Biology 559R: Introduction to Phylogenetic Comparative Methods

Topics for this week (Feb 24 & 26): • Ancestral state reconstruction (discrete) • Ancestral state reconstruction (continuous)

1 Ancestral Phenotypes

• The available information about past history of life is marked by its scarcity and incompleteness.

• To understand present diversity of life forms, we need to find a way to infer past forms.

• With very few examples, fossil record are not available for most taxa and some traits do not fossilize (e.g., behavioral traits, soft tissues).

Gingerich (1984) 2 Ancestral Phenotypes

• The current alternatives are to infer ancestral character states by mapping the phenotypes (or other characteristics, e.g., distributions) of living organisms onto phylogenies.

• The resulting outcomes of such reconstructions are hypotheses about ancient life forms and provide the basis to understand the path of evolution (decent with modification) to the extant species.

Ancestral calls of Physalaemus pustulosus using ancestral reconstructions (Ryan and Rand, 1999)

3 Ancestral Phenotypes

• These reconstructions also provide information about the number of times a character state has arisen independently and might support adaptation.

• Many factors determine the success of such reconstructions including a well-supported phylogeny, well-sampled extant living forms, and adequate character reconstruction methodologies

salamanders

biphasic life cycle (white) direct development(black)

stick insects: Whiting et al., 2003

4 Ancestral Phenotypes: Maximum Likelihood

• There many different methodologies for ancestral reconstruction: parsimony, maximum likelihood and Bayesian.

• ML approaches use an explicit model of character evolution to estimate the probabilities of all possible character state reconstructions at every node on the tree.

• In addition to the model of evolution, these probabilities are also determined by the distribution of character states in the terminal taxa.

• One interesting outcome from the ML approach, it is that the method considers every possible reconstruction and can estimate the relative probability of each character state at every node.

• This outcomes can be applied for hypothesis testing (e.g., fixing nodes to a particular state versus the inferred).

5 Ancestral Phenotypes: Maximum Likelihood

6 Ancestral Reconstruction (Discrete)

• Make a working directory (e.g., discrete) and select it as your working.

R top menus: Misc>Change Working Directory (select the directory that has our file) setwd("/Users/jcsantos/Desktop/R_class_winter_2015_home/0_Bio559R_course_final_files/ week_8/data/")!

• We need to download these files from the course website into our working directory: bird_data_pruned.txt bird_pruned_MB.newick tyrannidae_reduced_data.txt ß These are for your practice tyrannidae_reduced_tree.newick ß These are for your practice

• read 'ape’, ‘geiger’ and ‘phytools’ packages library(ape)! library(geiger)! library(phytools)! ! !

7 Ancestral Reconstruction (Discrete)

• Load our trees and data from last week and check concordance: ! bird_pruned_tree <- read.tree("bird_pruned_MB.newick")! bird_pruned_tree! birds_traits_data <- read.table("bird_data_pruned.txt", header = TRUE, sep = "\t")! birds_traits_data! ! using the 'read.tree' function of ape: ! bird_trees <-read.tree(file = "bird_tree.tree")! bird_trees #5 trees in loaded! bird_trees [[1]]# first tree a very large tree ! ! bird_tree_1 <- bird_trees [[1]]# assign first tree to object class phylo! ! • use the 'name.check' function to determine if we have concordance: ! name.check (bird_pruned_tree, birds_traits_data)! ! # [1] "OK" # we have concordance (i.e., we the same number of tips and data per tip)! ! !

8 Ancestral Reconstruction (Discrete)

• We need to make sure that our tree is ultrametric tree and preferably bifurcating ! is.ultrametric(bird_pruned_tree) # TRUE! ! is.binary.tree(bird_pruned_tree) ! # TRUE all nodes (including the root node) have exactly two descendant nodes!

• To plot, we need to sort variable values to match phylogeny names order match_birds_traits_data <- match(bird_pruned_tree$tip.label, rownames(birds_traits_data)) #use match function! sorted_birds_traits_data <- as.data.frame(birds_traits_data[match_birds_traits_data,])! sorted_birds_traits_data! !

