3 \title{Fits the Yule Model With Covariates}
5 yule.cov(phy, formula, data = NULL)
8 \item{phy}{an object of class \code{"phylo"}.}
9 \item{formula}{a formula specifying the model to be fitted.}
10 \item{data}{the name of the data frame where the variables in
11 \code{formula} are to be found; by default, the variables are looked
12 for in the global environment.}
15 This function fits by maximum likelihood the Yule model with
16 covariates, that is a birth-only model where speciation rate is
17 determined by a generalized linear model.
20 The model fitted is a generalization of the Yule model where the
21 speciation rate is determined by:
23 \deqn{\ln\frac{\lambda_i}{1 - \lambda_i} = \beta_1 x_{i1} + \beta_2 x_{i2}
24 + \dots + \alpha }{ln(li / (1 - li)) = b1 xi1 + b2 xi2 + ... a}
26 where \eqn{\lambda_i}{li} is the speciation rate for species i,
27 \eqn{x_{i1}, x_{i2}, \dots}{xi1, xi2, ...} are species-specific
28 variables, and \eqn{\beta_1, \beta_2, \dots, \alpha}{b1, b2, ..., a}
29 are parameters to be estimated. The term on the left-hand side above
30 is a logit function often used in generalized linear models for
31 binomial data (see \code{\link[stats]{family}}). The above model can
32 be written in matrix form:
34 \deqn{\mathrm{logit} \lambda_i = x_i' \beta}{logit li = xi' b}
36 The standard-errors of the parameters are computed with the second
37 derivatives of the log-likelihood function. (See References for other
38 details on the estimation procedure.)
40 The function needs three things:
42 \item a phylogenetic tree which may contain multichotomies;
44 \item a formula which specifies the predictors of the model described
45 above: this is given as a standard R formula and has no response (no
46 left-hand side term), for instance: \code{~ x + y}, it can include
47 interactions (\code{~ x + a * b}) (see \code{\link[stats]{formula}}
50 \item the predictors specified in the formula must be accessible to
51 the function (either in the global space, or though the \code{data}
52 option); they can be numeric vectors or factors. The length and the
53 order of these data are important: the number of values (length) must
54 be equal to the number of tips of the tree + the number of nodes. The
55 order is the following: first the values for the tips in the same
56 order than for the labels, then the values for the nodes sequentially
57 from the root to the most terminal nodes (i.e. in the order given by
60 The user must obtain the values for the nodes separately.
62 Note that the method in its present implementation assumes that the
63 change in a species trait is more or less continuous between two nodes
64 or between a node and a tip. Thus reconstructing the ancestral values
65 with a Brownian motion model may be consistent with the present
66 method. This can be done with the function \code{\link{pic}} but
67 currently needs some hacking!
70 A NULL value is returned, the results are simply printed.
73 Paradis, E. (2005) Statistical analysis of diversification with
74 species traits. \emph{Evolution}, \bold{59}, 1--12.
76 \author{Emmanuel Paradis \email{Emmanuel.Paradis@mpl.ird.fr}}
78 \code{\link{branching.times}}, \code{\link{diversi.gof}},
79 \code{\link{diversi.time}}, \code{\link{ltt.plot}},
80 \code{\link{birthdeath}}, \code{\link{bd.ext}}, \code{\link{yule}}
83 ### a simple example with some random data
85 x <- rnorm(45) # the tree has 23 tips and 22 nodes
86 ### the standard-error for x should be as large as
87 ### the estimated parameter
88 yule.cov(bird.orders, ~ x)
89 ### compare with the simple Yule model, eventually
90 ### with a likelihood ratio test
92 ### another example with a tree that has a multichotomy
93 ### but we cannot run yule() because of this!
95 y <- rnorm(272) # 137 tips + 135 nodes
96 yule.cov(bird.families, ~ y)