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Bayesian Estimators of Time to Most Recent Common Ancestry. Bruce Walsh. [email protected] Ecology and Evolutionary Biology. Adjunct Appointments Molecular and Cellular Biology Plant Sciences Epidemiology & Biostatistics Animal Sciences. Definitions.
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Bruce Walsh
Ecology and Evolutionary Biology
Adjunct Appointments
Molecular and Cellular Biology
Plant Sciences
Epidemiology & Biostatistics
Animal Sciences
MRCA  Most Recent Common Ancestor
TMRCA  Time to Most Recent Common Ancestor
Question: Given molecular marker information from
a pair of individuals, what is the estimated time back
to their most recent common ancestor?
With even a small number of highly polymorphic
autosomal markers, trivial to assess zero (subject/
biological sample) and one (parentoffspring) MRCA
Often we are very interested in MRCAs that are modest
(510 generations) or large (100’s to 10,000’s of
generations)
Unlinked autosomal markers simply don’t work over these
time scales.
Reason: IBD probabilities for individuals sharing a MRCA
5 or more generations ago are extremely small and hence
very hard to estimate (need VERY large number of
markers).
We need to distinguish between the MRCA for a pair
of individuals (MRCAI) and the MRCA for a particular
genetic marker G (MRCAG).
MRCAG varies between any two individuals over
recombination units.
For example, we could easily have for a pair of relatives
MRCA (mtDNA ) = 180 generations
MRCA (Y ) = 350 generations
MRCA (one aglobulin allele ) = 90 generations
MRCA (other a globulin allele ) = 400 generations
So how can we accurately estimate TMRCA for modest
to large number of generations?
Answer: Use a set of completely linked markers
With autosomes, unlinked markers assort each generation
leaving only a small amount of IBD information on each
marker, which we must then multiply together. IBD
information decays on the order of 1/2 each generation.
With completely linked marker loci, information on IBD does not assort away via recombination. IBD information decay is on the order of the mutation rate.
Estimates of human mutation rate in microsatellites
are fairly consistent over both the Y and the autosomes
Basic Structure of the Problem
What is the probability that the two marker
alleles at a haploid locus from two related
individuals agree given that their MRCA was
t generation ago?
Phrased another way, what is their probability
of identity by state (IBS), given they are identical
by descent (IBD) when their TMRCA is t
generations
The first step in answering this question is to
assume a particular mutational model
Our (initial) assumption will be the infinite alleles
model (IAM)
The key assumption of this model (originally due
to Kimura and Crow, 1964) is that each new mutation
gives rise to a new allele.
The IAM was the first populationgenetics model to
attempt to formally incorporate the structure of DNA
into a model
B
(1u)t
(1u) t
Pr(No mutation from MRCA>A) = (1u)t
Pr(No mutation from MRCA>B) = (1u)t
MRCA
MRCA
Key: Under the infinite alleles, two alleles that are
identical in state that are also ibd have not
experienced any mutations since their MRCA.
Let q(t) = Probability two alleles with a MRCA
t generations ago are identical in state
If u = per generation mutation rate, then
q(t) = (1u)2t
Building the Likelihood Function for n Loci





(

)


For any single marker locus, the probability of IBS
given a TMRCA of t generations is
q(t) = (1u)2t ≈ e2ut = et, t = 2ut
The probability that k of n marker loci are IBS is just
a Binomial distribution with success parameter q(t)
Likelihood function for t given k of n matches
)
(
)
^


^

(

)


ML Analysis of TMRCAIt would seem that we now have all the pieces in
hand for a likelihood analysis of TMRCA given
the marker data (k of n matches)
Likelihood function (t = 2ut)
MLE for t is solution of ∂ L/∂t = 0
p = fraction of matches
)

