The method is explained in the 1985 paper. It assumes a Brownian motion model. This model was introduced by Edwards and Cavalli-Sforza ( 1964; Cavalli-Sforza and Edwards, 1967) as an approximation to the evolution of gene frequencies. I have discussed (Felsenstein, 1973b, 1981c, 1985d, 1988b) the difficulties inherent in using it as a model for the evolution of quantitative characters. Chief among these is that the characters do not necessarily evolve independently or at equal rates. This program allows one to evaluate this, if there is independent information on the phylogeny. You can compute the variance of the contrasts for each character, as a measure of the variance accumulating per unit branch length. You can also test covariances of characters.

The input file is as described in the continuous characters programs documentation file above, for the case of continuous quantitative characters (not gene frequencies). Options are selected using a menu:

Continuous character Contrasts, version 3.5c Settings for this run: R Print out correlations and regressions? Yes M Analyze multiple trees? No 0 Terminal type (IBM PC, VT52, ANSI)? ANSI 1 Print out the data at start of run No 2 Print indications of progress of run Yes Are these settings correct? (type Y or the letter for one to change)M is similar to the usual multiple data sets input option, but is used here to allow multiple trees to be read from the treefile, not multiple data sets to be read from the input file. In this way you can use bootstrapping on the data that estimated these trees, get multiple bootstrap estimates of the tree, and then use the M option to make multiple analyses of the contrasts and the covariances, correlations, and regressions. In this way ( Felsenstein, 1988b) you can assess the effect of the inaccuracy of the trees on your estimates of these statistics.

R allows you to turn off or on the printing out of the statistics. If it is off only the contrasts will be printed out (unless option 1 is selected). With only the contrasts printed out, they are in a simple array that is in a form that many statistics packages should be able to read. The contrasts are rows, and each row has one contrast for each character. Any multivariate statistics package should be able to analyze these (but keep in mind that the contrasts have, by virtue of the way they are generated, expectation zero, so all regressions must pass through the origin).

The tree file should contain the desired tree or trees. These can be either in bifurcating form, or may have the bottommost fork be a trifurcation (it should not matter which of these ways you present the tree). The tree must, of course, have branch lengths.

If you have a molecular data set (for example) and also, on the same species, quantitative measurements, here is how you can allow for the uncertainty of yor estimate of the tree. Use SEQBOOT to generate multiple data sets from your molecular data. Then, whichever method you use to analyze it (the relevant ones are those that produce estimates of the branch lengths: DNAML, DNAMLK, FITCH, KITSCH, and NEIGHBOR -- the latter three require you to use DNADIST to turn the bootstrap data sets into multiple distance matrices), you should use the Multiple Data Sets option of that program. This will result in a tree file with many trees on it. Then use this tree file with the input file containing your continuous quantitative characters, choosing the Multiple Trees (M) option. You will get one set of contrasts and statistics for each tree in the tree file. At the moment there is no overall summary: you will have to tabulate these by hand. A similar process can be followed if you have restriction sites data (using RESTML) or gene frequencies data.

The statistics that are printed out include the covariances between all pairs of characters, the regressions of each character on each other (column j is regressed on row i), and the correlations between all pairs of characters. In assessing degress of freedom it is important to realize that each contrast was taken to have expectation zero, which is known because each contrast could as easily have been computed xi-xj instead of xj-xi. Thus there is no loss of a degree of freedom for estimation of a mean. The degrees of freedom is thus the same as the number of contrasts, namely one less than the number of species (tips). If you feed these contrasts into a multivariate statistics program make sure that it knows that each variable has expectation exactly zero.

A limitation of these programs is that they use species means for each quantitative character without attempting to correct for the finiteness of the sample size in the estimation of this mean. Thus the variability taken into account in the model is randomness of change in evolution, but not random sampling in the estimation of the species means. I hope to remedy this in the future. At the moment I do not have a good method of inputting individual measurements, just species means. Another limitation is the absence of a method for indicating missing data. The current program assumes all characters have been measured in all species.

The constants available for modification at the beginning of the program include the usual boolean contants for the terminal type plus "namelength", the length of species names.

The data set used as an example below is the example from a paper by Michael Lynch (1990), his characters having been log-transformed.

5 2 Homo 4.09434 4.74493 Pongo 3.61092 3.33220 Macaca 2.37024 3.36730 Ateles 2.02815 2.89037 Galago -1.46968 2.30259

((((Homo:0.21,Pongo:0.21):0.28,Macaca:0.49):0.13,Ateles:0.62):0.38,Galago:1.00);

Continuous character Contrasts, version 3.5c 5 Populations, 2 Characters Name Phenotypes ---- ---------- Homo 4.09434 4.74493 Pongo 3.61092 3.33220 Macaca 2.37024 3.36730 Ateles 2.02815 2.89037 Galago -1.46968 2.30259 Contrasts (columns are different characters) --------- -------- --- --------- ----------- 0.74593 2.17989 1.58474 0.71761 1.19293 0.86790 3.35832 0.89706 Covariance matrix ---------- ------ 3.9423 1.7028 1.7028 1.7062 Regressions (columns on rows) ----------- -------- -- ----- 1.0000 0.4319 0.9980 1.0000 Correlations ------------ 1.0000 0.6566 0.6566 1.0000

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