<![CDATA[
A method for studying molecular variation within a species, using electrophoresis data on DNA where 1 denotes the presence of a band and zero its absence.
<P>
ANALYSIS OF MOLECULAR VARIANCE
<P>
As the name suggests Analysis of Molecular Variance (AMOVA) is a
method for studying molecular variation within a species.<p>
Electrophoresis, one of the most widely used methods for studying the
structure of DNA, produces data of the form of 0s and 1s where 1 denotes
the presence of a band and zero its absence, e.g.
<p>
<table>
<tr> <th> Sample </th> <th> Band </th> </tr>
<tr> <td> A </td> <td> 0101... </td> </tr>
<tr> <td> B </td> <td> 1000... </td> </tr>
<tr> <td> C </td> <td> 0110... </td> </tr>
</table>
<p>
AMOVA works on such data to create a distance matrix between samples in
order to measure the genetic structure of the population from which the
samples are drawn. In statistical terms, AMOVA is a testing procedure
based on permutational analysis and involves few assumptions about the
statistical properties of the data.
<P>
USES OF AMOVA
<BR>
<table>
<tr>
<td> <img alt="humans" src="../images/people.jpg" WIDTH=209 HEIGHT=107> </td>
<td> Study of <a href="examp1.htm">human</a> evolution. </td>
</tr>
<tr>
<td> <img alt="buffalo" src="../images/grass.jpg" WIDTH=163 HEIGHT=132> </td>
<td> Investigating genetic variation in <a href="examp2.htm">Buffalograss.</a>
</td>
</tr>
<tr>
<td> <img alt="porpoise" src="../images/porpoise.jpg" WIDTH=177 HEIGHT=123> </td>
<td> Monitoring <a href="examp3.htm">Dall's
porpoise</a> in the Pacific . </td>
</tr>
</table>
<p>
<h5> <font color="FF0000">
Photographs from Microsoft Encarta
</font> </h5>
<P>
HUMAN mtDNA HAPLOTYPES
<BR>
Restriction haplotypes of human mtDNA were sampled from ten populations,
grouped into five regions. Altogether 672 mtDNAs were assayed, yielding 62
restriction sites of which 34 were polymorphic. <p>
<img alt="model" src="../images/human.gif" WIDTH=480 HEIGHT=219> <p>
<a href="ref.htm#excoffier"> Excoffier, Smouse & Quattro (1992)</a>
used this example to find significant differences
<UL>
<li> among populations within regions
<li> among regions, the African region having the biggest effect.
</UL>
<P>
RAPD VARIATION IN BUFFALOGRASS
<BR>
Buffalograss is native to the Great Plains of North America, where it is
important for rangeland forage, soil conservation and as turf. <p>
A survey by <a href="ref.htm#huff"> Huff, Peakall and Smouse (1993)</a> had
aimed to determine the pattern and extent of RAPD marker variation within and
among natural populations of diploid buffalograss. <p>
They examined 12 plants from each of two populations in each of two regions
(Texas and Central Mexico), i.e. 48 plants in all. <p>
<img alt="model" src="../images/buffalo.gif" WIDTH=350 HEIGHT=375> <p>
They found
<UL>
<LI>substantial variation in each of the four populations;
<LI>significant genetic differences between Texan and Mexican collections
<LI>significant differences between the populations within each of the regions.
</UL>
<P>
DALL'S PORPOISE mtDNA VARIATION
<BR>
mtDNA samples were taken from 101 Dall's porpoises, collected from three regions
in the northern Pacific, the Bering Sea, the Aleutians and the Western North
Pacific.<p> Digestion with 11 restriction enzymes yielded 34 distinct
restriction site haplotypes. <p>
<img alt="porpoises" src="../images/dall.gif" WIDTH=387 HEIGHT=203> <p>
<a href="ref.htm#mcmillan">McMillan & Bermingham (1996)</a> used AMOVA to find
a significant difference among the three populations, to which western North
Pacific made the greatest contribution.<p>
They concluded that separate management policies should be used for the three
populations.
