ouch (version 2.7-2)

hansen: Hansen model of evolution along a phylogenetic tree

Description

Fits the Ornstein-Uhlenbeck-based Hansen model to data. The fitting is done using optim or subplex.

Usage

hansen(data, tree, regimes, sqrt.alpha, sigma,
       fit = TRUE,
       method = c("Nelder-Mead","subplex","BFGS","L-BFGS-B"),
       hessian = FALSE, ...)

Arguments

data

Phenotypic data for extant species, i.e., species at the terminal twigs of the phylogenetic tree. This can either be a single named numeric vector, a list of nchar named vectors, or a data-frame containing nchar data variable

tree

A phylogenetic tree, specified as an ouchtree object.

regimes

A vector of codes, one for each node in the tree, specifying the selective regimes hypothesized to have been operative. Corresponding to each node, enter the code of the regime hypothesized for the branch segment terminating in that node. For the

sqrt.alpha, sigma

These are used to initialize the optimization algorithm. The selection strength matrix $\alpha$ and the random drift variance-covariance matrix $\sigma^2$ are parameterized by their matrix square roots. Specifically, these initial guesses are each

fit

If fit=TRUE, then the likelihood will be maximized. If fit=FALSE, the likelihood will be evaluated at the specified values of sqrt.alpha and sigma; the optima theta will be returned

method

The method to be used by the optimization algorithm, optim. See subplex and optim for information on the available options.

hessian

If hessian=TRUE, then the Hessian matrix will be computed by optim.

...

Additional arguments will be passed as control options to optim or subplex. See optim and subplex for informa

Value

• hansen returns an object of class hansentree. For details on the methods of that class, see hansentree.

Details

The Hansen model for the evolution of a multivariate trait $X$ along a lineage can be written as a stochastic differential equation (Ito diffusion) $$dX=\alpha(\theta(t)-X(t))dt+\sigma dB(t),$$ where $t$ is time along the lineage, $\theta(t)$ is the optimum trait value, $B(t)$ is a standard Wiener process (Brownian motion), and $\alpha$ and $\sigma$ are matrices quantifying, respectively, the strength of selection and random drift. Without loss of generality, one can assume $\sigma$ is lower-triangular. This is because only the infinitesimal variance-covariance matrix $\sigma^2=\sigma\sigma^T$ is identifiable, and for any admissible variance-covariance matrix, we can choose $\sigma$ to be lower-triangular. Moreover, if we view the basic model as describing evolution on a fitness landscape, then $\alpha$ will be symmetric and if we further restrict ourselves to the case of stabilizing selection, $\alpha$ will be positive definite as well. We make these assumptions and therefore can assume that the matrix $\alpha$ has a lower-triangular square root.

The hansen code uses unconstrained numerical optimization to maximize the likelihood. To do this, it parameterizes the $\alpha$ and $\sigma^2$ matrices in a special way: each matrix is parameterized by nchar*(nchar+1)/2 parameters, where nchar is the number of quantitative characters. Specifically, the parameters initialized by the sqrt.alpha argument of hansen are used to fill the nonzero entries of a lower-triangular matrix (in column-major order), which is then multiplied by its transpose to give the selection-strength matrix. The parameters specified in sigma fill the nonzero entries in the lower triangular $\sigma$ matrix. When hansen is executed, the numerical optimizer maximizes the likelihood over these parameters. The print, show, and summary methods for the resulting hansentree object display (among other things) the estimated $\alpha$ and $\sigma^2$ matrices. The coef method extracts a named list containing not only these matrices (given as the alpha.matrix and sigma.sq.matrix elements) but also the MLEs returned by the optimizer (as sqrt.alpha and sigma, respectively). The latter elements should not be interpreted, but can be used to restart the algorithm, etc.

References

Butler, M.A. and A.A. King (2004) Phylogenetic comparative analysis: a modeling approach for adaptive evolution. American Naturalist 164:683-695.

See Also

ouchtree, hansentree, optim, bimac, anolis.ssd

Examples

if (require(geiger)) {

### an example data set (Darwin's finches)
data(geospiza)
str(geospiza)
sapply(geospiza,class)

### check the correspondence between data and tree tips:
print(nc <- with(geospiza,name.check(geospiza.tree,geospiza.data)))
### looks like one of the terminal twigs has no data associated
### drop that tip:
tree <- with(geospiza,drop.tip(geospiza.tree,nc$Tree.not.data))
dat <- geospiza$geospiza.data

### make an ouchtree out of the phy-format tree
ot <- ape2ouch(tree)

### merge data with tree info
otd <- as(ot,"data.frame")
### in these data, it so happens that the rownames correspond to node names
### we will exploit this correspondence in the 'merge' operation:
dat$labels <- rownames(dat)
otd <- merge(otd,dat,by="labels",all=TRUE)
rownames(otd) <- otd$nodes
print(otd)
### this data-frame now contains the data as well as the tree geometry

### now remake the ouch tree
ot <- with(otd,ouchtree(nodes=nodes,ancestors=ancestors,times=times,labels=labels))

b1 <- brown(tree=ot,data=otd[c("tarsusL","beakD")])
summary(b1)

### evaluate an OU model with a single, global selective regime
otd$regimes <- as.factor("global")
h1 <- hansen(
             tree=ot,
             data=otd[c("tarsusL","beakD")],
             regimes=otd["regimes"],
             sqrt.alpha=c(1,0,1),
             sigma=c(1,0,1),
             maxit=10000
             )
summary(h1)
}