version 3.696

# Kitsch -- Fitch-Margoliash and Least Squares Methods
with Evolutionary Clock

© Copyright 1986-2014 by Joseph Felsenstein. All rights reserved. License terms here.

This program carries out the Fitch-Margoliash and Least Squares methods, plus a variety of others of the same family, with the assumption that all tip species are contemporaneous, and that there is an evolutionary clock (in effect, a molecular clock). This means that branches of the tree cannot be of arbitrary length, but are constrained so that the total length from the root of the tree to any species is the same. The quantity minimized is the same weighted sum of squares described in the Distance Matrix Methods documentation file.

The options are set using the menu:

    Fitch-Margoliash method with contemporary tips, version 3.696

    Settings for this run:
      D      Method (F-M, Minimum Evolution)?  Fitch-Margoliash
      U                 Search for best tree?  Yes
      P                                Power?  2.00000
      -      Negative branch lengths allowed?  No
      L         Lower-triangular data matrix?  No
      R         Upper-triangular data matrix?  No
      S                        Subreplicates?  No
      J     Randomize input order of species?  No. Use input order
      M           Analyze multiple data sets?  No
      0   Terminal type (IBM PC, ANSI, none)?  ANSI
      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

Most of the options are described in the Distance Matrix Programs documentation file.

The D (methods) option allows choice between the Fitch-Margoliash criterion and the Minimum Evolution method (Kidd and Sgaramella-Zonta, 1971; Rzhetsky and Nei, 1993). Minimum Evolution (not to be confused with parsimony) uses the Fitch-Margoliash criterion to fit branch lengths to each topology, but then chooses topologies based on their total branch length (rather than the goodness of fit sum of squares). There is no constraint on negative branch lengths in the Minimum Evolution method; it sometimes gives rather strange results, as it can like solutions that have large negative branch lengths, as these reduce the total sum of branch lengths!

Note that the User Trees (used by option U) must be rooted trees (with a bifurcation at their base). If you take a user tree from Fitch and try to evaluate it in Kitsch, it must first be rooted. This can be done using Retree. Of the options available in Fitch, the O option is not available, as Kitsch estimates a rooted tree which cannot be rerooted, and the G option is not available, as global rearrangement is the default condition anyway. It is also not possible to specify that specific branch lengths of a user tree be retained when it is read into Kitsch, unless all of them are present. In that case the tree should be properly clocklike. Readers who wonder why we have not provided the feature of holding some of the user tree branch lengths constant while iterating others are invited to tell us how they would do it. As you consider particular possible patterns of branch lengths you will find that the matter is not at all simple.

If you use a User Tree (option U) with branch lengths with Kitsch, and the tree is not clocklike, when two branch lengths give conflicting positions for a node, Kitsch will use the first of them and ignore the other. Thus the user tree:

         ((A:0.1,B:0.2):0.4,(C:0.06,D:0.01):43);

is nonclocklike, so it will be treated as if it were actually the tree:

         ((A:0.1,B:0.1):0.4,(C:0.06,D:0.06):44);

The input is exactly the same as described in the Distance Matrix Methods documentation file. The output is a rooted tree, together with the sum of squares, the number of tree topologies searched, and, if the power P is at its default value of 2.0, the Average Percent Standard Deviation is also supplied. The lengths of the branches of the tree are given in a table, that also shows for each branch the time at the upper end of the branch. "Time" here really means cumulative branch length from the root, going upwards (on the printed diagram, rightwards). For each branch, the "time" given is for the node at the right (upper) end of the branch. It is important to realize that the branch lengths are not exactly proportional to the lengths drawn on the printed tree diagram! In particular, short branches are exaggerated in the length on that diagram so that they are more visible.

The method may be considered as providing an estimate of the phylogeny. Alternatively, it can be considered as a phenetic clustering of the tip species. This method minimizes an objective function, the sum of squares, not only setting the levels of the clusters so as to do so, but rearranging the hierarchy of clusters to try to find alternative clusterings that give a lower overall sum of squares. When the power option P is set to a value of P = 0.0, so that we are minimizing a simple sum of squares of the differences between the observed distance matrix and the expected one, the method is very close in spirit to Unweighted Pair Group Arithmetic Average Clustering (UPGMA), also called Average-Linkage Clustering. If the topology of the tree is fixed and there turn out to be no branches of negative length, its result should be the same as UPGMA in that case. But since it tries alternative topologies and (unless the N option is set) it combines nodes that otherwise could result in a reversal of levels, it is possible for it to give a different, and better, result than simple sequential clustering. Of course UPGMA itself is available as an option in program Neighbor.

The U (User Tree) option requires a bifurcating tree, unlike Fitch, which requires an unrooted tree with a trifurcation at its base. Thus the tree shown below would be written:

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

If a tree with a trifurcation at the base is by mistake fed into the U option of Kitsch then some of its species (the entire rightmost furc, in fact) will be ignored and too small a tree read in. This should result in an error message and the program should stop. It is important to understand the difference between the User Tree formats for Kitsch and Fitch. You may want to use Retree to convert a user tree that is suitable for Fitch into one suitable for Kitsch or vice versa.

