version 3.69

# Gendist - Compute genetic distances from gene frequencies

© Copyright 1986-2008 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 computes any one of three measures of genetic distance from a set of gene frequencies in different populations (or species). The three are Nei's genetic distance (Nei, 1972), Cavalli-Sforza's chord measure (Cavalli- Sforza and Edwards, 1967) and Reynolds, Weir, and Cockerham's (1983) genetic distance. These are written to an output file in a format that can be read by the distance matrix phylogeny programs Fitch and Kitsch.

The three measures have somewhat different assumptions. All assume that all differences between populations arise from genetic drift. Nei's distance is formulated for an infinite isoalleles model of mutation, in which there is a rate of neutral mutation and each mutant is to a completely new allele. It is assumed that all loci have the same rate of neutral mutation, and that the genetic variability initially in the population is at equilibrium between mutation and genetic drift, with the effective population size of each population remaining constant.

Nei's distance is:

                                          \   \
                                          /_  /_  p1mi   p2mi
                                           m   i
               D  =  - ln  ( ------------------------------------- ).

                               \   \                \   \
                             [ /_  /_  p1mi2]1/2   [ /_  /_  p2mi2]1/2
                                m   i                m   i

where m is summed over loci, i over alleles at the m-th locus, and where

p1mi

is the frequency of the i-th allele at the m-th locus in population 1. Subject to the above assumptions, Nei's genetic distance is expected, for a sample of sufficiently many equivalent loci, to rise linearly with time.

The other two genetic distances assume that there is no mutation, and that all gene frequency changes are by genetic drift alone. However they do not assume that population sizes have remained constant and equal in all populations. They cope with changing population size by having expectations that rise linearly not with time, but with the sum over time of 1/N, where N is the effective population size. Thus if population size doubles, genetic drift will be taking place more slowly, and the genetic distance will be expected to be rising only half as fast with respect to time. Both genetic distances are different estimators of the same quantity under the same model.

Cavalli-Sforza's chord distance is given by

                       \               \                        \
         D2    =    4  /_  [  1   -    /_   p1mi1/2 p 2mi1/2]  /  /_  (am  - 1)
                        m               i                        m

where m indexes the loci, where i is summed over the alleles at the m-th locus, and where a is the number of alleles at the m-th locus. It can be shown that this distance always satisfies the triangle inequality. Note that as given here it is divided by the number of degrees of freedom, the sum of the numbers of alleles minus one. The quantity which is expected to rise linearly with amount of genetic drift (sum of 1/N over time) is D squared, the quantity computed above, and that is what is written out into the distance matrix.

Reynolds, Weir, and Cockerham's (1983) genetic distance is

                           \    \
                           /_   /_  [ p1mi     -  p2mi]2
                            m    i
           D2     =      --------------------------------------

                             \               \
                          2  /_   [  1   -   /_  p1mi    p2mi ]
                              m               i

where the notation is as before and D2 is the quantity that is expected to rise linearly with cumulated genetic drift.

Having computed one of these genetic distances, one which you feel is appropriate to the biology of the situation, you can use it as the input to the programs Fitch, Kitsch or Neighbor. Keep in mind that the statistical model in those programs implicitly assumes that the distances in the input table have independent errors. For any measure of genetic distance this will not be true, as bursts of random genetic drift, or sampling events in drawing the sample of individuals from each population, cause fluctuations of gene frequency that affect many distances simultaneously. While this is not expected to bias the estimate of the phylogeny, it does mean that the weighing of evidence from all the different distances in the table will not be done with maximal efficiency. One issue is which value of the P (Power) parameter should be used. This depends on how the variance of a distance rises with its expectation. For Cavalli-Sforza's chord distance, and for the Reynolds et. al. distance it can be shown that the variance of the distance will be proportional to the square of its expectation; this suggests a value of 2 for P, which the default value for Fitch and Kitsch (there is no P option in Neighbor).

If you think that the pure genetic drift model is appropriate, and are thus tempted to use the Cavalli-Sforza or Reynolds et. al. distances, you might consider using the maximum likelihood program Contml instead. It will correctly weigh the evidence in that case. Like those genetic distances, it uses approximations that break down as loci start to drift all the way to fixation. Although Nei's distance will not break down in that case, it makes other assumptions about equality of substitution rates at all loci and constancy of population sizes.

