

Linear Models Revisited

• Statistical linear models describe patterns in data with both a deterministic component (the relationships between variables) AND a process for how we expect stochastic variation (random error) to be generated $^1$

• The Linear Model: $\mathbf{Z}=\mathbf{X}\mathbf{\beta } +\mathbf{\epsilon}$

1: For a refreshing perspective on correlated errors in biology see Ives (2022). Methods Ecol. Evol.

• $\mathbf{X\beta}$ describes the predicted values $(\hat{\mathbf{Z}})$ given the independent variables $(\mathbf{X})$

• $\mathbf{\epsilon}$ captures the stochastic error in the processes generating $\mathbf{Z}$

• To this point (and in most statistics courses) we tend to focus on the deterministic component $(\mathbf{X\beta})$ and whether our model is an adequate description of variation in $\mathbf{Z}$ (i.e., by minimizing $\mathbf\epsilon$)

• But what about $\mathbf{\epsilon}$? What does it really represent and contain?

Thinking about the Error: $\large\mathbf{\epsilon}$

Visualizing $\large\mathbf{\epsilon}$ as a Matrix

• When random error is iid, this means that elements of $\epsilon$ are drawn from the same normal distribution: $\epsilon\small\sim\mathcal{N}(0,\sigma^2)$

NOTE: In the PCM literature, $\mathbf\Omega$ is often termed C or $\mathbf\Sigma$. In this workshop we use $\mathbf\Omega$ so as not to confuse this matrix with a trait covariance matrix $\mathbf\Sigma$.

Modeling $\large\mathbf{\epsilon}$ with Heterogeneous Variance

• Imagine taking repeated caliper measurements of body size on 4 specimens

  • Error is expected to be proportionately greater when measuring smaller individuals (it is harder to measure smaller specimens as accurately)

• This means the expected variance differs among observations

• Now $\mathbf\Omega$ models independent observations with heterogeneous random variance (i.e., $\sigma^2$ varies across observations)

Modeling $\large\mathbf{\epsilon}$ with Correlated Observations

• Now imagine we measure 2 species from each of two genera

  • We expect that values are more similar between congeneric species (due to phylogenetic relatedness)

• This means the expected covariance is not always zero

• Now $\mathbf\Omega$ models random variation in non-independent observations

• For actual data, two questions remain:

  1. How do we generate $\mathbf\Omega$?
  2. What do we do with it?

Generating Object Covariance Matrices $\large(\mathbf\Omega)$

• $\mathbf\Omega$ is an $N\times N$ matrix describing the expected covariation of the random errors among observations

  • If $\epsilon$ is iid, then $\mathbf\Omega$ is an identity matrix
  • If $\epsilon$ is NOT iid, then $\mathbf\Omega$ displays some other pattern, which was caused by 'something'

• Biology informs us of what that something is (phylogeny, space, time)

• In this case, generating $\mathbf\Omega$ requires knowledge of the relationships among observations, and a model of how trait variation is expected to accumulate

• Let's look at this for the case of a phylogeny, and what are termed phylogenetic comparative methods (PCMs)

  • But first, we will describe the model of evolutionary change (Brownian motion)

Brownian Motion and PCMs

From Brownian Motion to $\large\mathbf\Omega$

• Imagine a trait evolving through time
  • From the root of the phylogeny, its value changes following a BM process
  • When speciation occurs, traits in each species evolve via BM

• The result is that traits tend to be more similar in more closely related species
  • Using BM, we can estimate $\mathbf\Omega$

Phylogenetic Covariance Matrices $(\large\mathbf\Omega)$

• Under Brownian motion, the expected amount of change in trait values is proportional to time

  • Species evolve 'together' on their parental branch until they speciate
  • Thus, the more time before they speciate, the more similar their traits are expected to be (less time evolving separately means less opportunity to diverge)

• This gives us the expected object covariance $(\mathbf\Omega)$ under Brownian motion $^1$

1: Other evolutionary models, such as OU, could also be used.

Phylogenetic Covariance Matrices $(\large\mathbf\Omega)$: Example

• What is the expected covariation for species related by a phylogeny under BM?

