cauphylm {cauphy} | R Documentation |

## Phylogenetic Regression using a Cauchy Process

### Description

Perform a phylogenetic regression using the Cauchy Process, by numerical optimization.

### Usage

```
cauphylm(
formula,
data = list(),
phy,
model = c("cauchy", "lambda"),
lower.bound = list(disp = 0, lambda = 0),
upper.bound = list(disp = Inf, lambda = NULL),
starting.value = list(disp = NULL, lambda = NULL),
hessian = FALSE
)
```

### Arguments

`formula` |
a model formula. |

`data` |
a data frame containing variables in the model. If not found in data, the variables are taken from current environment. |

`phy` |
a phylogenetic tree of class |

`model` |
a model for the trait evolution. One of |

`lower.bound` |
named list with lower bound values for the parameters. See Details for the default values. |

`upper.bound` |
named list with upper bound values for the parameters. See Details for the default values. |

`starting.value` |
named list initial values for the parameters. See Details for the default values. |

`hessian` |
if |

### Details

This function fits a Cauchy Process on the phylogeny, using maximum likelihood
and the `"fixed.root"`

method (see `fitCauchy`

).
It further assumes that the root value `x0`

is a linear combination of the
covariables in `formula`

.
The corresponding regression model is:

`Y = X \beta + E,`

with:

`Y`

the vector of traits at the tips of the tree;

`X`

the regression matrix of covariables in

`formula`

;`\beta`

the vector of coefficients;

`E`

a centered error vector that is Cauchy distributed, and can be seen as the result of a Cauchy process starting at 0 at the root, and with a dispersion

`disp`

(see`fitCauchy`

).

Unless specified by the user, the initial values for the parameters are taken according to the following heuristics:

`coefficients`

:-
`\beta`

are obtained from a robust regression using`lmrob.S`

; `disp`

:is initialized from the trait centered and normalized by tip heights, with one of the following statistics, taken from Rousseeuw & Croux 1993:

`IQR`

:half of the inter-quartile range (see

`IQR`

);`MAD`

:median absolute deviation with constant equal to 1 (see

`mad`

);`Sn`

:Sn statistics with constant 0.7071 (see

`Sn`

);`Qn`

:Qn statistics with constant 1.2071 (see

`Qn`

).

Unless specified by the user, `disp`

is taken positive unbounded.

The function uses `nloptr`

for the numerical optimization of the
(restricted) likelihood, computed with function `logDensityTipsCauchy`

.
It uses algorithms `BOBYQA`

and `MLSL_LDS`

for local and global optimization.

If `model="lambda"`

, the CP is fit on a tree with branch lengths re-scaled
using the Pagel's lambda transform (see `transf.branch.lengths`

),
and the `lambda`

value is estimated using numerical optimization.
The default initial value for the `lambda`

parameter is computed using adequate robust moments.
The default maximum value is computed using `phytools:::maxLambda`

,
and is the ratio between the maximum height of a tip node over the maximum height of an internal node.
This can be larger than 1.
The default minimum value is 0.

### Value

`coefficients` |
the named vector of estimated coefficients. |

`disp` |
the maximum likelihood estimate of the dispersion parameter. |

`logLik` |
the maximum of the log likelihood. |

`p` |
the number of all parameters of the model. |

`aic` |
AIC value of the model. |

`fitted.values` |
fitted values |

`residuals` |
raw residuals |

`y` |
response |

`X` |
design matrix |

`n` |
number of observations (tips in the tree) |

`d` |
number of dependent variables |

`formula` |
the model formula |

`call` |
the original call to the function |

`model` |
the phylogenetic model for the covariance |

`phy` |
the phylogenetic tree |

`lambda` |
the ml estimate of the lambda parameter (for |

### References

Bastide, P. and Didier, G. 2023. The Cauchy Process on Phylogenies: a Tractable Model for Pulsed Evolution. Systematic Biology. doi:10.1093/sysbio/syad053.

Rothenberg T. J., Fisher F. M., Tilanus C. B. 1964. A Note on Estimation from a Cauchy Sample. Journal of the American Statistical Association. 59:460–463.

Rousseeuw P.J., Croux C. 1993. Alternatives to the Median Absolute Deviation. Journal of the American Statistical Association. 88:1273–1283.

### See Also

`fitCauchy`

, `confint.cauphylm`

, `ancestral`

, `increment`

, `logDensityTipsCauchy`

, `phylolm`

### Examples

```
# Simulate tree and data
set.seed(1289)
phy <- ape::rphylo(20, 0.1, 0)
error <- rTraitCauchy(n = 1, phy = phy, model = "cauchy",
parameters = list(root.value = 0, disp = 0.1))
x1 <- ape::rTraitCont(phy, model = "BM", sigma = 0.1, root.value = 0)
trait <- 3 + 2*x1 + error
# Fit the data
fit <- cauphylm(trait ~ x1, phy = phy)
fit
# Approximate confidence intervals
confint(fit)
```

*cauphy*version 1.0.2 Index]