9 Ancestral Reconstruction (Discrete)

• Prepare data for plotting the Food discrete trait. We are going to use colored label for discrete traits

#Food! Food_birds_label <- character(length(sorted_birds_traits_MR$Food)) ! names(Food_birds_label) <- rownames(sorted_birds_traits_MR)! ! #get the unique states for Food! ! unique(sorted_birds_traits_MR$Food)! #[1] vegetarian carnivore omnivore ! ! #label states with colors for plot! ! Food_birds_label[sorted_birds_traits_MR$Food=="vegetarian"] <- "green3" #assign color! Food_birds_label[sorted_birds_traits_MR$Food=="carnivore"] <- "red" #assign color! Food_birds_label[sorted_birds_traits_MR$Food=="omnivore"] <- "darkorchid" #assign color! Food_birds_label!

10 Ancestral Reconstruction (Discrete)

• Let’s plot the bird tree as a ‘phylogram’ type format without names:! ! plot(bird_pruned_tree, cex=1, show.tip.label = FALSE) #Plot a tree that leaves some room between the tree tips and taxon labels so that we can plot habitat use in this space! points(rep(122.5, nrow(sorted_birds_traits_data)), 1:nrow(sorted_birds_traits_data), pch=22, bg=Food_birds_label[bird_pruned_tree$tip.label], lwd = 0.05, col="white", cex=1.5)! !

11 Ancestral reconstruction using ‘ape’

• We can use the ace (Ancestral Character Estimation) function of ape to reconstruct ancestral character states using maximum likelihood:

Let’s explore ape: ?ace!

# ace(x, phy, ! # type = "discrete", ! # method = "ML", # only option for discrete characters, following Pagel 1994! # CI = TRUE, # 95% confidence intervals! # model = "ER", # Four option, equal rates (ER), ! # symmetrical (SYM), ! # All-rates different (ARD)! # scaled = TRUE, ! # kappa = 1, # if branch lengths needs transformation, ! # a positive value giving the exponent for such transformation! # corStruct = NULL, ! # ip = 0.1,! # use.expm = FALSE,! # use.eigen = TRUE, ! # marginal = FALSE) # returns the node marginal reconstruction (likelihood values) ! # at each node! • Let's explore the "ER" model: ! food_bird_ace_ER <- ace(birds_traits_data$Food, bird_pruned_tree, type="discrete", model = "ER", marginal = TRUE) #Equal rates! food_bird_ace_ER! 12 Ancestral reconstruction using ‘ape’ • Let's explore the "ER" model: ! # Ancestral Character Estimation! ! This is the model that was calculated:! ! # Call: ace(x = birds_traits_data$Food, phy = bird_pruned_tree, type = "discrete", model = "ER”)! ! The log-likelihood of our model: # Log-likelihood: -406.578 ! ! The instantaneous rate matrix (Q)– more on this later– for transformation between states:! ! # Rate index matrix:! # carnivore omnivore vegetarian! # carnivore . 1 1! # omnivore 1 . 1! # vegetarian 1 1 .! ! This is the parameter values including its standard error: ! # Parameter estimates:! # rate index estimate std-err! # 1 0.0083 7e-04! ! This is the marginal reconstructions at the root (all should add to 1):! ! # Scaled likelihoods at the root (type '...$lik.anc' to get them for all nodes):! # carnivore omnivore vegetarian ! 13 # 0.5489050 0.1842301 0.2668649 ! Ancestral reconstruction using ‘ape’ • Let's explore the "ER" model: ! # Ancestral Character Estimation! ! This is the model that was calculated:! ! # Call: ace(x = birds_traits_data$Food, phy = bird_pruned_tree, type = "discrete", model = "ER”)! ! The log-likelihood of our model: # Log-likelihood: -406.578 ! ! The instantaneous rate matrix (Q)– more on this later– for transformation between states:! ! # Rate index matrix:! # carnivore omnivore vegetarian! # carnivore . 1 1! # omnivore 1 . 1! # vegetarian 1 1 .! ! This is the parameter values including its standard error: ! # Parameter estimates:! # rate index estimate std-err! # 1 0.0083 7e-04! ! This is the marginal reconstructions at the root (each row should add to 1):! ! # Scaled likelihoods at the root (type '...$lik.anc' to get them for all nodes):! # carnivore omnivore vegetarian ! 14 # 0.5489050 0.1842301 0.2668649 ! Ancestral reconstruction using ‘ape’

• you can call the marginal probabilities at any node along the tree ! food_bird_ace_ER$lik.anc# Marginal values for each node, each row should add to ~1.0! ! # carnivore omnivore vegetarian! # [1,] 5.489050e-01 1.842301e-01 2.668649e-01! # [2,] 3.832150e-01 3.246254e-01 2.921596e-01! # [3,] 5.018158e-01 1.908888e-01 3.072954e-01! # [4,] 7.351661e-01 1.519711e-01 1.128628e-01! # [5,] 8.025606e-01 1.880971e-01 9.342243e-03! !