• Now we have $\mathbf\Omega$ for our species

• The next question is, what do we do with it?

From Ordinary Least Squares to Generalized Least Squares

• Linear models with iid error: $\epsilon\small\sim\mathcal{N}(0,\sigma^2)$, are called Ordinary Least Squares methods

  • These include all the methods we have discussed previously
  • ANOVA, regression, ANCOVA, factorial ANOVA, multiple regression, pairwise comparisons, trajectory analysis, are part of this paradigm

• With $\mathbf\Omega$ we move from iid to correlated error

• Linear models with correlated error: $\epsilon\small\sim\mathcal{N}(0,\sigma^2\mathbf\Omega)$, are called Generalized Least Squares methods

  • With GLS, we can perform versions of ANOVA, regression, ANCOVA, multiple regression, pairwise comparisons, trajectory analysis, etc.
  • These are the SAME statistical models as before, but now account for the correlations among observations as described by $\mathbf\Omega$
  • Because $\mathbf\Omega$ describes covariation of the errors, we must adjust our linear model algebra to accommodate it

• Phylogenetic Comparative Methods (PCMs) are GLS approaches

PCMs: General Model

• Most PCMs use GLS model: $\mathbf{Z}=\mathbf{X}\mathbf{\beta} +\mathbf{\epsilon}$

  • $\mathbf{X\beta}$ describes the predicted values $(\hat{\mathbf{Z}})$ given the independent variable $(\mathbf{X})$
  • $\epsilon\sim\mathcal{N}(0,\sigma^2\mathbf\Omega)$ captures the stochastic error in the processes generating $\mathbf{Z}$ given $\mathbf\Omega$

• Model design $(\small\mathbf{X})$ describes the type of analysis $^1$

• The Phylogenetic Comparative Method Toolkit is a set of procedures for evaluating different hypotheses

1: reviewed in: Adams and Collyer (2018;2019)

PCMs: An Incomplete Historical Walk

• The conceptual development of PCMs

Phylogenetic Comparative Method Toolkit

1: Phylogenetic Least Squares: Regression (Concept)

• Evaluate $\mathbf{Z}=\mathbf{X}\mathbf{\beta } +\mathbf{\epsilon}$ in a phylogenetic context

• The workhorse of PCMs

1: Phylogenetic Least Squares: Regression and ANOVA

• EVERYTHING is the same as before, only the phylogeny is incorporated in the analysis!

1: A Visual of Phylogenetic Least Squares

1: Phylogenetic Least Squares Regression: Example

• Is there evolutionary allometry of head shape (does shape covary with size across species while accounting for phylogeny)?

## 
## No curves detected; all points appear to be fixed landmarks.

1: Phylogenetic Least Squares Regression: Example (Cont.)

• Yes, there is significant evolutionary allometry across the Plethodon phylogeny

pgls.reg <- procD.pgls(f1 = shape~svl, phy=plethtree, data=gdf, print.progress = FALSE)
summary(pgls.reg)

## 
## Analysis of Variance, using Residual Randomization
## Permutation procedure: Randomization of null model residuals 
## Number of permutations: 1000 
## Estimation method: Generalized Least-Squares (via OLS projection) 
## Sums of Squares and Cross-products: Type I 
## Effect sizes (Z) based on F distributions
## 
##           Df        SS         MS     Rsq      F      Z Pr(>F)  
## svl        1 0.0006581 0.00065811 0.07046 3.0323 1.9503  0.028 *
## Residuals 40 0.0086814 0.00021704 0.92954                       
## Total     41 0.0093395                                          
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Call: procD.lm(f1 = shape ~ svl, iter = iter, seed = seed, RRPP = TRUE,  
##     SS.type = SS.type, effect.type = effect.type, int.first = int.first,  
##     Cov = Cov, data = data, print.progress = print.progress)

1: Phylogenetic Least Squares Regression: Visualization

• Now let's visualize these results

plot.out <- plot(pgls.reg, type = "regression", 
                 predictor = svl, reg.type = "RegScore", pch=19, col = "black")
abline(lm(plot.out$RegScore~svl))

preds <- shape.predictor(pgls.reg$GM$pgls.fitted, 
  x= svl, Intercept = TRUE, predmin = min(svl), predmax = max(svl))

1: Phylogenetic Least Squares ANOVA: Example

• Does head shape in Plethodon differ between high and low elevation species?

