version 3.5c

             CONTRAST -- Computes contrasts for comparative method

(c) Copyright  1991-1993  by  Joseph  Felsenstein  and  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 implements the contrasts calculation  described  in  my  1985
paper  on  the comparative method (Felsenstein, 1985d).  It reads in a data set
of the standard  quantitative  characters  sort,  and  also  a  tree  from  the
treefile.   It then forms the contrasts between species that, according to that
tree, are statistically independent.  This is done  for  each  character.   The
contrasts  are  all  standardized  by branch lengths (actually, square roots of
branch lengths).

     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

     The input file is as described in the continuous characters  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.

--------------------- TEST SET INPUT ------------------------------------

    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

--------------------- TEST SET INPUT TREEFILE ---------------------------


--------------- TEST SET OUTPUT (with all numerical options on ) -------------

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


    1.0000    0.6566
    0.6566    1.0000