<P>
ANALOGY WITH ANOVA
<BR>
AMOVA is like an hierarchical analysis of variance in that it
separates and tests tiers of genetic diversity:
<UL>
<LI>Diversity among groups of populations
<LI>Diversity among the populations within groups
<LI>Diversity among the individuals within a population
</UL>
When studying molecular variation, it is customary to examine
<a href="slide1.htm">haplotypes</a> and so there is no variation within
individuals. <p>
AMOVA differs from analysis of variance in that:
<UL>
<LI>AMOVA can accommodate different evolutionary assumptions without
modifying the basic structure of the analysis
<LI>Hypotheses are tested using permutational methods so that no Normal
distribution assumption is required.
</UL>
<P>
HAPLOID DATA
<BR>
With haploid data each individual is represented by one haplotype. <p>
A haplotype may be obtained from:
<UL>
<LI>Restriction Fragment Length Polymorphism<A HREF="glossary.htm"> (RFLP)</A>
data;
<LI>Random Amplified Polymorphic DNA<A HREF="glossary.htm"> (RAPD)</A> data;
<LI>DNA sequences
</UL>
A haplotype for an individual is represented by a vector p. <p>
For instance
<UL>
<LI>For RFLP data p has a 1 if a restriction site is present and 0 if absent.
<LI>For RAPD data p has a 1 if an homologous band is present and 0 if absent.
</UL>
Other representations of p are possible.
<P>
DISTANCES BETWEEN HAPLOTYPES
<BR>
We are interested in the genetic distance between pairs of haplotypes. <p>
Let Xj be a Boolean vector, of length S, representing the jth
haplotype.<br>S is the number of restriction sites or marker bands. <p>
The squared distance between haplotypes hj and hk is: <p>
<img alt="distance" src="../images/distan.gif" WIDTH=182 HEIGHT=45> <p>
The <a href="slide2a.htm">total sum of squares</a> of the distances between
all pairs of haplotypes
is equivalent to the conventional sum of squares of deviations of
multidimensional vectors from the centroid of a multidimensional space.
<p>
The analysis of molecular variance is essentially an analysis of variance
using squared distances between pairs of haplotypes as the data.<p>
HIERARCHICAL MODEL
<BR>
Now extend the notation to include the population structure. <p>
If X(jig) indexes haplotype j in subpopulation i in group g, it can be
represented by the linear model <p>
<img alt="model" src="../images/image4d.gif" WIDTH=187 HEIGHT=37> <p>
where
<UL>
<LI>X is the unknown expectation
<LI>a is the group effect with variance
<img alt="model" align=top src="../images/image4a.gif" WIDTH=27 HEIGHT=21>, g = 1 ... G
<LI>b is the subpopulation effect within a group with variance
<img alt="model" align=top src="../images/image4b.gif" WIDTH=29 HEIGHT=23>, i = 1 ... Ig
<LI>c is the individual effect within a subpopulation with variance
<img alt="model" align=top src="../images/image4c.gif" WIDTH=27 HEIGHT=24>, j = 1 ... Nig
</UL>
These effects are assumed to be additive, random and uncorrelated. <p>
ANALYSIS
<BR>
<img alt="model" src="../images/image4e.gif" WIDTH=566 HEIGHT=262><p>
The <a href="slide4a.htm">sums of squares</a> of deviations (SSD) are functions
of the squared distances between pairs of haplotypes.
Note that MSD = SSD/df<p>
The terms for <a href="slide4b.htm">n, n', n''</a> in the Expected(MSD)
represent the average sample sizes of particular hierarchical levels.
<P>
DISTANCE METRICS
<BR>
The analysis of molecular variance is performed using the distances between
pairs of haplotypes as data
, and with W as a <a href="slide5b.htm">weight matrix</a>, as follows.<p>
<img alt="distance" src="../images/distan.gif" WIDTH=182 HEIGHT=45>
<p>
Distance can be defined in a variety of ways.
<UL>
<LI><a href="slide5a.htm">Euclidean</a> distance--
a simple count of the number of differences between two haplotypes. <p>
<LI>All haplotypes are <a href="slide5g.htm">equidistant</a>. They are merely
recorded as being different or the same.<p>
<LI>Distance is measured along a <a href="slide5c.htm">network</a>.<p>
<LI>Distance is measured along a network but the distances are weighted
according to the <a href="slide5d.htm">nucleotide diversity</a> between
adjacent types in the network.<p>
</UL>
Unweighted Euclidean and network distance metrics are best understood by way of
an <a href="slide5e.htm">illustration</a>.<p>
AMOVA is not strictly rigorous with a non-Euclidean metric.
