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Polymorphic

What is it?What is it?

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Pop A Pop B

XMonomorphic

(if referring to single-locus data)

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Why is it important?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

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TMRCA dictates diversityTMRCA dictates diversity

More diversity

Large sum of branch lengths

More time for mutations to accumulate

Tim

e

Less diversity

Small sum of branch lengths

Less time for mutations to accumulate

Factors which increase TMRCA?…

Large N (constant through time)

No bottlenecks (N varies through time)

Recent admixture

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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

How do we measure diversity?How do we measure diversity?

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hh interpretation I (Probability) 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

Aii

D

Aii

pP

pP

2

2

1same) theNOT are chromos 2(

same) theare chromos 2(

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hh interpretation I (Probability) 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

7.0same) theNOT are chromos 2(

3.0same) theare chromos 2(

P

P

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)

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hh interpretation II (Variance) 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…

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VarianceVarianceIn statistics, the most widely used measure of diversity is

variance

(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 4

0.0

0.1

0.2

0.3

0.4

x

dn

orm

(x)

E[X] X

Deviation from the mean

One example value of X from its distribution

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

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hh interpretation II (Variance) 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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EstimatingEstimating h h

By definition,

Where p i is the true population frequency of Allele i

m

iiph

1

21

Hat means this is an estimate

But this estimate is biased – i.e truehhE ]ˆ[

truehhE ]ˆ[bias

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

m

iixh

1

21ˆ

An obvious estimate of h is therefore:

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Deriving an unbiased estimate ofDeriving an unbiased estimate of h hThe 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

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Deriving an unbiased estimate ofDeriving an unbiased estimate of h hAdjust to avoid self-matches…

Each number in the grid below represents a different chromosome in the sample

1 2 3 4 5 6 7 8 9 n

1 2

3 4

5 6

7 8

9 n

a1 = 3

a2 = 3

a3 = 4

Area of “box” = n 2

Unadjusted frequency of ‘same’ matches:

(a12 + a2

2 + a32)/n 2

Adjusted frequency of ‘same’ matches:

(a12 + a2

2 + a32 – n) / (n 2 – n)

Adjusted frequency of ‘different’ matches:

1 – (a12 + a2

2 + a32 – n) / (n 2 – n)

m

iiunb x

n

nh

1

21)1(

ˆSome algebra results in:

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a1

a2

a3

The sampling distribution ofThe sampling distribution of h hunbunb

‘True’ h has no variance – there is only one unique value for each population

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 0.8 1.0

01

23

4

x

db

eta

(x, 9

, 2

)

‘true’ h = 0.9

DiversitDiversityy ^

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

x

db

eta

(x, 7

, 2

)

The sampling distribution ofThe sampling distribution of h hbiasedbiased

‘true’ h = 0.9


Estimating the sampling distribution of Estimating the sampling distribution of h h by bootstrappingby 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

unbhFor each resample, we calculate and use the values over many resamples to build up the bootstrap distribution for

unbh

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^

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

x

db

eta

(x, 7

, 2

)

The bootstrap distribution ofThe bootstrap distribution of h hunbunb

Because bootstrapping resamples the sample, and not the population, the resulting bootstrap distribution is biased

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)

‘true’ h = 0.9


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