R: Ornstein-Uhlenbeck Model for Continuous Characters

compar.ou {ape} | R Documentation

## Ornstein–Uhlenbeck Model for Continuous Characters

### Description

This function fits an Ornstein–Uhlenbeck model giving a phylogenetic tree, and a continuous character. The user specifies the node(s) where the optimum changes. The parameters are estimated by maximum likelihood; their standard-errors are computed assuming normality of these estimates.

### Usage

    compar.ou(x, phy, node = NULL, alpha = NULL)

### Arguments

`x` | a numeric vector giving the values of a continuous character.
`phy` | an object of class `"phylo"`.
`node` | a vector giving the number(s) of the node(s) where the parameter ‘theta’ (the trait optimum) is assumed to change. The node(s) can be specified with their labels if `phy` has node labels. By default there is no change (same optimum thoughout lineages).
`alpha` | the value of alpha to be used when fitting the model. By default, this parameter is estimated (see details).

### Details

The Ornstein–Uhlenbeck (OU) process can be seen as a generalization of the Brownian motion process. In the latter, characters are assumed to evolve randomly under a random walk, that is change is equally likely in any direction. In the OU model, change is more likely towards the direction of an optimum (denoted theta) with a strength controlled by a parameter denoted alpha.

The present function fits a model where the optimum parameter theta, is allowed to vary throughout the tree. This is specified with the argument `node`: theta changes after each node whose number is given there. Note that the optimum changes only for the lineages which are descendants of this node.

Hansen (1997) recommends to not estimate alpha together with the other parameters. The present function allows this by giving a numeric value to the argument `alpha`. By default, this parameter is estimated, but this seems to yield very large standard-errors, thus validating Hansen's recommendation. In practice, a “poor man estimation” of alpha can be done by repeating the function call with different values of `alpha`, and selecting the one that minimizes the deviance (see Hansen 1997 for an example).

If `x` has names, its values are matched to the tip labels of `phy`, otherwise its values are taken to be in the same order than the tip labels of `phy`.

The user must be careful here since the function requires that both series of names perfectly match, so this operation may fail if there is a typing or syntax error. If both series of names do not match, the values in the `x` are taken to be in the same order than the tip labels of `phy`, and a warning message is issued.

### Value

an object of class `"compar.ou"` which is list with the following components:

`deviance` | the deviance (= -2 * loglik).
`para` | a data frame with the maximum likelihood estimates and their standard-errors.
`call` | the function call.

### Note

The inversion of the variance-covariance matrix in the likelihood function appeared as somehow problematic. The present implementation uses a Cholevski decomposition with the function `chol2inv` instead of the usual function `solve`.

### Author(s)

Emmanuel Paradis

### References

Hansen, T. F. (1997) Stabilizing selection and the comparative analysis of adaptation. Evolution, 51, 1341–1351.

### See Also

`ace`, `compar.lynch`, `corBrownian`, `corMartins`, `pic`

### Examples

    data(bird.orders)
    ### This is likely to give you estimates close to 0, 1, and 0
    ### for alpha, sigma^2, and theta, respectively:
    compar.ou(x <- rnorm(23), bird.orders)
    ### Much better with a fixed alpha:
    compar.ou(x, bird.orders, alpha = 0.1)
    ### Let us 'mimick' the effect of different optima
    ### for the two clades of birds...
    x <- c(rnorm(5, 0), rnorm(18, 5))
    ### ... the model with two optima:
    compar.ou(x, bird.orders, node = 25, alpha = .1)
    ### ... and the model with a single optimum:
    compar.ou(x, bird.orders, node = NULL, alpha = .1)
    ### => Compare both models with the difference in deviances
    ##     which follows a chi^2 with df = 1.

[Package ape version 5.6 Index]