# compar.ou

##### Ornstein--Uhlenbeck Model for Continuous Characters

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.

- Keywords
- models

##### 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 character optimum) is assumed to change. 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`

.

##### References

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

##### See Also

##### 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(rnorm(23), bird.orders)
### Much better with a fixed alpha:
compar.ou(rnorm(23), 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 = -2, 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
## with follows a chi^2 with df = 1.
```

*Documentation reproduced from package ape, version 2.1-1, License: GPL (>= 2)*