Diversity
What is it?
Pop A
Pop B
****
********
*********
*********
*********
****
****
********
*********
*********
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****

Polymorphic
(if referring to
single-locus data)
X
Monomorphic
Diversity
Why is it important?
Diversity is a reflection of
1) Demographic history
2) Mutational processes
Almost without exception, when comparing two populations,
we can assume that mutational processes are the same in
both pops.
Hence differences in diversity indicate differences in
demographic history
TMRCA dictates diversity
Time
Diversity
More diversity
Less diversity
Large sum of branch lengths
Small sum of branch lengths
More time for mutations to
accumulate
Less time for mutations to
accumulate
Factors which increase TMRCA?…
Large N (constant through time)
No bottlenecks (N varies through time)
Recent admixture
Diversity
How do we measure diversity?
Hence, finding differences in diversity can give us
clues about differences in the demographic history
of two populations
So, how do we measure diversity?
Depends on what we’re measuring.
The simplest data are simple categories: Allele A, B, C, D etc.
Even here, there is more than one way to measure diversity
- Number of different alleles
- Genetic diversity, h
Problem: depends on sample size
Despite appearing more complicated,
has the advantage of interpretability
Diversity
h interpretation I (Probability)
h is the probability that two chromosomes picked at random
from the population will be different (using the given
genetic markers)
Allele
Frequency
A
pA
pA2 = probability that 2 randomly-chosen chromosomes are both AA
B
pB
pB2 = probability that 2 randomly-chosen chromosomes are both BB
C
pC
pC2 = probability that 2 randomly-chosen chromosomes are both CC
D
pD
pD2 = probability that 2 randomly-chosen chromosomes are both DD
D
 P(2 chromos are the same)   pi2
i A
D
 P(2 chromos are NOT the same)  1   pi2
i A
Diversity
h interpretation I (Probability)
h is the probability that two chromosomes picked at random
from the population will be different (using the given
genetic markers)
e.g….
Allele
Frequency
A
0.3
0.09 = probability that 2 randomly-chosen chromosomes are both AA
B
0.2
0.04 = probability that 2 randomly-chosen chromosomes are both BB
C
0.1
0.01 = probability that 2 randomly-chosen chromosomes are both CC
D
0.4
0.16 = probability that 2 randomly-chosen chromosomes are both DD
 P(2 chromos are the same)  0.3
 P(2 chromos are NOT the same)  0.7
In diploid systems, chromosomes naturally come in pairs. Here, h is also the
“expected heterozygosity” – i.e. the expected frequency of heterozygotes if
alleles joined at random (Hardy Weinberg Equilibrium)
Diversity
h interpretation II (Variance)
You may wonder why we use h in haploid systems, when chromosomes
do not come naturally in pairs
The answer is that h is still a good measure of diversity, and that
thinking about pairs of chromosomes is still a natural way to think
about the problem
h is twice the “within-population variance”, when defined as follows…
Diversity
Variance
In statistics, the most widely used measure of diversity is variance
0.2
Deviation from the mean
One example value of X from its
distribution
0.0
0.1
dnorm(x)
0.3
0.4
(Note: standard deviation is derived from the variance with a 1-to-1
correspondence, so mathematically contains the same information
(it is the square root of the variance))
-4
-2
0
2
x
X
E[X]
4
Variance is the expected squared deviation from the mean
Var(X) = E[ (X – E[X])2 ]
A little-known fact is the variance is also half the expected squared
difference between 2 randomly-sampled X values
Var(X) = E[ (X –X’)2 ]/2 = E[diff 2]/2
Diversity
h interpretation II (Variance)
Going back to diversity in a population, let us define diff=0 if 2 chromos
are the same, and define diff=1 if 2 chromos are different
What is E[ diff 2 ] for 2 randomly-drawn chromosomes?
= Fr(same) x 02 + Fr(different) x 12
Hence, by defining variance in terms of difference between 2 objects,
and defining diff=0 for ‘same’ and diff=1 for ‘different’, we gain a
mathematical 1-to-1 relationship between h and variance
h = Fr(different) = E[ diff
2
] = 2*variance
This is more nifty than it may at first appear, because variance is a
concept normally applied to a scalar variable X, whereas h applies
to a vector of frequency variables p1, p2, p3 … pm (where m is the
number of different alleles in the population)
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Estimating h
m
By definition,
h  1   pi2
i 1
Where p i is the true population frequency of Allele i
In practice, we never know p i , only an estimate x i based on sample
counts:
x i = a i /n
where a i = number of Allele i in sample and n = total sample size
An obvious estimate of h is therefore:
m
hˆ  1   xi2
Hat means this is an estimate
i 1
But this estimate is biased – i.e
E[hˆ]  htrue
bias  E[hˆ]  htrue
Diversity
Deriving an unbiased estimate of h
The following is is not a full explanation, but hopefully will give the gist of it
Remember that h can be derived by thinking about picking 2 chromosomes at
random from the true population
The true population, for this purpose, is assumed to be infinite so that it is
impossible to pick the same chromosome twice
To mimic this situation in the sample we have taken, we must arrange things so
that the two chromosomes are picked without replacement from the sample
Diversity
Deriving an unbiased estimate of h
Adjust to avoid self-matches…
a1
a2
7 8 9 n
a3
1 2 3 4 5 6
Each number in the grid below represents a different chromosome in the sample
a1 = 3
1 2 3 4 5 6 7 8 9 n
a2 = 3
a3 = 4
Area of “box” = n
2
Unadjusted frequency of ‘same’ matches:
(a12 + a22 + a32)/n
2
Adjusted frequency of ‘same’ matches:
(a12 + a22 + a32 – n) / (n 2 – n)
Adjusted frequency of ‘different’ matches:
1 – (a12 + a22 + a32 – n) / (n 2 – n)
Some algebra results in:
m
n

2
ˆ
hunb 
1   xi 
(n  1)  i 1 
Diversity
^
The sampling distribution of hunb
‘True’ h has no variance – there is only one unique value for each population
2
1
0
dbeta(x, 9, 2)
3
4
Estimated h does have a variance – you will get a slightly different value every
time you sample n chromosomes from the population, because the sample will
be different
0.0
0.2
0.4
0.6
x
0.8
1.0
‘true’ h = 0.9
^
0.0
0.5
1.0
1.5
2.0
2.5
3.0
The sampling distribution of hbiased
dbeta(x, 7, 2)
Diversity
0.0
0.2
0.4
0.6
x
0.8
1.0
‘true’ h = 0.9
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^
Estimating the sampling distribution of h by
bootstrapping
What is bootstrapping?
In bootstrapping, we assume that the estimated allele frequencies x i ARE the
‘true‘ frequencies p i
We now resample “fake” samples of size n from this imaginary population, lots of
times
For each resample, we calculate ĥunb and use the values over many resamples
to build up the bootstrap distribution for ĥ
unb
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^
The bootstrap distribution of hunb
Because bootstrapping resamples the sample, and not the population, the
resulting bootstrap distribution is biased
1.5
1.0
0.5
0.0
dbeta(x, 7, 2)
2.0
2.5
3.0
In fact, there is no absolutely watertight way of testing for the difference between
two h values. For this reason, I use a double-conservative procedure (see
http://www.tcga.ucl.ac.uk/software)
0.0
0.2
0.4
0.6
x
0.8
1.0
‘true’ h = 0.9