1: Phylogenetic Least Squares ANOVA: Example (Cont.)

• No, head shape does not differ across elevational groups when phylogeny is considered

pgls.aov <- procD.pgls(f1 = shape~elev, phy=plethtree, data=gdf, print.progress = FALSE)
summary(pgls.aov)

## 
## Analysis of Variance, using Residual Randomization
## Permutation procedure: Randomization of null model residuals 
## Number of permutations: 1000 
## Estimation method: Generalized Least-Squares (via OLS projection) 
## Sums of Squares and Cross-products: Type I 
## Effect sizes (Z) based on F distributions
## 
##           Df        SS         MS     Rsq     F      Z Pr(>F)
## elev       1 0.0004220 0.00042201 0.04519 1.893 1.3479    0.1
## Residuals 40 0.0089175 0.00022294 0.95481                    
## Total     41 0.0093395                                       
## 
## Call: procD.lm(f1 = shape ~ elev, iter = iter, seed = seed, RRPP = TRUE,  
##     SS.type = SS.type, effect.type = effect.type, int.first = int.first,  
##     Cov = Cov, data = data, print.progress = print.progress)

1: Phylogenetic Least Squares ANOVA: Visualization

• Now let's visualize these results

plot.res <- gm.prcomp(shape,phy=plethtree)
plot(plot.res,phylo = FALSE, pch=21, bg=gdf$elev, cex=2)
legend("topleft", pch=21, pt.bg = unique(gdf$elev), legend = levels(gdf$elev))

preds <- shape.predictor(arrayspecs(pgls.aov$pgls.fitted, 11,2), x= pgls.aov$X[,-1],
           Intercept = TRUE, Low = Low, High = High)

1: Phylogenetic ANOVA: Group Aggregation

• How groups are distributed on the phylogeny can make a difference: Adams and Collyer. Evol. (2018)

• Too few group 'state' changes across phylogeny results in lower power

1: Phylogenetic Least Squares: Implementations

• Three implementations are found in the literature

• All yield equivalent results if performed properly

1B: Phylogenetic Partial Least Squares $^1$

• A related approach assesses the covariation between blocks of variables while accounting for phylogenetic non-independence

1B: Phylogenetic Partial Least Squares: Example

• Is there phenotypic integration between the cranium and mandible in Plethodon salamanders?

land.gps<-c("A","A","A","A","A","B","B","B","B","B","B")
PLS.Y <- phylo.integration(A = gdf$shape, partition.gp = land.gps, phy= plethtree, print.progress = FALSE)
summary(PLS.Y)

## 
## Call:
## phylo.integration(A = gdf$shape, phy = plethtree, partition.gp = land.gps,  
##     print.progress = FALSE) 
## 
## 
## 
## r-PLS: 0.843
## 
## Effect Size (Z): 4.7366
## 
## P-value: 0.001
## 
## Based on 1000 random permutations

1B: Phylogenetic Partial Least Squares: Visualization

• Now let's visualize these results

plot(PLS.Y)

2: Phylogenetic Signal

• The degree to which phenotypic similarity associates with phylogenetic relatedness

2: Phylogenetic Signal Summary Measures

• Kappa statistic: Blomberg et al. Evol. (2003)

$$\small{K=\frac{(\mathbf{Z}-E(\mathbf{Z}))^T(\mathbf{Z}-E(\mathbf{Z}))}{(\mathbf{Z}-E(\mathbf{Z}))^T\mathbf\Omega^{-1}(\mathbf{Z}-E(\mathbf{Z}))}/\frac{tr(\mathbf\Omega)-N(\mathbf{1}^T\mathbf{\Omega1})^{-1}}{N-1}}$$

• $\lambda$ statistic: Pagel Nature (1999)

$$\small{logL=log(\frac{(\mathbf{Z}-E(\mathbf{Z}))^T\mathbf\Omega^\lambda{^{-1}}(\mathbf{Z}-E(\mathbf{Z})))}{\sqrt{(2\pi)^{Np}\times|\mathbf\Omega^\lambda|}}}$$

• Branch-length scaling during ML fit of data to phylogeny

• Both are for univariate response data only!

