# Maximum likelhood for models of trait evoluion.

### Description

Fits likelihood models for various models of continuous character evolution. Model fitting is based on maximum-likelihood evaluation using phylogenetically independent contrasts. This is exactly equivalent to, but substantially faster than, GLS approaches.

### Usage

1 2 3 |

### Arguments

`y` |
A matrix of trait values. |

`phy` |
An object of class "phylo" (see ape package). |

`model` |
The model of trait evolution (see details). |

`modelCIs` |
Logical - estimate confidence intervals for parameter estimates. |

`nodeIDs` |
Integer - ancestral nodes of clades. |

`rateType` |
If model="clade", a vector specifying if rate shift occurs in a clade ("clade") or on the single branch leading to a clade ("branch"). |

`minCladeSize` |
Integer - minimum clade size for inferred rate shift (where model="medusa"). |

`nSplits` |
Integer - number of rate shifts to apply for model="medusa" |

`restrictNode` |
List defining monophyletic groups within which no further rate shifts are searched for. |

`lowerBound` |
Minimum value for parameter estimates |

`upperBound` |
Maximum value for parameter estimates |

`tol` |
Tolerance (minimum branch length) to exclude branches from trait MEDUSA search. Primarily intended to prevent inference of rate shifts at randomly resolved polytomies. |

`n.cores` |
Integer. Set number of computing cores when running model="medusa" |

### Details

This function finds the maximum likelihood parameter values for continuous character evolution. The function returns the maximum-likihood parameter estimates for the following models.

model="bm"- Brownian motion (constant rates random walk).

model="kappa" - fits Pagel's kappa by raising all branch lengths to the power kappa. As kappa approaches zero, trait change becomes focused at branching events. For complete phylogenies, if kappa approaches zero this infers speciational trait change. Default bounds from ~0 - 1.

model="lambda" - fits Pagel's lambda to estimate phylogenetic signal by multiplying all internal branches of the tree by lambda, leaving tip branches as their original length (root to tip distances are unchanged). Default bounds from ~0 - 1.

model="delta" - fits Pagel's delta by raising all node depths to the power delta. If delta <1, trait evolution is concentrated early in the tree whereas if delta >1 trait evolution is concentrated towards the tips. Values of delta above one can be difficult to fit reliably. Default bounds from ~0 - 5.

model="OU" - fits an Ornstein-Uhlenbeck model - a random walk with a central tendency proportional to alpha. High values of alpha can be interpreted as evidence of evolutionary constraints, stabilising selection or weak phylogenetic signal. It is often difficult to distinguish among these possibilities. Default bounds from ~0 - 10.

model="psi" - fits a model to assess to the relative contributions of speciation and gradual evolution to a trait's evolutionary rate (Ingram 2010).

model="free" - fits Mooers et al's free model where each branch has its own rate of trait evolution. This can be a useful exploratory analysis but it is slow due to the number of parameters, particularly for large trees. Default rate parameter bounds from ~0 - 200.

model="clade" - fits a model where particular clades are a priori hypothesised to have different rates of trait evolution (see O'Meara et al. 2006; Thomas et al. 2006, 2009). Clades are specified using nodeIDs and are defined as the mrca node. Default rate parameter bounds from ~0 - 200.

model="tm1" - fits "clade" models without any a priori assertion of the location of phenotypic diversification rate shifts. It uses the same AIC approach as the runMedusa function in the geiger package (runMedusa tests for shifts in the rate of lineage diversification). The algorithm first fits a constant-rate Brownian model to the data, it then works iteratively through the phylogeny fitting a two-rate model at each node in turn. Each two-rate model is compared to the constant rate model and the best two-rate model is retained. Keeping the location of this rate shift intact, it then repeats the procedure for a three-rate model and so on. The maximum number of rate shifts can be specified a priori using nSplits. Limits can be applied to the size (species richness) of clades on which to infer new rate shifts using minCladeSize. This can be useful to enable large trees to be handled but should be used cautiously since specifiying a large minimum clade size may result in biologically interesting nested rate shifts being missed. Equally, very small clade sizes may provide poor estimates of rate that may not be informative. Limits on the search can also be placed using restrictNode. This requires a list where each element of the list is a vector of tip names that define monophyletic groups. Rate shifts will not be searched for within any of the defined groups. Default rate parameter bounds from ~0 - 200.

model="tm2" - this model is similar to "tm1", however, at each node it assesses the fit of two models. The first model is exactly as per "tm1". The second model infers a rate shift on the single branch descending directly from a node but not on any of the descending branches thereafter. Only the best fitting single-branch or whole clade model is retained for the next iteration. If a single-branch shift is favoured, this infers either that there was a rapid shift in trait value along the stem leading to the crown group, or that the members of the clade have undergone parallel shifts. In either case, this can be considered as a change in mean, though separating a single early shift from a clade-parallel shift is not possible with this method.

