version 3.6

# CONTML - Gene Frequencies and Continuous Characters Maximum Likelihood method

© Copyright 1986-2002 by the University of Washington. Written by Joseph Felsenstein. Permission is granted to copy this document provided that no fee is charged for it and that this copyright notice is not removed.

This program estimates phylogenies by the restricted maximum likelihood method based on the Brownian motion model. It is based on the model of Edwards and Cavalli-Sforza (1964; Cavalli-Sforza and Edwards, 1967). Gomberg (1966), Felsenstein (1973b, 1981c) and Thompson (1975) have done extensive further work leading to efficient algorithms. CONTML uses restricted maximum likelihood estimation (REML), which is the criterion used by Felsenstein (1973b). The actual algorithm is an iterative EM Algorithm (Dempster, Laird, and Rubin, 1977) which is guaranteed to always give increasing likelihoods. The algorithm is described in detail in a paper of mine (Felsenstein, 1981c), which you should definitely consult if you are going to use this program. Some simulation tests of it are given by Rohlf and Wooten (1988) and Kim and Burgman (1988).

The default (gene frequency) mode treats the input as gene frequencies at a series of loci, and square-root-transforms the allele frequencies (constructing the frequency of the missing allele at each locus first). This enables us to use the Brownian motion model on the resulting coordinates, in an approximation equivalent to using Cavalli-Sforza and Edwards's (1967) chord measure of genetic distance and taking that to give distance between particles undergoing pure Brownian motion. It assumes that each locus evolves independently by pure genetic drift.

The alternative continuous characters mode (menu option C) treats the input as a series of coordinates of each species in N dimensions. It assumes that we have transformed the characters to remove correlations and to standardize their variances.

The input file is as described in the continuous characters documentation file above. Options are selected using a menu:

    Continuous character Maximum Likelihood method version 3.6a3

    Settings for this run:
      U                       Search for best tree?  Yes
      C  Gene frequencies or continuous characters?  Gene frequencies
      A   Input file has all alleles at each locus?  No, one allele missing at each
      O                              Outgroup root?  No, use as outgroup species 1
      G                      Global rearrangements?  No
      J           Randomize input order of species?  No. Use input order
      M                 Analyze multiple data sets?  No
      0         Terminal type (IBM PC, ANSI, none)?  (none)
      1          Print out the data at start of run  No
      2        Print indications of progress of run  Yes
      3                              Print out tree  Yes
      4             Write out trees onto tree file?  Yes

      Y to accept these or type the letter for one to change

Option U is the usual User Tree option. Options C (Continuous Characters) and A (All alleles present) have been described in the Gene Frequencies and Continuous Characters Programs documentation file. The options G, J, O and M are the usual Global Rearrangements, Jumble order of species, Outgroup root, and Multiple Data Sets options.

The M (Multiple data sets) option does not allow multiple sets of weights instead of multiple data sets, as there are no weights in this program.

The G and J options have no effect if the User Tree option is selected. User trees are given with a trifurcation (three-way split) at the base. They can start from any interior node. Thus the tree:

         A
         !
         *--B
         !
         *-----C
         !
         *--D
         !
         E

can be represented by any of the following:

         (A,B,(C,(D,E)));
         ((A,B),C,(D,E));
         (((A,B),C),D,E);

(there are of course 69 other representations as well obtained from these by swapping the order of branches at an interior node).

The output has a standard appearance. The topology of the tree is given by an unrooted tree diagram. The lengths (in time or in expected amounts of variance) are given in a table below the topology, and a rough confidence interval given for each length. Negative lower bounds on length indicate that rearrangements may be acceptable.

The units of length are amounts of expected accumulated variance (not time). The log likelihood (natural log) of each tree is also given, and it is indicated how many topologies have been tried. The tree does not necessarily have all tips contemporary, and the log likelihood may be either positive or negative (this simply corresponds to whether the density function does or does not exceed 1) and a negative log likelihood does not indicate any error. The log likelihood allows various formal likelihood ratio hypothesis tests. The description of the tree includes approximate standard errors on the lengths of segments of the tree. These are calculated by considering only the curvature of the likelihood surface as the length of the segment is varied, holding all other lengths constant. As such they are most probably underestimates of the variance, and hence may give too much confidence in the given tree.

One should use caution in interpreting the likelihoods that are printed out. If the model is wrong, it will not be possible to use the likelihoods to make formal statistical statements. Thus, if gene frequencies are being analyzed, but the gene frequencies change not only by genetic drift, but also by mutation, the model is not correct. It would be as well-justified in this case to use GENDIST to compute the Nei (1972) genetic distance and then use FITCH, KITSCH or NEIGHBOR to make a tree. If continuous characters are being analyzed, but if the characters have not been transformed to new coordinates that evolve independently and at equal rates, then the model is also violated and no statistical analysis is possible.

If the U (User Tree) option is used and more than one tree is supplied, the program also performs a statistical test of each of these trees against the one with highest likelihood. If there are two user trees, the test done is one which is due to Kishino and Hasegawa (1989), a version of a test originally introduced by Templeton (1983). In this implementation it uses the mean and variance of log-likelihood differences between trees, taken across loci. If the two trees means are more than 1.96 standard deviations different then the trees are declared significantly different. This use of the empirical variance of log-likelihood differences is more robust and nonparametric than the classical likelihood ratio test, and may to some extent compensate for the any lack of realism in the model underlying this program.