2: Multivariate Phylogenetic Signal $^1$

• A multivariate generalization of Kappa has been proposed: $\small{K}_{mult}$
  • Approach based on the distance-covariance equivalency
  • Uses distances between species in the original and phylo-transformed dataspaces

$$K_{mult}=\frac{\mathbf{D}_{\mathbf{Z},E(\mathbf{Z})}^{T}\mathbf{D}_{\mathbf{Z},E(\mathbf{Z})}}{\mathbf{PD}_{\tilde{\mathbf{Z}},0}^{T}\mathbf{PD}_{\tilde{\mathbf{Z}},0}}/\frac{tr(\mathbf\Omega)-N(\mathbf{1}^{T}\mathbf{\Omega1})^{-1}}{N-1}$$

• $\small\mathbf{D}_{\mathbf{Z},E(\mathbf{Z})}$ is the distance from each object to the phylogenetic mean
• $\small\mathbf{PD}_{\tilde{\mathbf{Z}},0}$ is the distance between each object in the phylogenetically-transformed space (found as: $\small\tilde{\mathbf{Z}}=\mathbf{P}(\mathbf{Z-E(\mathbf{Z})})$) and the origin

This is the same as:

$$K_{mult}=\frac{tr\left((\mathbf{Z}-E(\mathbf{Z}))^T(\mathbf{Z}-E(\mathbf{Z})\right)}{tr\left((\mathbf{Z}-E(\mathbf{Z}))^T\mathbf\Omega^{-1}(\mathbf{Z}-E(\mathbf{Z}))\right)}/\frac{tr(\mathbf\Omega)-N(\mathbf{1}^T\mathbf{\Omega1})^{-1}}{N-1}$$

1: Adams (2014). Syst. Biol.

2: Multivariate Phylogenetic Signal: Example

Does head shape in in Plethodon display phylogenetic signal?

PS.shape <- physignal(A=shape,phy=plethtree,iter=999, print.progress = FALSE)
summary(PS.shape)

## 
## Call:
## physignal(A = shape, phy = plethtree, iter = 999, print.progress = FALSE) 
## 
## 
## 
## Observed Phylogenetic Signal (K): 0.392
## 
## P-value: 0.001
## 
## Based on 1000 random permutations
## 
##  Use physignal.z to estimate effect size.

plot(PS.shape)

2: Phylogenetic Signal: An Effect Size $^1$

• Optimizing $\Omega$ by $\lambda$ in the equation of $Kappa$ provides a direct link between the two statistics $^2$

• This formulation lends itself nicely to evaluation via RRPP

  • RRPP many times, obtaining $logL$

  • Calculate Effect size as:

  $$\small{Z=\frac{\theta_{obs}-\mu_\theta}{\sigma_\theta}}$$ where $\theta_{obs}=f(logL_\Omega)$

• One can now compare effect sizes to evaluate the strength of phylogenetic signal across datasets:

$$\small{Z_{12}=\frac{(\theta_{obs_1}-\mu_{\theta_1})-(\theta_{obs_2}-\mu_{\theta_2})}{\sqrt{\sigma^2_{\theta_1}+\sigma^2_{\theta_2}}}}$$

2: Comparing Phylogenetic Signal: Example

From Collyer et al. (2022). Methods Ecol. Evol.

3: Phylogenetic Ordination

• Ordinations that incorporate phylogenetic relatedness

• Intent is to provide visual insights into macroevolutionary patterns

• Three approaches:

1. Phylomorphospace: project phylogeny into PCA space (using ancestral states)

2. Phylogenetic PCA (pPCA): account for phylogeny in PCA computations

   • Rotates data 'away' from phylogenetic signal (i.e., $pPC_1$ is uncorrelated with phylogenetic signal)

3. Phylogenetically-Aligned Components Analysis (PACA)

   • Rotates data 'towards' phylogenetic signal (i.e., $PACA_1$ maximizes phylogenetic signal)

3: Phylogenetic Ordination $^1$

• Recall the general ordination equation (Shape Statistics I lecture):

$$\mathbf{I}^T_{n \times n} \mathbf{Z}=\mathbf{UDV}^T$$

where: $\mathbf{A}$ = $\mathbf{I}_{n \times n}$. This is PCA, and finds the principal directions of variation in the data independent of any other association.

