Hansen model of evolution along a phylogenetic tree

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Description

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

Usage

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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 variables. There must be an entry per variable for every node in the tree; use NA to represent missing data. If the data are supplied as one or more named vectors, the names attributes are taken to correspond to the node names specified when the ouchtree was constructed (see ouchtree). If the data are supplied as a data-frame, the rownames serve that purpose.

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 root node, because it has no branch segment terminating on it, the regime specification is irrelevant. If there are nchar quantitative characters, then one can specify a single set of regimes for all characters or a list of nchar regime specifications, one for each character.

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 packed into lower-triangular matrices (column by column). The product of this matrix with its transpose is the alpha or sigma^2 matrix. See Details, below.

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 as well.

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 information on the available options.

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%*%transpose(sigma) 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.

Value

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

Author(s)

Aaron A. King <kingaa at umich dot edu>

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

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## Not run: 
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$tree_not_data))
dat <- as.data.frame(geospiza$dat)


### 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)
}

## End(Not run)