15 Ancestral reconstruction using ‘ape’ • Let's explore the ”SYM" model: ! food_bird_ace_SYM <- ace(birds_traits_data$Food, bird_pruned_tree, type="discrete", model = "SYM") #Symmetrical ! food_bird_ace_SYM! ! # Ancestral Character Estimation! #! # Call: ace(x = birds_traits_data$Food, phy = bird_pruned_tree, type = "discrete", model = "SYM")! #! # Log-likelihood: -401.9363 ! #! # Rate index matrix:! # carnivore omnivore vegetarian! # carnivore . 1 2! # omnivore 1 . 3! # vegetarian 2 3 .! #! # Parameter estimates:! # rate index estimate std-err! # 1 0.0111 0.0016! # 2 0.0044 0.0009! # 3 0.0129 0.0024! #! # Scaled likelihoods at the root (type '...$lik.anc' to get them for all nodes):! # carnivore omnivore vegetarian ! # 0.5452463 0.2518925 0.2028612 ! ! 16 Ancestral reconstruction using ‘ape’ • Let's explore the ”SYM" model: ! food_bird_ace_SYM <- ace(birds_traits_data$Food, bird_pruned_tree, type="discrete", model = "SYM") #Symmetrical ! food_bird_ace_SYM! ! # Ancestral Character Estimation! #! # Call: ace(x = birds_traits_data$Food, phy = bird_pruned_tree, type = "discrete", model = "SYM")! #! # Log-likelihood: -401.9363 ! #! # Rate index matrix:! # carnivore omnivore vegetarian! # carnivore . 1 2! # omnivore 1 . 3! # vegetarian 2 3 .! #! instantaneous rate matrix (Q) # Parameter estimates:! # rate index estimate std-err! # 1 0.0111 0.0016! # 2 0.0044 0.0009! # 3 0.0129 0.0024! #! # Scaled likelihoods at the root (type '...$lik.anc' to get them for all nodes):! # carnivore omnivore vegetarian ! # 0.5452463 0.2518925 0.2028612 ! ! 17 Ancestral reconstruction using ‘ape’ • Let's explore the ”ARD" model: ! food_bird_ace_ARD <- ace(birds_traits_data$Food, bird_pruned_tree, type="discrete", model = "ARD") #All-rates different! food_bird_ace_ARD! ! # Ancestral Character Estimation! #! # Call: ace(x = birds_traits_data$Food, phy = bird_pruned_tree, type = "discrete", model = "ARD")! #! # Log-likelihood: -386.3962 ! #! # Rate index matrix:! # carnivore omnivore vegetarian! # carnivore . 3 5! # omnivore 1 . 6! # vegetarian 2 4 .! #! # Parameter estimates:! # rate index estimate std-err! # 1 0.0174 0.0025! # 2 0.0095 0.0017! # 3 0.0064 0.0014! # 4 0.0056 0.0017! # 5 0.0005 0.0006! # 6 0.0153 0.0024! #! # Scaled likelihoods at the root (type '...$lik.anc' to get them for all nodes):! # carnivore omnivore vegetarian ! 18 # 0.0379084 0.2905876 0.6715040 ! Ancestral reconstruction using ‘ape’ • Let's explore the ”ARD" model: ! food_bird_ace_ARD <- ace(birds_traits_data$Food, bird_pruned_tree, type="discrete", model = "ARD") #All-rates different! food_bird_ace_ARD! ! # Ancestral Character Estimation! #! # Call: ace(x = birds_traits_data$Food, phy = bird_pruned_tree, type = "discrete", model = "ARD")! #! # Log-likelihood: -386.3962 ! #! # Rate index matrix:! # carnivore omnivore vegetarian! # carnivore . 3 5! # omnivore 1 . 6! # vegetarian 2 4 .! #! instantaneous rate matrix (Q) # Parameter estimates:! # rate index estimate std-err! # 1 0.0174 0.0025! # 2 0.0095 0.0017! # 3 0.0064 0.0014! # 4 0.0056 0.0017! # 5 0.0005 0.0006! # 6 0.0153 0.0024! #! # Scaled likelihoods at the root (type '...$lik.anc' to get them for all nodes):! # carnivore omnivore vegetarian ! 19 # 0.0379084 0.2905876 0.6715040 ! Ancestral reconstruction using ‘ape’: Plot ancestral states