1: See Collyer and Adams (2021). Methods Ecol. Evol.

Mathematically the difference between these phylogenetic ordination procedures is that they align the data relative to different matrices in $\mathbf{A}$.

3: Phylogenetic Ordination: Phylomorphospace (PCA)

• Phylomorphospace is a PCA with the phylogeny superimposed (via projection of ancestral states)

$$\mathbf{I}^T_{n \times n} \mathbf{Z}=\mathbf{UDV}^T$$

plot.pca <- gm.prcomp(shape,phy=plethtree)
plot(plot.pca,phylo = TRUE, pch=21, bg=gdf$elev, cex=2, phylo.par = list(tip.labels = FALSE, node.labels = FALSE) )
legend("topleft", pch=21, pt.bg = unique(gdf$elev), legend = levels(gdf$elev))

3: Phylogenetic Ordination: Phylogenetic PCA (pPCA)

• Aligns data to directions independent of phylogenetic signal

$$\mathbf{T^T_\Omega}\mathbf{Z} =\mathbf{UDV}^T$$

plot.ppca <- gm.prcomp(shape,phy=plethtree, GLS = TRUE, transform = FALSE)
plot(plot.ppca,phylo = TRUE, pch=21, bg=gdf$elev, cex=2, phylo.par = list(tip.labels = FALSE, node.labels = FALSE) )
legend("topleft", pch=21, pt.bg = unique(gdf$elev), legend = levels(gdf$elev))

3: Phylogenetic Ordination: Phylogenetically Aligned Components (PACA)

• Aligns data to directions that maximize phylogenetic signal

$$\mathbf{\Omega^T}\mathbf{Z} =\mathbf{UDV}^T$$

plot.paca <- gm.prcomp(shape,phy=plethtree, align.to.phy = TRUE)
plot(plot.paca,phylo = TRUE, pch=21, bg=gdf$elev, cex=2, phylo.par = list(tip.labels = FALSE, node.labels = FALSE) )
legend("topleft", pch=21, pt.bg = unique(gdf$elev), legend = levels(gdf$elev))

3: Phylogenetic Ordination: Side By Side

3: Phylogenetic Ordination: Comments

• Methods provide contrasting views of the data
  • Phylomorphospace is a 'pure' view of the space
  • pPCA and PACA rotate relative to the phylogeny (albeit differently)

• Utilizing several may be helpful

4: Comparing Evolutionary Models

• What evolutionary model best describes trait variation?

4: Comparing Evolutionary Rates Among Clades

Is there evidence for multiple evolutionary rates on the phylogeny?

• Fit data to phylogeny under single-rate and multi-rate models, where $$\small\log{L}=-\frac{np}{2}ln{2\pi}-\frac{n}{2}ln{\begin{vmatrix}{(\mathbf{\sigma^2\Omega})}\end{vmatrix}}-\frac{1}{2}tr{(\mathbf{Z}-E(\mathbf{Z}))^{T}(\mathbf{\sigma^2\Omega})^{-1}(\mathbf{Z}-E(\mathbf{Z}))}$$ Perform LRT, AIC, simulation, or permutation-based (RRPP) evaluation

4: Comparing Multivariate Evolutionary Rates Among Clades

• Approach has been extended to multivariate traits (using distance-based procedure of Adams 2014)
• $\sigma^2_{mult}=\mathbf{PD}_{\tilde{\mathbf{Z}},0}^{T}\mathbf{PD}_{\tilde{\mathbf{Z}},0}/N-1$
  • RRPP used for model evaluation

Does head shape in high-elevation Plethodon species evolve more rapidly than in low-elevation species?