<P>
WEIGHT MATRIX
<BR>
W is a weight matrix of dimension, S = number of restriction sites
or marker bands. W has several forms:
<UL>
<LI>Identity, indicating that all sites or bands are independent and given equal
weight;
<LI>Diagonal, indicating that all sites or bands are independent of one another
but some are given more weight than others;
<LI>Non-zero off-diagonal elements, indicating that sites are not independent.
</UL>
<P>
EQUIDISTANCE
<BR>
Use this method for allozymes and other protein systems, for antigenic
systems and for molecular fingerprint analysis.<p>
<A href="ref.htm#peakall"> Peakall, Smouse and Huff (1995)</A> use this method
to analyse allozyme data with multiple alleles.<p>
In this case AMOVA is mathematically identical to <A href="ref.htm#weir">
Weir and Cockerham's (1984)</A> analysis of variance for multiple alleles
except that they use a different testing procedure.
<P>
EXAMPLES
<BR>
Two ways of constructing haplotypes from restriction sites.<p>
Example 1: three restriction sites, 1 = presence, 0 = absence<br>
There are five distinct haplotypes in the data set.<p>
Example 2: Four mutational events obtained from the network constructed from
the p vectors, 1 = presence, 0 = absence of a mutational event. There are
still five distinct haplotypes.<p>
<img alt="model" src="../images/image6a.gif" WIDTH=513 HEIGHT=687> <p>
PHI STATISTICS
<BR>
The variance components from the analysis are used to estimate
<img alt="model" src="../images/image5a.gif">
statistics which are similar
to <a href="slide6a.htm">F statistics</a>. <p>
<img alt="model" src="../images/image5b.gif" WIDTH=30 HEIGHT=19> is the correlation of random pairs
of haplotypes drawn from a group relative to the correlation of pairs of random
haplotypes drawn from the whole population. <p>
<img alt="model" src="../images/image5c.gif" WIDTH=181 HEIGHT=56> <p>
<img alt="model" src="../images/image5d.gif" WIDTH=29 HEIGHT=19> is the correlation of random pairs
of haplotypes drawn from a subpopulation relative to the correlation of pairs
of random haplotypes drawn from the whole region, averaged over all
subpopulations. <p>
<img alt="model" src="../images/image5e.gif" WIDTH=148 HEIGHT=54> <p>
<img alt="model" src="../images/image5f.gif" WIDTH=30 HEIGHT=19> is the correlation of random pairs
of haplotypes drawn from within subpopulations relative to the correlation of
pairs of random haplotypes drawn from the whole population. <p>
<img alt="model" src="../images/image5g.gif" WIDTH=170 HEIGHT=56> <p>
Note that <img alt="model" src="../images/image5h.gif" WIDTH=187 HEIGHT=19>
<P>
POPULATION STRUCTURE
<BR>
Consider several subpopulations of the same species separated geographically.
<p>
In the absence of selection and with random mating, two trends are expected:
<UL>
<LI>gene frequencies for the total population are expected to remain constant
over generations;
<LI>the variance of gene frequencies is expected to increase over time because
of differentiation among subpopulations.
</UL>
Wright's F statistics <A HREF="ref.htm#wright"> (Wright, 1965)</A>, quantify
the differentiation among subpopulations and among individuals. <p>
<A HREF="ref.htm#cockerham">Cockerham (1973)</A> showed how F statistics
could be estimated from an analysis of variance comparing mean gene
frequencies. <p>
F statistics apply to two possible alleles at a particular locus. <p>
<A HREF="ref.htm#long"> Long (1986)</A> extended Cockerham's analysis to
several loci each having two or more possible alleles. This method was
particularly suited to the analysis of <A HREF="glossary.htm">allozyme </A>
data.
<P>
POPLAR TREE EXAMPLE
<BR>
Poplar trees are grown commercially throughout the United Kingdom.