• We can plot the marginal likelihoods for each state reconstruction at each node using pie diagrams: ! #Food! # We need to sort variable values to match phylogeny names order! ! match_birds_data <- match(bird_pruned_tree$tip.label, rownames(birds_traits_data)) #use match function! sorted_birds_data <- as.data.frame(birds_traits_data[match_birds_data,])! sorted_birds_data! ! #labels! ! Food_birds_label <- character(length(sorted_birds_data$Food)) ! names(Food_birds_label) <- rownames(sorted_birds_data)! ! #get the unique states for Food! ! unique(sorted_birds_data$Food)! #[1] vegetarian carnivore omnivore ! ! #label states with colors for plot! ! Food_birds_label[sorted_birds_data$Food=="vegetarian"] <- "vegetarian" #assign color! Food_birds_label[sorted_birds_data$Food=="carnivore"] <- "carnivore" #assign color! Food_birds_label[sorted_birds_data$Food=="omnivore"] <- "omnivore" #assign color! Food_birds_label! ! 20 Ancestral reconstruction using ‘ape’: Plot ancestral states

• To create a binary matrix for our characters (useful for drawing) we use 'to.matrix’ function of phytools ! food_pie_matrix <- to.matrix(Food_birds_label, sort(unique(Food_birds_label)))! food_pie_matrix! ! carnivore omnivore vegetarian! Nothoprocta_perdicaria 0 0 1! Apteryx_australis 1 0 0! Apteryx_haastii 1 0 0! Apteryx_owenii 1 0 0! Dromaius_novaehollandiae 0 1 0! Struthio_camelus 0 1 0! Callipepla_californica 0 0 1! Callipepla_gambelii 0 0 1! Coturnix_chinensis 0 0 1! Coturnix_pectoralis 0 0 1! Coturnix_japonica 0 0 1! Coturnix_coturnix 0 0 1! Alectoris_graeca 0 0 1! ...! ! !

21 Ancestral reconstruction using ‘ape’: Plot ancestral states

• Plot our tree and its ancestral reconstruction (ER – model): ! plot(bird_pruned_tree, show.node.label=FALSE, show.tip.label = FALSE, label.offset=3, cex=0.4, type = "fan") ! tiplabels(pie = food_pie_matrix[bird_pruned_tree$tip.label, ], piecol = c("red", "darkorchid", "green3"), cex = 0.12)! nodelabels(pie=food_bird_ace_ER$lik.anc, piecol=c("red", "darkorchid", "green3"), cex = 0.12)! axisPhylo()! !

22 Ancestral reconstruction using ‘ape’: Plot ancestral states

• Plot our tree and its ancestral reconstruction (SYM and ARD – models): ! plot(bird_pruned_tree, show.node.label=FALSE, show.tip.label = FALSE, label.offset=3, cex=0.4) ! tiplabels(pie = food_pie_matrix[bird_pruned_tree$tip.label, ], piecol = c("red", "darkorchid", "green3"), cex = 0.12)! nodelabels(pie=food_bird_ace_SYM$lik.anc, piecol=c("red", "darkorchid", "green3"), cex = 0.12)! ! plot(bird_pruned_tree, show.node.label=FALSE, show.tip.label = FALSE, label.offset=3, cex=0.4) ! tiplabels(pie = food_pie_matrix[bird_pruned_tree$tip.label, ], piecol = c("red", "darkorchid", "green3"), cex = 0.12)! nodelabels(pie=food_bird_ace_ARD$lik.anc, piecol=c("red", "darkorchid", "green3"), cex = 0.12)! axisPhylo()!

23 Fit models of trait (discrete) evolution using ‘geiger’

• This is one of the best and more flexible methods. It is implemented under the ‘fitDiscrete’ function in 'geiger’. This method provides a set of statistics that allows us to decide which model best describes the evolution of our trait of interest.