ER<-compare.evol.rates(A=gdf$shape, phy=plethtree,gp=gdf$elev,iter=999, method = 'permutation',print.progress = FALSE)
summary(ER)

## 
## Call:
## 
## 
## Observed Rate Ratio: 1.0133
## 
## P-value: 0.9645
## 
## Effect Size: -1.5892
## 
## Based on 1000 random permutations
## 
## The rate for group High is 1.00093880316058e-05  
## 
## The rate for group Low is 1.01425751777744e-05

4: Comparing Evolutionary Rates Among Traits

• Is there evidence that one trait evolves faster than another?

4: Comparing Evolutionary Rates Among Traits: Example

4: Comparing Evolutionary Rates Among Multivariate Traits

• Approach has been extended to multivariate traits (using distance-based procedure of Adams 2014)
• $\sigma^2_{mult}=\mathbf{PD}_{\tilde{\mathbf{Z}},0}^{T}\mathbf{PD}_{\tilde{\mathbf{Z}},0}/N-1$
  • Rate-ratio used as test measure: $R=\frac{max(\sigma^2_{mult})}{min(\sigma^2_{mult})}$
  • Permutation tests used for statistical evaluation

EMR <- compare.multi.evol.rates(A=gdf$shape, phy=plethtree, gp=c(rep(1,5),rep(2,6)), print.progress = FALSE)
summary(EMR)

## 
## Call:
## 
## 
## Observed Rate Ratio: 1.0826
## 
## P-value: 0.975
## 
## Effect Size: -1.943
## 
## Based on 1000 random permutations
## 
## The rate for group 1 is 9.67170500545254e-06  
## 
## The rate for group 2 is 1.04710160170648e-05

4: Comparing Multivariate Evolutionary Rates: Other Approaches $^1$

• Likelihood-based model comparisons comparing rates for multivariate data have also been proposed

• logL (various authors): Methods work so long as $N>>p$.

  • When this is not the case, $|\mathbf{R\otimes\Omega}|$ is singular, and the logL cannot be completed.

1: see Adams and Collyer (2018;2019)

• PCL (Pairwise Composite Likelihood: Goolsby 2016)

  • Method obtains set of logL for pairs of traits and sums them

  • Method has high type I error rate which differs by the rotation of the data!

4: Comparing Evolutionary Models: Evolutionary 'Modes'

• Compare models that describe the manner in which trait variation accumulates over evolutionary time

• BM1, BMM, OU (Ornstein-Uhlenbeck), Early Burst (EB), ACDC, etc.

• Fit data to phylogeny under differing evolutionary models and compare fit (LRT, AIC, simulation, RRPP, etc.)

4: Comparing Evolutionary 'Modes': Univariate Example

• How did Anolis body size groups evolve?
  • 5 models: BM, $\small{OU}_1$, $\small{OU}_3$ $\small{OU}_4$ (3 group+anc), $\small{OU}_{LP}$ (3 gp + history of colonization)

• $\small{OU}_{LP}$ (3 gp + col. hist.) best explains body size evolution

4: Comparing Evolutionary 'Modes': Multivariate Data

• Several methods proposed for comparing multivariate evolutionary models: $logL_{mult}$, $PCL$, $\sum{logL_{indiv}}$

• Unfortunately, nearly all current implementations for multivariate OU models display high misspecification rates

• More analytical development needed here

4: Comparing Evolutionary 'Modes': Recent Developments

• New developments in $logL_{mult}$
• Bartoszek et al. (2024) updated algorithms in mvSlouch $^1$

• Misspecification levels now appropriate, and method rotation invariant

• Provides a path forward for multivariate OU models (though still requires $n>p$ and no redundant dimensions)

1: took our suggestions to heart!

Phylogenetic Comparative Methods: Conclusions

• Phylogenetic comparative methods provide useful tools to account for phylogenetic non-independence among observations

• Hypotheses one can evaluate with shape data include:

  • phylogenetic ANOVA

  • Phylogenetic regression

  • Phylogenetic covariation (PLS)

  • Phylogenetic ordination (PCA, pPCA, PACA)

  • Phylogenetic signal (development regarding $\lambda_{mult}$ ongoing)

  • Comparing evolutionary rates

  • Comparing evolutionary modes (some advance: requires more analytical development)