If there are more than two trees, the test done is an extension of the KHT test, due to Shimodaira and Hasegawa (1999). They pointed out that a correction for the number of trees was necessary, and they introduced a resampling method to make this correction. The version used here is a multivariate normal approximation to their test; it is due to Shimodaira (1998). The variances and covariances of the sum of log likelihoods across loci are computed for all pairs of trees. To test whether the difference between each tree and the best one is larger than could have been expected if they all had the same expected log-likelihood, log-likelihoods for all trees are sampled with these covariances and equal means (Shimodaira and Hasegawa's "least favorable hypothesis"), and a P value is computed from the fraction of times the difference between the tree's value and the highest log-likelihood exceeds that actually observed. Note that this sampling needs random numbers, and so the program will prompt the user for a random number seed if one has not already been supplied. With the two-tree KHT test no random numbers are used.

In either the KHT or the SH test the program prints out a table of the log-likelihoods of each tree, the differences of each from the highest one, the variance of that quantity as determined by the log-likelihood differences at individual sites, and a conclusion as to whether that tree is or is not significantly worse than the best one.

One problem which sometimes arises is that the program is fed two species (or populations) with identical transformed gene frequencies: this can happen if sample sizes are small and/or many loci are monomorphic. In this case the program "gets its knickers in a twist" and can divide by zero, usually causing a crash. If you suspect that this has happened, check for two species with identical coordinates. If you find them, eliminate one from the problem: the two must always show up as being at the same point on the tree anyway.

The constants available for modification at the beginning of the program include "epsilon1", a small quantity used in the iterations of branch lengths, "epsilon2", another not quite so small quantity used to check whether gene frequencies that were fed in for all alleles do not add up to 1, "smoothings", the number of passes through a given tree in the iterative likelihood maximization for a given topology, "maxtrees", the maximum number of user trees that will be used for the Kishino-Hasegawa-Templeton test, and "namelength", the length of species names. There is no provision in this program for saving multiple trees that are tied for having the highest likelihood, mostly because an exact tie is unlikely anyway.

The algorithm does not run as quickly as the discrete character methods but is not enormously slower. Like them, its execution time should rise as the cube of the number of species.

### TEST DATA SET

This data set was compiled by me from the compilation of human gene frequencies by Mourant (1976). It appeared in a paper of mine (Felsenstein, 1981c) on maximum likelihood phylogenies from gene frequencies. The names of the loci and alleles are given in that paper.

        5    10
    2 2 2 2 2 2 2 2 2 2
    European   0.2868 0.5684 0.4422 0.4286 0.3828 0.7285 0.6386 0.0205
    0.8055 0.5043
    African    0.1356 0.4840 0.0602 0.0397 0.5977 0.9675 0.9511 0.0600
    0.7582 0.6207
    Chinese    0.1628 0.5958 0.7298 1.0000 0.3811 0.7986 0.7782 0.0726
    0.7482 0.7334
    American   0.0144 0.6990 0.3280 0.7421 0.6606 0.8603 0.7924 0.0000
    0.8086 0.8636
    Australian 0.1211 0.2274 0.5821 1.0000 0.2018 0.9000 0.9837 0.0396
    0.9097 0.2976

### TEST SET OUTPUT (WITH ALL NUMERICAL OPTIONS TURNED ON)

    Continuous character Maximum Likelihood method version 3.6a3

       5 Populations,   10 Loci

    Numbers of alleles at the loci:
    ------- -- ------- -- --- -----

       2   2   2   2   2   2   2   2   2   2

    Name                 Gene Frequencies
    ----                 ---- -----------

      locus:         1         2         3         4         5         6
                     7         8         9        10

    European     0.28680   0.56840   0.44220   0.42860   0.38280   0.72850
                 0.63860   0.02050   0.80550   0.50430
    African      0.13560   0.48400   0.06020   0.03970   0.59770   0.96750
                 0.95110   0.06000   0.75820   0.62070
    Chinese      0.16280   0.59580   0.72980   1.00000   0.38110   0.79860
                 0.77820   0.07260   0.74820   0.73340
    American     0.01440   0.69900   0.32800   0.74210   0.66060   0.86030
                 0.79240   0.00000   0.80860   0.86360
    Australian   0.12110   0.22740   0.58210   1.00000   0.20180   0.90000
                 0.98370   0.03960   0.90970   0.29760

      +----------------------------------African
      !
      !              +--------American
      1--------------2
      !              !                    +-----------------------Australian
      !              +--------------------3
      !                                   +Chinese
      !
      +--European

    remember: this is an unrooted tree!

    Ln Likelihood =    33.29060

    Between     And             Length      Approx. Confidence Limits
    -------     ---             ------      ------- ---------- ------
      1       African           0.08464   (     0.02351,     0.17917)
      1          2              0.03569   (    -0.00262,     0.09493)
      2       American          0.02094   (    -0.00904,     0.06731)
      2          3              0.05098   (     0.00555,     0.12124)
      3       Australian        0.05959   (     0.01775,     0.12430)
      3       Chinese           0.00221   (    -0.02034,     0.03710)
      1       European          0.00624   (    -0.01948,     0.04601)