Notice: We also need to understand the biology of our system of study.

• First, we need to prepare our data using the function 'treedata'

Food_birds_geiger <- treedata(bird_pruned_tree, birds_traits_data)! Food_birds_geiger$phy # a class phylo! Food_birds_geiger$data # a matrix!

• Let's explore the 'fitDiscrete' function ! ?fitDiscrete!

24 Fit models of trait (discrete) evolution using ‘geiger’

• The 'fitDiscrete' allows to fit models for discrete data. This function returns parameter estimates and the likelihood of the model for a single trait (univariate) datasets.

• All of the models are continuous-time Markov models of trait evolution.

• This type of models allow modeling a random process where the probability of change between character states depends only on the current state.

• For example, if we have a binary trait (i.e., 0-absent, 1-present) there should be at any time a chance a character change of state from 0-absent to 1-present or vice versa. This is represented by the instantaneous rate matrix (Q)

25 Fit models of trait (discrete) evolution using ‘geiger’

Basic models

ER is an equal-rates model and a single parameter governs all transition rates at the same rate: 0 <-> 1, 0 <-> 2, 1 <-> 2

SYM is a symmetric model where forward and reverse transitions share the same parameter rate-1: 0 <-> 1, rate-2: 0 <-> 2, rate-3: 1 <-> 2

ARD is an all-rates-different model where each rate is a unique parameter all rates are different

26 Fit models of trait (discrete) evolution using ‘geiger’

• Let's estimate the ER model food_bird_ace_ER_geiger <- fitDiscrete(bird_pruned_tree, Food_birds_geiger$data [,13], type="discrete", model = "ER") ! food_bird_ace_ER_geiger!

# GEIGER-fitted comparative model of discrete data! # fitted Q matrix: <-- This is the instantaneous matrix! # carnivore omnivore vegetarian! # carnivore -0.010154425 0.005077212 0.005077212! # omnivore 0.005077212 -0.010154425 0.005077212! # vegetarian 0.005077212 0.005077212 -0.010154425! #! # model summary: <-- This is set of statistics to compare models! # !log-likelihood = -326.374043! # !AIC = 654.748086! # !AICc = 654.756994! # !free parameters = 1! #! # Convergence diagnostics: <-- Report in the convergence ! # !optimization iterations = 100! # !failed iterations = 0! # !frequency of best fit = 1.00! #! # object summary: <-- objects that can be called for analyses! # !'lik' -- likelihood function! # !'bnd' -- bounds for likelihood search! # !'res' -- optimization iteration summary! # !'opt' -- maximum likelihood parameter estimates! 27 Fit models of trait (discrete) evolution using ‘geiger’

• Let's estimate the SYM model ! food_bird_ace_SYM_geiger <- fitDiscrete(bird_pruned_tree, Food_birds_geiger$data [,13], type="discrete", model = "SYM")! food_bird_ace_SYM_geiger! ! # GEIGER-fitted comparative model of discrete data! # fitted Q matrix:! # carnivore omnivore vegetarian! # carnivore -0.007866836 0.005917884 0.001948951! # omnivore 0.005917884 -0.017907770 0.011989886! # vegetarian 0.001948951 0.011989886 -0.013938837! #! # model summary:! # !log-likelihood = -315.365882! # !AIC = 636.731764! # !AICc = 636.785455! # !free parameters = 3! #! #Convergence diagnostics:! # !optimization iterations = 100! # !failed iterations = 0! # !frequency of best fit = 0.06! #! # object summary:! # !'lik' -- likelihood function! # !'bnd' -- bounds for likelihood search! # !'res' -- optimization iteration summary! # !'opt' -- maximum likelihood parameter estimates! 28 Fit models of trait (discrete) evolution using ‘geiger’

