Go to web page outlining major aspects of analyzing genetic population structure (WAAP.html)
(some important measures to calculate, very basic intro. to the practicalities of running
a few of the many software choices)
Taxonomy of genetic distance measures.
We began our study of population genetics by developing
the concept of hetero and homozygosity from HardyWeinberg principles.
We used a HardyWeinberg approach as one way to get at a measure of subpopulation
differentiation in terms of Fstatistics. The Fstatistics
provide a view of the variance structure of populations, and can provide
an overall comparison of the degree to which populations are structured
F_{ST} = 0 meaning no structure, no differentiation, and
F_{ST} = 1 meaning completely differentiated;
F_{IS} = 0 meaning
neither inbreeding nor outbreeding (i.e., meeting the random mating HardyWeinberg expectation),
F_{IS} = 1 meaning completely inbred,
F_{IS} = 1 meaning completely outbred.
Go to web page describing how to calculate F_{ST} from heterozygosities (FST.html)
Fstatistics do not, however, easily allow pairwise comparisons among subpopulations or populations. That is, we can assess pairwise F_{ST} between populations, but those pairwise "distances" take account only of the data for the two populations concerned, not all the data simultaneously. We would like a way to quantify the degree to which A differs from B, B from C, and A from C from the entire pool of data. We can do this in two major ways  with or without underlying biological models. The latter (no biological assumptions or model) are also known as geometric distances. These geometric distances include Rogers’ and CavalliSforza chord distances. Distance measures that do make biological assumptions include Reynolds’ and Nei’s distances. Let’s examine each in turn.
1) Distance methods with no biological assumptions. A locusspecific, codominant marker population genetic data set, such as the bear one you have used for homeworks, consists of a set of individual and populationindexed gene frequencies at one or more loci. We can analyze these data as a set of numbers without making any biological assumptions. Approaches could include principal components analyses (PCA), Euclidean distances or somewhat more complex geometric distances. Many of these will allow us to create a sort of abstract "map" of the populations in one, two, three or more dimensions (obviously, maps with dimensionalities > 3 are hard to visualize). Some of these maps can be condensed into matrices of distances. Here’s an example using real microsatellite data for Western ScrubJays (Aphelocoma californica).Table 6.1. CavalliSforza chord distances for five populations of Western ScrubJays, Aphelocoma californica.
WOb3  WSp3  WCal  WOoc  WSp2  
WOb3 
0

