R: Ancestral Character Estimation

ace {ape} | R Documentation

## Ancestral Character Estimation

### Description

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, or the Akaike information criterion of a fitted object. If no such values are available, `NULL` is returned.

`anova` is another generic function which 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.

### Usage

    ace(x, phy, type = "continuous", method = if (type == "continuous")
       "REML" else "ML", CI = TRUE,
        model = if (type == "continuous") "BM" else "ER",
        scaled = TRUE, kappa = 1, corStruct = NULL, ip = 0.1,
        use.expm = FALSE, use.eigen = TRUE, marginal = FALSE)
    ## 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).
`use.expm` | a logical specifying whether to use the package expm to compute the matrix exponential (relevant only if `type = "d"`). If `FALSE`, the function `matexpo` from ape is used (see details). This option is ignored if `use.eigen = TRUE` (see next).
`use.eigen` | a logical (relevant if `type = "d"`); if `TRUE` then the probability matrix is computed with an eigen decomposition instead of a matrix exponential (see details).
`marginal` | a logical (relevant if `type = "d"`). By default, the joint reconstruction of the ancestral states are done. Set this option to `TRUE` if you want the marginal reconstruction (see details.)
`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 residual maximum likelihood (the default), maximum likelihood (Felsenstein 1973, 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 last case, the specification of `phy` and `model` are actually ignored: it is instead given through a correlation structure with the option `corStruct`.

In the setting `method = "ML"` and `model = "BM"` (this used to be the default until ape 3.0-7) 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.

For discrete characters (`type = "discrete"`), only maximum likelihood estimation is available (Pagel 1994) (see `MPR` for an alternative method). 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.

By default, the likelihood of the different ancestral states of discrete characters are computed with a joint estimation procedure using a procedure similar to the one described in Pupko et al. (2000). If `marginal = TRUE`, a marginal estimation procedure is used (this was the only choice until ape 3.1-1). With this method, the likelihood values at a given node are computed using only the information from the tips (and branches) descending from this node. With the joint estimation, all information is used for each node. The difference between these two methods is further explained in Felsenstein (2004, pp. 259-260) and in Yang (2006, pp. 121-126). The present implementation of the joint estimation uses a “two-pass” algorithm which is much faster than stochastic mapping while the estimates of both methods are very close.

With discrete characters it is necessary to compute the exponential of the rate matrix. The only possibility until ape 3.0-7 was the function `matexpo` in ape. If `use.expm = TRUE` and `use.eigen = FALSE`, the function `expm`, in the package of the same name, is used. `matexpo` is faster but quite inaccurate for large and/or asymmetric matrices. In case of doubt, use the latter. Since ape 3.0-10, it is possible to use an eigen decomposition avoiding the need to compute the matrix exponential; see details in Lebl (2013, sect. 3.8.3). This is much faster and is now the default.

### Value

an object of class `"ace"` 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.

### Note

Liam Revell points out that for discrete characters the ancestral likelihood values returned with `marginal = FALSE` are actually the marginal estimates, while setting `marginal = TRUE` returns the conditional (scaled) likelihoods of the subtree:

http://blog.phytools.org/2015/05/about-how-acemarginaltrue-does-not.html

### Author(s)

Emmanuel Paradis, Ben Bolker

### 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. (1973) Maximum likelihood estimation of evolutionary trees from continuous characters. American Journal of Human Genetics, 25, 471–492.

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

Felsenstein, J. (2004) Inferring Phylogenies. Sunderland: Sinauer Associates.

Lebl, J. (2013) Notes on Diffy Qs: Differential Equations for Engineers. http://www.jirka.org/diffyqs/.

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.

Pupko, T., Pe'er, I, Shamir, R., and Graur, D. (2000) A fast algorithm for joint reconstruction of ancestral amino acid sequences. Molecular Biology and Evolution, 17, 890–896.

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

Yang, Z. (2006) Computational Molecular Evolution. Oxford: Oxford University Press.

### See Also

`MPR`, `corBrownian`, `compar.ou`, `anova`

Reconstruction of ancestral sequences can be done with the package phangorn (see function `?ancestral.pml`).

### Examples

    ### 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)

[Package ape version 4.1 Index]