The Forestry Commission has the task of ensuring that clones can be
unambiguously identified. In the past, this identification was done using
morphological characters but nowadays is performed using molecular data.<p>
<table>
<tr>
<td> <img src="../images/beaupre.jpg" WIDTH=151 HEIGHT=204> </td>
<td> A 4-year-old Beaupre hybrid </td>
</tr>
<tr>
<td> <img src="../images/mature.jpg" WIDTH=123 HEIGHT=198> </td>
<td> A mature poplar </td>
</tr>
</table> <p>
<a href="ref.htm#cottrell"> Cottrell, Forrest and White (1996)</a> have provided
RAPD <a href="slide7a.htm">data</a> for three species and several hybrids.<p>
Unweighted <a href="slide7b.htm">Euclidean distances</a> have been used in the
analysis.
<P>
POPLAR TREE DATA
<BR>
The data show the presence (1) or absence (0) of
RAPD bands in each clone examined. 14 clones for which some band data were
missing, have been excluded.
Species and hybrids for which only one sample was available have not been used
in this analysis.<p>
The first column is a code for the clone. The second column is an abbreviation
of either the species name or the cross used to produce a hybrid. There is one
column for each marker band.
POPLAR TREE EUCLIDEAN DISTANCES
<BR>
The matrix below lists the clones across the top and down the left side.
The body of the matrix contains the squared Euclidean distances between pairs
of clones.
<p>
So the distance between clones 1 and 3 is 8 while the distance between clones
3 and 4 is 5. The distance between a clone and itself is 0 and so all
diagonal elements are 0.
<p>
Some Observations :
<p>
The first nine clones are all P. trichocarpa and are closer to one another than
to any of clones 11-15 (P.nigra) or clones 16-17 (P.deltoides). <p>
Clones 19-23 and 27-29 are formed from crosses between P.trichocarpa and P.
deltoides. These are closer to each of the parent species than the parents are
to one another.
SOFTWARE FOR AMOVA ANALYSIS
<BR>
AMOVA is a computer program developed by
<a TARGET="_top" href="http://anthropologie.unige.ch/~laurent/">
Laurent Excoffier</a>
at the
<a TARGET="_top" href="http://acasun1.unige.ch/LGB/">Genetics and Biometry Laboratory</a>
of the
<a TARGET="_top" href="http://acasun1.unige.ch/">Department of Anthropology
& Ecology </a> in the University of Geneva.
<p>
The program is a very useful tool for the analysis of population
genetic structure at the molecular level. It runs under PC Windows and
is currently available as freeware from :<br>
<a TARGET="_top" href="http://acasun1.unige.ch/LGB/software/win/amova/">
http://acasun1.unige.ch/LGB/software/win/amova/ </a> <p>
To run the AMOVA program it is necessary to construct
<UL>
<LI>a <a href="slide8a.htm">group</a> file
<LI>a <a href="slide8b.htm">distance</a> file
<LI>several <a href="slide8c.htm">population</a> files
</UL>
There are <a href="slide8d.htm">settings</a> for choosing various options
in the analysis.<p>
Using the input files and settings as illustrated an <a href="slide8e.htm">
output</a> file is obtained.
<p>
Since developing AMOVA the Genetics & Biometry Laboratory has
produced a much more sophisticated program -
<a TARGET="_top" href="http://anthropologie.unige.ch/arlequin">ARLEQUIN </a> -
which can handle data from RFLPs, DNA sequences, microsatellites
as well as standard multi-locus or allele frequency data.
<P>
SUMMARY
<BR>
Analysis of Molecular Variance is a method for analysing population variation
using molecular data. <p>
It may be used on RFLP, RAPD, protein or allozyme data.<p>
A variety of distance metrics may be used for establishing distances between
pairs of chromosomes but AMOVA is strictly valid for squared Euclidean
distances only. <p>
AMOVA assumes that the restriction sites, RAPD markers, or allozymes are
independent.<p>
AMOVA provides a way of estimating <img alt="Phi" src="../images/image5a.gif" WIDTH=15 HEIGHT=16>
statistics which are an extension of
Wright's F statistics.
<P>
]]>