# ace

##### Ancestral Character Estimation

This function estimates ancestral character states, and the associated uncertainty, for continuous and discrete characters.

`logLik`

, `deviance`

, and `AIC`

are generic functions
used to extract the log-likelihood, the deviance (-2*logLik), or the
Akaike information criterion of a tree. If no such values are
available, `NULL`

is returned.

`anova`

is another generic function that is used to compare
nested models: the significance of the additional parameter(s) is
tested with likelihood ratio tests. You must ensure that the models
are effectively nested (if they are not, the results will be
meaningless). It is better to list the models from the smallest to the
largest.

- Keywords
- models

##### Usage

```
ace(x, phy, type = "continuous", method = "ML", CI = TRUE,
model = if (type == "continuous") "BM" else "ER",
scaled = TRUE, kappa = 1, corStruct = NULL, ip = 0.1)
## S3 method for class 'ace':
print(x, digits = 4, ...)
## S3 method for class 'ace':
logLik(object, ...)
## S3 method for class 'ace':
deviance(object, ...)
## S3 method for class 'ace':
AIC(object, ..., k = 2)
## S3 method for class 'ace':
anova(object, ...)
```

##### Arguments

- x
- a vector or a factor; an object of class
`"ace"`

in the case of`print`

. - phy
- an object of class
`"phylo"`

. - type
- the variable type; either
`"continuous"`

or`"discrete"`

(or an abbreviation of these). - method
- a character specifying the method used for
estimation. Four choices are possible:
`"ML"`

,`"REML"`

,`"pic"`

, or`"GLS"`

. - CI
- a logical specifying whether to return the 95% confidence intervals of the ancestral state estimates (for continuous characters) or the likelihood of the different states (for discrete ones).
- model
- a character specifying the model (ignored if
`method = "GLS"`

), or a numeric matrix if`type = "discrete"`

(see details). - scaled
- a logical specifying whether to scale the contrast
estimate (used only if
`method = "pic"`

). - kappa
- a positive value giving the exponent transformation of the branch lengths (see details).
- corStruct
- if
`method = "GLS"`

, specifies the correlation structure to be used (this also gives the assumed model). - ip
- the initial value(s) used for the ML estimation procedure
when
`type == "discrete"`

(possibly recycled). - digits
- the number of digits to be printed.
- object
- an object of class
`"ace"`

. - k
- a numeric value giving the penalty per estimated parameter;
the default is
`k = 2`

which is the classical Akaike information criterion. - ...
- further arguments passed to or from other methods.

##### Details

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.

##### Value

- a list with the following elements:
ace if `type = "continuous"`

, the estimates of the ancestral character values.CI95 if `type = "continuous"`

, the estimated 95% confidence intervals.sigma2 if `type = "continuous"`

,`model = "BM"`

, and`method = "ML"`

, the maximum likelihood estimate of the Brownian parameter.rates if `type = "discrete"`

, the maximum likelihood estimates of the transition rates.se if `type = "discrete"`

, the standard-errors of estimated rates.index.matrix if `type = "discrete"`

, gives the indices of the`rates`

in the rate matrix.loglik if `method = "ML"`

, the maximum log-likelihood.lik.anc if `type = "discrete"`

, the scaled likelihoods of each ancestral state.call the function call.

##### References

Cunningham, C. W., Omland, K. E. and Oakley, T. H. (1998)
Reconstructing ancestral character states: a critical
reappraisal. *Trends in Ecology & Evolution*, **13**,
361--366.

Felsenstein, J. (1985) Phylogenies and the comparative
method. *American Naturalist*, **125**, 1--15.

Martins, E. P. and Hansen, T. F. (1997) Phylogenies and the
comparative method: a general approach to incorporating phylogenetic
information into the analysis of interspecific data. *American
Naturalist*, **149**, 646--667.

Pagel, M. (1994) Detecting correlated evolution on phylogenies: a
general method for the comparative analysis of discrete
characters. *Proceedings of the Royal Society of London. Series
B. Biological Sciences*, **255**, 37--45.

Schluter, D., Price, T., Mooers, A. O. and Ludwig, D. (1997)
Likelihood of ancestor states in adaptive radiation. *Evolution*,
**51**, 1699--1711.

##### See Also

##### Examples

```
### Just some random data...
data(bird.orders)
x <- rnorm(23)
### Compare the three methods for continuous characters:
ace(x, bird.orders)
ace(x, bird.orders, method = "pic")
ace(x, bird.orders, method = "GLS",
corStruct = corBrownian(1, bird.orders))
### For discrete characters:
x <- factor(c(rep(0, 5), rep(1, 18)))
ans <- ace(x, bird.orders, type = "d")
#### Showing the likelihoods on each node:
plot(bird.orders, type = "c", FALSE, label.offset = 1)
co <- c("blue", "yellow")
tiplabels(pch = 22, bg = co[as.numeric(x)], cex = 2, adj = 1)
nodelabels(thermo = ans$lik.anc, piecol = co, cex = 0.75)
### An example of the use of the argument `ip':
tr <- character(4)
tr[1] <- "((((t10:5.03,t2:5.03):2.74,(t9:4.17,"
tr[2] <- "t5:4.17):3.60):2.80,(t3:4.05,t7:"
tr[3] <- "4.05):6.53):2.32,((t6:4.38,t1:4.38):"
tr[4] <- "2.18,(t8:2.17,t4:2.17):4.39):6.33);"
tr <- read.tree(text = paste(tr, collapse = ""))
y <- c(rep(1, 6), rep(2, 4))
### The default `ip = 0.1' makes ace fails:
ace(y, tr, type = "d")
ace(y, tr, type = "d", ip = 0.01)
### Surprisingly, using an initial value farther to the
### MLE than the default one works:
ace(y, tr, type = "d", ip = 0.3)
```

*Documentation reproduced from package ape, version 2.7-3, License: GPL (>= 2)*