### Value

Returns the maximum log-likelihood and parameter estimates (with 95 percent confidence intervals if specified). If model="bm" instead returns the Brownian (co)variance and log-likelihood.

`traitMedusaObject ` |
A list in which the first element contains a matrix summarising the parameter estimates and node ids, log-likelihoods, number of parameters (k), AIC and AICc for the best one-rate model, two-rate model, three rate model and so on. The second element is a sub-list where the first element contains all two-rate models, the second element contains all three-rate models and so on. This can be summarised using traitMedusaSummary. The third element is the input trait data. The fourth element is the input phylogeny. |

### Note

Confidence intervals are based on the assumption of an asymptotic Chi-square distribution. For multi-parameter models (e.g. rate shift models with more than two rates) the confidence intervals are approximate and are calculated for each parameter in turn while holding all other parameters at their maximum likelihood value.

### Author(s)

Gavin Thomas

### References

Alfaro ME, Santini F, Brock CD, Alamillo H, Dornburg A, Carnevale G, Rabosky D & Harmon LJ. 2009. Nine exceptional radiations plus high turnover explain species diversity in jawed vertebrates. PNAS 106, 13410-13414.

Felsenstein J. 1973. Maximum-likelihood estimation of evolutionary trees from continuous characters. Am. J. Hum. Genet. 25, 471-492.

Felsenstein J. 1985. Phylogenies and the comparative method. American Naturalist 125, 1-15.

Freckleton RP & Jetz W. 2009. Space versus phylogeny: disentangling phylogenetic and spatial signals in comparative data. Proc. Roy. Soc. B 276, 21-30.

Ingram T. 2010. Speciation along a depth gradient in a marine adaptive radiation. Proceeding of the Royal Society B. In press.

Mooers AO, Vamosi S, & Schluter D. 1999. Using phylogenies to test macroevolutionary models of trait evolution: sexual selection and speciation in Cranes (Gruinae). American Naturalist 154, 249-259.

O'Meara BC, Ane C, Sanderson MJ & Wainwright PC. 2006. Testing for different rates of continuous trait evolution using likelihood. Evolution 60, 922-933

Pagel M. 1997. Inferring evolutionary processes from phylogenies. Zoologica Scripta 26, 331-348.

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

Thomas GH, Meiri S, & Phillimore AB. 2009. Body size diversification in Anolis: novel environments and island effects. Evolution 63, 2017-2030.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 | ```
# Data and phylogeny
data(anolis.tree)
data(anolis.data)
# anolis.data is not matrix and contains missing data so put together matrix of # relevant traits (here female and male snout vent lengths) and remove species
# with missing data from the matrix and phylogeny
anolisSVL <- data.matrix(anolis.data)[,c(5,6)]
anolisSVL[,1] <- log(anolisSVL[,1])
anolisSVL[,2] <- log(anolisSVL[,2])
tree <- drop.tip(anolis.tree, names(attr(na.omit(anolisSVL), "na.action")))
anolisSVL <- na.omit(anolisSVL)
# Brownian motion model
transformPhylo.ML(anolisSVL, phy=tree, model="bm")
# Pagel's kappa
transformPhylo.ML(anolisSVL, phy=tree, model="kappa")
# The upper confidence interval for kappa is outside the bounds so try increasing the upper bound
transformPhylo.ML(anolisSVL, phy=tree, model="kappa", upperBound=1.5)
# Lambda, delta, OU, psi
transformPhylo.ML(anolisSVL, phy=tree, model="lambda")
transformPhylo.ML(anolisSVL, phy=tree, model="delta")
transformPhylo.ML(anolisSVL, phy=tree, model="OU")
transformPhylo.ML(anolisSVL, phy=tree, model="psi")
# Test for different rates in different clades - here with 2 hypothesised unusual rates compared to the background
# This fits the non-censored model of O'Meara et al. (2006)
transformPhylo.ML(anolisSVL, phy=tree, model="clade", nodeIDs=c(178, 199))
# Identify rate shifts and print and plot results with upto three rate shifts and minimum clade size of 20.
# Not run
#anolisSVL_MEDUSA <- transformPhylo.ML(anolisSVL, phy=tree, model="tm1", #minCladeSize=10, nSplits=2)
#anolisSVL_MEDUSA_out <- traitMedusaSummary(anolisSVL_MEDUSA, cutoff=3, #AICc=FALSE)
#colours <- plotPhylo.motmot(phy=tree, traitMedusaObject=anolisSVL_MEDUSA_out, #reconType = "rates", type = "fan", cex=0.6, edge.width=3)
``` |