If type = "continuous"
, the default model is Brownian motion
where characters evolve randomly following a random walk. This model
can be fitted by maximum likelihood (the default, Schluter et
al. 1997), least squares (method = "pic"
, Felsenstein 1985), or
generalized least squares (method = "GLS"
, Martins and Hansen
1997, Cunningham et al. 1998). In the latter case, the specification
of phy
and model
are actually ignored: it is instead
given through a correlation structure with the option
corStruct
. In the default setting (i.e., method = "ML"
and model =
"BM"
) the maximum likelihood estimation is done simultaneously on the
ancestral values and the variance of the Brownian motion process;
these estimates are then used to compute the confidence intervals in
the standard way. The REML method first estimates the ancestral value
at the root (aka, the phylogenetic mean), then the variance of the
Brownian motion process is estimated by optimizing the residual
log-likelihood. The ancestral values are finally inferred from the
likelihood function giving these two parameters. If method =
"pic"
or "GLS"
, the confidence intervals are computed using
the expected variances under the model, so they depend only on the
tree.
It could be shown that, with a continous character, REML results in
unbiased estimates of the variance of the Brownian motion process
while ML gives a downward bias. Therefore, the former is recommanded
over the latter, even though it is not the default.
For discrete characters (type = "discrete"
), only maximum
likelihood estimation is available (Pagel 1994). The model is
specified through a numeric matrix with integer values taken as
indices of the parameters. The numbers of rows and of columns of this
matrix must be equal, and are taken to give the number of states of
the character. For instance, matrix(c(0, 1, 1, 0), 2)
will
represent a model with two character states and equal rates of
transition, matrix(c(0, 1, 2, 0), 2)
a model with unequal
rates, matrix(c(0, 1, 1, 1, 0, 1, 1, 1, 0), 3)
a model with
three states and equal rates of transition (the diagonal is always
ignored). There are short-cuts to specify these models: "ER"
is
an equal-rates model (e.g., the first and third examples above),
"ARD"
is an all-rates-different model (the second example), and
"SYM"
is a symmetrical model (e.g., matrix(c(0, 1, 2, 1,
0, 3, 2, 3, 0), 3)
). If a short-cut is used, the number of states
is determined from the data.