An important use of this method will be to do a formal statistical test of the evolutionary clock hypothesis. This can be done by comparing the sums of squares achieved by Fitch and by Kitsch, BUT SOME CAVEATS ARE NECESSARY. First, the assumption is that the observed distances are truly independent, that no original data item contributes to more than one of them (not counting the two reciprocal distances from i to j and from j to i). THIS WILL NOT HOLD IF THE DISTANCES ARE OBTAINED FROM GENE FREQUENCIES, FROM MORPHOLOGICAL CHARACTERS, OR FROM MOLECULAR SEQUENCES. It may be invalid even for immunological distances and levels of DNA hybridization, provided that the use of common standard for all members of a row or column allows an error in the measurement of the standard to affect all these distances simultaneously. It will also be invalid if the numbers have been collected in experimental groups, each measured by taking differences from a common standard which itself is measured with error. Only if the numbers in different cells are measured from independent standards can we depend on the statistical model. The details of the test and the assumptions are discussed in my review paper on distance methods (Felsenstein, 1984a). For further and sometimes irrelevant controversy on these matters see the papers by Farris (1981, 1985, 1986) and myself (Felsenstein, 1986, 1988b).

A second caveat is that the distances must be expected to rise linearly with time, not according to any other curve. Thus it may be necessary to transform the distances to achieve an expected linearity. If the distances have an upper limit beyond which they could not go, this is a signal that linearity may not hold. It is also VERY important to choose the power P at a value that results in the standard deviation of the variation of the observed from the expected distances being the P/2-th power of the expected distance.

To carry out the test, fit the same data with both Fitch and Kitsch, and record the two sums of squares. If the topology has turned out the same, we have N = n(n-1)/2 distances which have been fit with 2n-3 parameters in Fitch, and with n-1 parameters in Kitsch. Then the difference between S(K) and S(F) has d1 = n-2 degrees of freedom. It is statistically independent of the value of S(F), which has d2 = N-(2n-3) degrees of freedom. The ratio of mean squares

          [S(K)-S(F)]/d1
         ----------------
              S(F)/d2

should, under the evolutionary clock, have an F distribution with n-2 and N-(2n-3) degrees of freedom respectively. The test desired is that the F ratio is in the upper tail (say the upper 5%) of its distribution. If the S (subreplication) option is in effect, the above degrees of freedom must be modified by noting that N is not n(n-1)/2 but is the sum of the numbers of replicates of all cells in the distance matrix read in, which may be either square or triangular. A further explanation of the statistical test of the clock is given in a paper of mine (Felsenstein, 1986).

The program uses a similar tree construction method to the other programs in the package and, like them, is not guaranteed to give the best-fitting tree. The assignment of the branch lengths for a given topology is a least squares fit, subject to the constraints against negative branch lengths, and should not be able to be improved upon. Kitsch runs more quickly than Fitch.

The constant available for modification at the beginning of the program is "epsilon", which defines a small quantity needed in some of the calculations. There is no feature saving multiple trees tied for best, because exact ties are not expected, except in cases where it should be obvious from the tree printed out what is the nature of the tie (as when an interior branch is of length zero).

### TEST DATA SET

        7
    Bovine      0.0000  1.6866  1.7198  1.6606  1.5243  1.6043  1.5905
    Mouse       1.6866  0.0000  1.5232  1.4841  1.4465  1.4389  1.4629
    Gibbon      1.7198  1.5232  0.0000  0.7115  0.5958  0.6179  0.5583
    Orang       1.6606  1.4841  0.7115  0.0000  0.4631  0.5061  0.4710
    Gorilla     1.5243  1.4465  0.5958  0.4631  0.0000  0.3484  0.3083
    Chimp       1.6043  1.4389  0.6179  0.5061  0.3484  0.0000  0.2692
    Human       1.5905  1.4629  0.5583  0.4710  0.3083  0.2692  0.0000

### TEST SET OUTPUT FILE (with all numerical options on)

       7 Populations

    Fitch-Margoliash method with contemporary tips, version 3.69

                      __ __             2
                      \  \   (Obs - Exp)
    Sum of squares =  /_ /_  ------------
                                    2
                       i  j      Obs

    negative branch lengths not allowed

    Name                       Distances
    ----                       ---------

    Bovine        0.00000   1.68660   1.71980   1.66060   1.52430   1.60430
                  1.59050
    Mouse         1.68660   0.00000   1.52320   1.48410   1.44650   1.43890
                  1.46290
    Gibbon        1.71980   1.52320   0.00000   0.71150   0.59580   0.61790
                  0.55830
    Orang         1.66060   1.48410   0.71150   0.00000   0.46310   0.50610
                  0.47100
    Gorilla       1.52430   1.44650   0.59580   0.46310   0.00000   0.34840
                  0.30830
    Chimp         1.60430   1.43890   0.61790   0.50610   0.34840   0.00000
                  0.26920
    Human         1.59050   1.46290   0.55830   0.47100   0.30830   0.26920
                  0.00000

                                               +-------Human
                                             +-6
                                        +----5 +-------Chimp
                                        !    !
                                    +---4    +---------Gorilla
                                    !   !
           +------------------------3   +--------------Orang
           !                        !
      +----2                        +------------------Gibbon
      !    !
    --1    +-------------------------------------------Mouse
      !
      +------------------------------------------------Bovine

    Sum of squares =      0.107

    Average percent standard deviation =   5.16213

    From     To            Length          Height
    ----     --            ------          ------

       6   Human           0.13460         0.81285
       5      6            0.02836         0.67825
       6   Chimp           0.13460         0.81285
       4      5            0.07638         0.64990
       5   Gorilla         0.16296         0.81285
       3      4            0.06639         0.57352
       4   Orang           0.23933         0.81285
       2      3            0.42923         0.50713
       3   Gibbon          0.30572         0.81285
       1      2            0.07790         0.07790
       2   Mouse           0.73495         0.81285
       1   Bovine          0.81285         0.81285