The most important thing to remember is that genetic distance is not an abstract, idealized measure of "differentness". It is an estimate of a parameter (time or cumulated inverse effective population size) of the model which is thought to have generated the differences we see. As an estimate, it has statistical properties that can be assessed, and we should never have to choose between genetic distances based on their aesthetic properties, or on the personal prestige of their originators. Considering them as estimates focuses us on the questions which genetic distances are intended to answer, for if there are none there is no reason to compute them. For further perspective on genetic distances, I recommend my own paper evaluating different genetic distances (Felsenstein, 1985c), Reynolds, Weir, and Cockerham (1983), and the material in Nei's book (Nei, 1987).

## INPUT FORMAT

The input to this program is standard and is as described in the Gene Frequencies and Continuous Characters Programs documentation file above. It consists of the number of populations (or species), the number of loci, and after that a line containing the numbers of alleles at each of the loci. Then the gene frequencies follow in standard format.

The options are selected using a menu:

    Genetic Distance Matrix program, version 3.69

    Settings for this run:
      A   Input file contains all alleles at each locus?  One omitted at each locus
      N                        Use Nei genetic distance?  Yes
      C                Use Cavalli-Sforza chord measure?  No
      R                   Use Reynolds genetic distance?  No
      L                         Form of distance matrix?  Square
      M                      Analyze multiple data sets?  No
      0              Terminal type (IBM PC, ANSI, none)?  ANSI
      1            Print indications of progress of run?  Yes

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

The A (All alleles) option is described in the Gene Frequencies and Continuous Characters Programs documentation file. As with Contml, it is the signal that all alleles are represented in the gene frequency input, without one being left out per locus. C, N, and R are the signals to use the Cavalli-Sforza, Nei, or Reynolds et. al. genetic distances respectively. The Nei distance is the default, and it will be computed if none of these options is explicitly invoked. The L option is the signal that the distance matrix is to be written out in Lower triangular form. The M option is the usual Multiple Data Sets option, useful for doing bootstrap analyses with the distance matrix programs. It allows multiple data sets, but does not allow multiple sets of weights (since there is no provision for weighting in this program).

## OUTPUT FORMAT

The output file simply contains on its first line the number of species (or populations). Each species (or population) starts a new line, with its name printed out first, and then and there are up to nine genetic distances printed on each line, in the standard format used as input by the distance matrix programs. The output, in its default form, is ready to be used in the distance matrix programs.

## CONSTANTS

A constant "epsilong" is available to be changed by the user if the program is recompiled which defines a small quantity that is used when checking whether allele frequencies at a locus sum to more than one: if all alleles are input (option A) and the sum differs from 1 by more than epsilong, or if not all alleles are input and the sum is greater than 1 by more then epsilon, the program will see this as an error and stop. You may find this causes difficulties if you gene frequencies have been rounded. I have tried to keep epsilong from being too small to prevent such problems.

## RUN TIMES

The program is quite fast and the user should effectively never be limited by the amount of time it takes. All that the program has to do is read in the gene frequency data and then, for each pair of species, compute a genetic distance formula for each pair of species. This should require an amount of effort proportional to the total number of alleles over loci, and to the square of the number of populations.

## FUTURE OF THIS PROGRAM

The main change that will be made to this program in the future is to add provisions for taking into account the sample size for each population. The genetic distance formulas have been modified by their inventors to correct for the inaccuracy of the estimate of the genetic distances, which on the whole should artificially increase the distance between populations by a small amount dependent on the sample sizes. The main difficulty with doing this is that I have not yet settled on a format for putting the sample size in the input data along with the gene frequency data for a species or population.

I may also include other distance measures, but only if I think their use is justified. There are many very arbitrary genetic distances, and I am reluctant to include most of them.

### TEST DATA SET

        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

        5
    European    0.000000  0.078002  0.080749  0.066805  0.103014
    African     0.078002  0.000000  0.234698  0.104975  0.227281
    Chinese     0.080749  0.234698  0.000000  0.053879  0.063275
    American    0.066805  0.104975  0.053879  0.000000  0.134756
    Australian  0.103014  0.227281  0.063275  0.134756  0.000000