• Let's estimate the ARD model ! food_bird_ace_ARD_geiger <- fitDiscrete(bird_pruned_tree, Food_birds_geiger$data [,13], type="discrete", model = "ARD")! food_bird_ace_ARD_geiger ! ! # GEIGER-fitted comparative model of discrete data! # fitted Q matrix:! # carnivore omnivore vegetarian! # carnivore -0.002923285 0.002923285 8.825334e-20! # omnivore 0.017452911 -0.032781504 1.532859e-02! # vegetarian 0.001587524 0.006893189 -8.480713e-03! #! # model summary:! # !log-likelihood = -304.427936! # !AIC = 620.855872! # !AICc = 621.045062! # !free parameters = 6! #! #Convergence diagnostics:! # !optimization iterations = 100! # !failed iterations = 0! # !frequency of best fit = 0.06! #! # object summary:! # !'lik' -- likelihood function! # !'bnd' -- bounds for likelihood search! # !'res' -- optimization iteration summary! # !'opt' -- maximum likelihood parameter estimates! 29 Fit models of trait (discrete) evolution using ‘geiger’

• Let's compare models based on the summaries ! # parameters! # !free parameters = 1 #food_bird_ace_ER_geiger -- most simple! # !free parameters = 3 #food_bird_ace_SYM_geiger ! # !free parameters = 6 #food_bird_ace_ADR_geiger -- most complex! ! # log-likelihood! # !log-likelihood = -326.374043 #food_bird_ace_ER_geiger -- worst score! # !log-likelihood = -315.365882 #food_bird_ace_SYM_geiger! # !log-likelihood = -304.427936 #food_bird_ace_ADR_geiger -- best score! ! # AICc! # !AICc = 654.756994 #food_bird_ace_ER_geiger -- worst score! # !AICc = 636.785455 #food_bird_ace_SYM_geiger! # !AICc = 621.045062 #food_bird_ace_ADR_geiger -- best score! ! !

30 Fit models of trait (discrete) evolution using ‘geiger’

• For model comparison, we prefer a simple model with the highest likelihood (the lowest –lnL) and the lowest AIC score ! # For AICc consider:! ! # |AICc_model1 - AICc_model2| < 2-3 (no support for the more complex model)! # |AICc_model1 - AICc_model2| = 4-9 (some support for the more complex model)! # |AICc_model1 - AICc_model2| > 10 (so support for the simple model)! ! ER_versus_SYM <- abs(food_bird_ace_ER_geiger$opt$aicc - food_bird_ace_SYM_geiger$opt$aicc)! ER_versus_SYM # 17.97154 -- no support from ER! ! ER_versus_ARD <- abs(food_bird_ace_ER_geiger$opt$aicc - food_bird_ace_ARD_geiger$opt$aicc)! ER_versus_ARD # 33.71193 -- no support from ER! ! SYM_versus_ARD <- abs(food_bird_ace_SYM_geiger$opt$aicc - food_bird_ace_ARD_geiger$opt$aicc)! SYM_versus_ARD # 15.74039 -- no support from SYM! ! #Delta AICc suggest that ARD is the best model! !

31 Fit models of trait (discrete) evolution using ‘geiger’

• We can compare the likelihood of the two models using a likelihood ratio test. You can calculate the p-value of the likelihood ratio test in R by comparing it to the chi- square distribution with the degrees of freedom (i.e., difference in number of parameters for the two models).

##### ER versus SYM! ! d_ER_versus_SYM <- abs(2*(food_bird_ace_ER_geiger$opt$lnL-food_bird_ace_SYM_geiger$opt$lnL))! d_ER_versus_SYM ! ! #[1] 22.01632! ! p_value_ER_versus_SYM <- pchisq(d_ER_versus_SYM, 3-1, lower.tail=FALSE)! p_value_ER_versus_SYM! ! #[1] 1.656595e-05 -- Rejects ER-model! ! ##### ER versus ARD! ! d_ER_versus_ARD <- abs(2*(food_bird_ace_ER_geiger$opt$lnL-food_bird_ace_ARD_geiger$opt$lnL))! d_ER_versus_ARD ! ! #[1] 43.89221! ! p_value_ER_versus_ARD <- pchisq(d_ER_versus_ARD, 6-1, lower.tail=FALSE)! p_value_ER_versus_ARD! ! #[1] 2.435899e-08 -- Rejects ER-model! 32 Fit models of trait (discrete) evolution using ‘geiger’

• We can compare the likelihood of the two models using a likelihood ratio test. You can calculate the p-value of the likelihood ratio test in R by comparing it to the chi- square distribution with the degrees of freedom (i.e., difference in number of parameters for the two models). ! ##### SYM versus ARD! ! d_SYM_versus_ARD <- abs(2*(food_bird_ace_SYM_geiger$opt$lnL-food_bird_ace_ARD_geiger$opt $lnL))! d_SYM_versus_ARD ! ! #[1] 21.87589! ! p_value_SYM_versus_ARD <- pchisq(d_SYM_versus_ARD, 6-3, lower.tail=FALSE)! p_value_SYM_versus_ARD! ! #[1] 6.922698e-05 -- Rejects SYM-model! ! ! We can conclude that ARD (i.e., all rates are different for the transition between character states) does a better job than the other models to trace the evolution of food habit in birds !