0.0332

0.0492

0.0428

0.0466

WSp3 
0.0332

0

0.0488

0.0645

0.0449

WCal 
0.0492

0.0488

0

0.0617

0.0533

WOoc 
0.0428

0.0645

0.0617

0

0.058

WSp2 
0.0466

0.0449

0.0533

0.058

0

How did we get these CavalliSforza distances? They are simply a geometric view of the distances between multidimensional points on a hypersphere (a sphere with > 3 dimensions). Say we have two subpopulations S1 and S2 assayed at a single locus with alleles i = 1 to k. The formal definition is:
Eqn 6.1
That is, we take the square root of the frequency of allele 1 in S1 times that of allele 1 in S2 and repeat and sum that quantity for all k alleles. That gives us Cos (y) which we can plug into the squareroot term on the RHS (right hand side) of Eqn 5.1 above. I don’t expect you to use or memorize this  just to see that it is a purely numerical/geometric approach. If we were doing it in 3 dimensions it would be akin to figuring out the distance from New York to London along the surface of the globe (called the chord distance). It can be fairly easily incorporated into a numbercrunching computer program that will produce output like the table of CavalliSforza distances shown above. Those distances, in matrix form, can then be used as input for phylogenetic treebuilding routines such as the UPGMA, FitchMargoliash and neighborjoining approaches we used in the homeworks.
The CavalliSforza chord distance was an early measure and is still used (in fact I see it gaining ground for use with microsatellites). Another geometric distance that was widely used with allozymes (but I have not seen used with microsatellite data) is Rogers’ distance (Wright, 1978). One reason the CavalliSforza distance may be in greater current use is that it was specifically evaluated (and performed well) in simulations of treebuilding algorithms by Takezaki and Nei (1996). [For all we know Roger’s distance may perform equally well or better under circumstances that would apply well to the questions people like me seek to address  but since no one has done such a study, people like me will tend to go with one that has a documented good track record]. A very important part of the robustness of a distance measure is its performance under a variety of conditions. It is always best if we can compare several distance measures under conditions in which we know what the answer should be. Paetkau et al. (1997) provide an evaluation of various distance measures that apply to distance measures potentially useful for microsatellite analysis of bear populations.
2) Distance methods with biological assumptions. With a little luck (or a lot of hard work), we know something about the evolutionary forces (most importantly here mutation and drift, since we assume we are using markers that are not subject to natural selection) driving genetic change in the system we're interested in. If so, it seems reasonable to take advantage of that knowledge by incorporating it into building a distance model. After all, we expect models with greater realism to perform better (albeit at the cost of greater complexity, usually). Several distance measures incorporate assumptions about the importance of drift and mutation as forces of change:
Reynolds’ distance or the "coancestry" distance (Reynolds
et al., 1983; see Weir, 1996, p. 167)
Nei’s distance (Nei 1972, 1978)
Models using a stepwise mutation model (SMM) specifically
developed for microsatellites (e.g., dm^{2}[delta
mu squared] of Goldstein et al., 1995).
The problem with making assumptions is that violations
can cause errors. Empirically, it appears that many of the stepwise mutation
models for microsatellites do not perform well when analyzing many (most?)
data sets, especially those where small population sizes mean that drift
has played at least as large a role as mutation. Reynolds’ distance, which
was derived for allozyme data on small (e.g., vertebrate) populations assumes
a primary role for drift and is an infinitealleles model (an allele can
change from any given state into any other given state). Reynolds’ reliance
on "drift only" seemed inappropriate for microsatellites, which
have:
a) a mutation rate that appeared clearly much larger than that
of allozymes (1 mutation per 1,000 or 10,000 replication events for microsatellites
vs. 1 mutation per 1,000,000 replication events for allozymes). [But that
may be based on very long repeats in highly polymorphic human populations].
b) a mutation process that would seemingly not fit the infinitealleles
model because mutations generally occur in "stepwise" fashion by adding
or deleting one of a series of beads (AC_{10} goes to AC_{9} or AC_{11}, where the
subscript refers to the number of AC repeat units).
[See my web page http://www.uwyo.edu/zoology/mcdonald/dna.htm for a
quick overview of microsatellites].
Nevertheless, Reynolds' distance and its neglect of the importance
of mutation, may work better than we would have expected (at least
in some species/populations) for two reasons:
a) small population sizes (= high potential for drift)
b) "missing steps" because drift creates a "chunky" distribution of
alleles instead of the smooth bell curve we would expect under a strict
stepwise process.
Fig. 6.1. A microsatellite allele frequency
distribution under a strict stepwise mutation model (SMM). The Xaxis
shows the number of repeat units (e.g., AC_{8} to AC_{19}),
while the Yaxis shows the number of alleles. Starting with either
a 13 or 14 repeat chain as the ancestor, we tend to accumulate more
alleles at sizes close to the starting point because of equal likelihood
of additions or subtractions and because larger changes (a variant of "mutations of large effect") will tend to be rare (we think).
Fig. 6.2. An allele frequency distribution that has been greatly affected by drift and may better fit an infinitealleles model (IAM). Even if the mutations that generated the original variation did occur in stepwise fashion, drift has removed some allele sizes (e.g., the 10repeat category) while randomly selecting others to be greatly overrepresented (e.g., 12, 15 and 17). This sort of "chunky" distribution may be quite common in many natural populations of vertebrates (where effective population sizes, N_{e}, are always small or at least often fluctuate to low numbers).
Download a pdf of equations for various measures of genetic distance. This pdf was once on the NC State website (statistical genetics home base for Weir and Cockerham).
http://statgen.ncsu.edu/brcwebsite/software_BRC.php
{The NC site has several software programs and a link to many others at: http://www.nslijgenetics.org/soft/}
Goldstein, D. B., A.R. Linares, L.L. CavalliSforza, and M.W. Feldman. 1995. Genetic absolute dating based on microsatellites and the origin of modern humans. PNAS USA 92: 67236727.
Nei, M. 1972. Genetic distance between populations. Am. Nat. 106: 283292.
Nei, M. 1978. Estimation of average heterozygosity and genetic distance from a small number of individuals. Genetics 76: 379390.
Paetkau, D., L.P. Waits, P.L. Clarkson, L. Craighead, and C. Strobeck. 1997. An empirical evaluation of genetic distance statistics using microsatellite data from bear (Ursidae) populations. Genetics 147:19431957.
Reynolds, J., B.S. Weir, and C.C. Cockerham. 1983. Estimation of the coancestry coefficient: Basis for a shortterm genetic distance. Genetics 105: 767779.
Takezaki, N., and M. Nei. 1996. Genetic distances and reconstruction of phylogenetic trees from microsatellite DNA. Genetics 144: 389399.
Weir, Bruce S. 1996. Genetic Data Analysis II: Methods for discrete population genetic data (2nd. ed.). Sinauer Assoc., Sunderland, MA.
Wright, S. 1978. Evolution and the Genetics of Populations, Vol. 4: Variability Within and Among Natural Populations. University of Chicago Press, Chicago.