33 Fit models of trait (discrete) evolution using ‘geiger’

• Let's explore our best model (the ARD model) ! food_bird_ace_ARD_geiger <- fitDiscrete(bird_pruned_tree, Food_birds_geiger$data [,13], type="discrete", model = "ARD")! food_bird_ace_ARD_geiger ! ! # GEIGER-fitted comparative model of discrete data! # fitted Q matrix:! # carnivore omnivore vegetarian! # carnivore -0.002923285 0.002923285 8.825334e-20! The transition from meat eaters # omnivore 0.017452911 -0.032781504 1.532859e-02! to vegetarians is very rare # vegetarian 0.001587524 0.006893189 -8.480713e-03! #! # model summary:! # !log-likelihood = -304.427936! # !AIC = 620.855872! # !AICc = 621.045062! # !free parameters = 6! #! #Convergence diagnostics:! # !optimization iterations = 100! # !failed iterations = 0! # !frequency of best fit = 0.06! #! # object summary:! # !'lik' -- likelihood function! # !'bnd' -- bounds for likelihood search! # !'res' -- optimization iteration summary! # !'opt' -- maximum likelihood parameter estimates! 34 Phylogenetic Signal for Discrete Characters

• We can use Pagel's lambda to assesses the phylogenetic signal (tendency of closely related species to resemble each other for an specific trait).

Pagel M. 1999. Inferring the historical patterns of biological evolution. Nature 401:877-884.

• Lambda parameter is also defined as a tree transformation parameter that multiply the internal branches of the tree by values between 0 and 1.

A lambda of value equal or near 1 indicates strong phylogenetic signal (traits evolve in close association with the phylogeny)

A lambda of value equal or near 0 indicates no signal (traits are independent of the phylogeny)

• In consequence, we can determine the magnitude of the signal by estimating the lambda parameter in our tree and then compare the model fit using log-lik with that of model where the tree was transformed to a polytomy (i.e., branches multiplied by lambda = 0) !

35 Phylogenetic Signal for Discrete Characters

• Let's estimate our lambda 0 tree bird_pruned_tree_lambda_0 <- rescale(bird_pruned_tree, model = "lambda", 0)!

• Let's plot this tree and the original bird tree par(mfrow=c(1,2))! plot(bird_pruned_tree, edge.width = 1, show.tip.label = FALSE) #smaller tree! add.scale.bar(cex = 0.7, font = 2, col = "red")! plot(bird_pruned_tree_lambda_0, edge.width = 1, show.tip.label = FALSE) #smaller tree! add.scale.bar(cex = 0.7, font = 2, col = "red")!

36 Phylogenetic Signal for Discrete Characters Our tree No signal (lambda = 0)

37 Phylogenetic Signal for Discrete Characters

• Let's fit the lambda transformation to determine if there is phylogenetic signal ! # Fit lambda to our tree! food_bird_ace_ARD_geiger_lambda <- fitDiscrete(bird_pruned_tree, Food_birds_geiger$data [, 13], type="discrete", model = "ARD", transform = "lambda”, niter = 500)! ! # GEIGER-fitted comparative model of discrete data! # fitted Q matrix:! # carnivore omnivore vegetarian! # carnivore -1.950375e-03 0.001950375 4.052955e-16! # omnivore 1.255327e-02 -0.024693463 1.214019e-02! # vegetarian 1.481506e-16 0.001482542 -1.482542e-03! #! # fitted ‘lambda’ model parameter:! # !lambda = 0.906137! #! # model summary:! # !log-likelihood = -290.226291! # !AIC = 594.452583! # !AICc = 594.705404! # !free parameters = 7! #! # Convergence diagnostics:! # !optimization iterations = 100! # !failed iterations = 63! # !frequency of best fit = NA!

38 Phylogenetic Signal for Discrete Characters

• Let's fit the lambda transformation to determine if there is signal ! # Fit the lamda 0 tree (no signal)! ! food_bird_ace_ARD_geiger_lambda_0 <- fitDiscrete(bird_pruned_tree_lambda_0, Food_birds_geiger $data [,13], type="discrete", model = "ARD", transform = "lambda", niter = 500)! ! #GEIGER-fitted comparative model of discrete data! # fitted Q matrix:! # carnivore omnivore vegetarian! # carnivore -0.0069920757 0.0029614817 0.0040305940! # omnivore 0.0005590967 -0.0013650835 0.0008059868! # vegetarian 0.0005886789 0.0007984343 -0.0013871132! #! # fitted ‘lambda’ model parameter:! # !lambda = 0.069683! #! # model summary:! # !log-likelihood = -479.923127! # !AIC = 973.846253! # !AICc = 974.099075! # !free parameters = 7! #! #Convergence diagnostics:! # !optimization iterations = 100! # !failed iterations = 0! # !frequency of best fit = 0.45!

39 Phylogenetic Signal for Discrete Characters

• For model comparison, the model with the lowest AIC score is preferred ! our_tree_versus_lambda_0_aicc <- abs(food_bird_ace_ARD_geiger_lambda_0$opt$aicc - food_bird_ace_ARD_geiger_lambda$opt$aicc)! our_tree_versus_lambda_0_aicc # 379.3937 -- lambda 0 tree (no signal) is very unlikely! ! ##### our tree versus lambda 0! ! d_our_tree_versus_lambda_0 <- abs(2*(food_bird_ace_ARD_geiger_lambda_0$opt$lnL- food_bird_ace_ARD_geiger_lambda$opt$lnL))! d_our_tree_versus_lambda_0! ! #[1] 379.3937! ! p_our_tree_versus_lambda_0 <- pchisq(d_our_tree_versus_lambda_0, 1, lower.tail=FALSE)! p_our_tree_versus_lambda_0! ! #[1] 1.686434e-84 -- Rejects no signal model! ! ! • The food trait present a strong phylogenetic signal (i.e., birds that eat similar food tend to be related)!

40 Testing if rates have increased/decreased Discrete Characters

• We can test whether rates of change have increased or slowed over evolutionary time using 'fitDiscrete'.

• For this purpose, we use the delta model.

The delta model fits the relative contributions of early versus late evolution of a character state in our tree to the covariance of species trait values.

If delta > 1, recent evolution has been relatively fast If delta < 1, recent evolution has been relatively slow

If rates decrease over time it is a signature of adaptive radiation if the traits are ecologically relevant. ! food_bird_ace_ARD_geiger_delta <- fitDiscrete(bird_pruned_tree, Food_birds_geiger$data [,13], type="discrete", model = "ARD", transform = "delta")! ! !

41 Testing if rates have increased/decreased Discrete Characters

• Let’s explore the results ! food_bird_ace_ARD_geiger_delta <- fitDiscrete(bird_pruned_tree, Food_birds_geiger$data [,13], type="discrete", model = "ARD", transform = "delta")! ! ! #GEIGER-fitted comparative model of discrete data! # fitted Q matrix:! # carnivore omnivore vegetarian! # carnivore -0.0018102415 0.001810241 2.114637e-38! # omnivore 0.0120647007 -0.022792976 1.072828e-02! # vegetarian 0.0004968849 0.003132820 -3.629705e-03! #! # fitted ‘delta’ model parameter:! # !delta = 1.691392! #! # model summary:! # !log-likelihood = -303.756563! # !AIC = 621.513126! # !AICc = 621.765947! # !free parameters = 7! #! # Convergence diagnostics:! # !optimization iterations = 100! # !failed iterations = 0! # !frequency of best fit = 0.01! !

42 Testing if rates have increased/decreased Discrete Characters

• We compare the AICc values ! # !AICc = 621.765947 for delta! # !AICc = 654.756994 for equal rates! ! # For model comparison, the model with the lowest AIC score is preferred! # difference in AICc is 32.991047. Then delta model is supported! ! # fitted ‘delta’ model parameter:! # !delta = 1.691392! ! The delta > 1 suggest that recent evolution has been relatively fast

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