(
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^

Var( t ) =
^
In particular, the MLE for t becomes
Likewise, the precision of this estimator follows
for the (negative) inverse of the 2nd derivative
of the loglikelihood function evaluated at the
MLE,
Finally, hypothesis testing, say Ho: MRCA = t0, is
easily accomplished by comparing 2* the natural
log of the ratio of the value of the likelihood function
at t = t0 over the value of the likelihood function at the
MLE t = t
^
Likewise, we can (numerically) easily find 1LOD
support intervals for t and hence construct
approximate 95% confidence intervals to TMRCA
The resulting log likelihood ratio LR is (asymptotically)
distributed as a chisquare distribution with one degree of
freedom
The ML machinery has seem to have done its job,
giving us an estimate, its approximate sampling
error, approximate confidence intervals, and a scheme
for hypothesis testing.
Hence, all seems well.
Problem: Look at k=n (= complete match at all markers).
MLE (TMRCA) = 0 (independent of n)
Var(MLE) = 0 (ouch!)
What about oneLOD support intervals (95% CI) ?
With n=k, the value of the likelihood function is
L(t) = (1u)2tn ≈ e2tun
L has a maximum value of one under the MLE
Hence, any value of t that gives a likelihood value of
0.1 or larger is in the oneLOD support interval
Solving, the oneLOD support interval is from t=0 to
t = (1/2n) [ Ln(10)/Ln(1u) ] ≈(1/n) [ Ln(10)/(2u) ]
For u = 0.002, CI is (0, 575/n)
MLE(t) = 0 for all values on n
1 LOD ≈ t = 29
1 LOD ≈ t = 58
1 LOD ≈ t = 115
0.1 of max value (1) of
likelihood function
With n=k, likelihood function reduces to
L(t) = (1u)2tn ≈ e2tun
n=5
n=10
L(t)
n=20
t
(Plots for u = 0.002)
What about Hypothesis testing?
Again recall that for k =n that the likelihood at t = t0 is
L(t0) ≈ Exp(2t0un)
Hence, the LR test statistic for Ho: t = t0 is just
LR = 2 ln [ L(t0)/ L(0) ]
= 2 ln [ Exp(2t0un) / 1 ]
= 4t0un
Thus the probability for the test
that TMRCA = t0 is just Pr( c12 > 4t0un)
The expressions developed for the sampling variance,
approximate confidence intervals, and hypothesis
testing are all largesample approximations
Problem 1: Here our sample size is the number of
markers scored in the two individuals. Not likely to
be large.
Problem 2: These expressions are obtained by taking
appropriate limits of the likelihood function. If the
ML is exactly at the boundary of the admissible space
on the likelihood surface, this limit may not formally
exist, and hence the above approximations are incorrect.
q given x
Likelihood function for q
Given the data x
priordistribution for q
The appropriate constant so that the posterior
integrates to one.
The Solution? Go BayesianAn extension of likelihood is Bayesian statistics
Instead of simply estimating a point estimate (e.g., the
MLE), the goal is the estimate the entire distributionfor the unknown parameter q given the data x
p(q  x) = C * l(x  q) p(q)
Why Bayesian?
• Exact for any sample size
• Marginal posteriors
• Efficient use of any prior information
• MCMC (such as Gibbs sampling) methods
The first step in any Bayesian analysis is choice of
an appropriate prior distribution p(t)  our thoughts on
the distribution of TMRCA in the absence of any of
the marker data
Standard approach: Use a flat or uninformative prior,
with p(t) = a constant over the admissible range of the
parameter. Can cause problems if the likelihood function
is unbounded (integrates to infinity)
In our case, populationgenetic theory provides the
prior: under very general settings, the time to MRCA
for a pair of individuals follows a geometric distribution




Previous likelihood function (ignoring constants
that cancel when we compute the normalizing factor
C)
Prior
In particular, for a haploid gene, TMRCA follows
a geometric distribution with mean 1/Ne.
Hence, our prior is just
p(t) = l(1l)t≈l elt, where l = 1/Ne
Hence, we can use an exponential prior with
hyperparameter (the parameter fully
characterizing the distribution) l = 1/Ne.
The posterior thus becomes







The Normalizing constantwhere
I ensures that the posterior distribution integrates
to one, and hence is formally a probability distribution



What is the effect of the hyperparameter?If 2uk >> l, then essentially no dependence on the
actual value of l chosen.
Hence, if 2Neuk >> 1, essentially no dependence on
(hyperparameter) assumptions of the prior.
For a typical microsatellite rate of u = 0.002, this is just
Nek >> 250, which is a very weak assumption. For example,
with k =10 matches, Ne >> 25. Even with only 1 match (k=1),
just require Ne >> 250.




(




=








Closedform Solutions for the Posterior DistributionComplete analytic solutions to the prior can be obtained
by using a series expansion (of the (1ex)n term) to give
Each term is just a * ebt, which is easily integrated









Hence, the complete analytic solution of the posterior is
Suppose k = n (no mismatches)
In this case, the prior is simply an exponential
distribution with mean 2un + l,




Analysis of n = k caseMean TMRCA and its variance:
Cumulative probability:
In particular, the time Ta satisfying P(t < Ta) = a is
For a flat prior (l = 0), the 95% (oneside) confidence
interval is thus given by ln(0.5)/(2nu) ≈ 1.50/(nu)
Hence, under a Bayesian analysis for u = 0.02, the
95% upper confidence interval is given by ≈ 749/n
Recall that the oneLOD support interval (approximate
95% CI) under an ML analysis is ≈ 575/n
The ML solution’s asymptotic approximation significantly
underestimates the true interval relative to the
exact analysis under a full Bayesian model
n = 10, t0.95 = 75
Under a Bayesian analysis, we look
at the posterior probability
distribution (likelihood adjusted
to integrate to one) and find the
values that give an area of 0.95
Under ML, we plot the
likelihood function and look
for the 0.1 value
Pr(TMRCA < t)
n = 10, area to
left of t=75 = 0.95
n = 20, area to
left of t=38 = 0.95
t
Why the difference?
Posteriors for n = 20
Posteriors for n = 50
n = 100
93
n = 100
0.30
92
0.060
94
91
100
90
0.25
0.050
99
0.20
0.040
p( t  k )
p( t  k )
98
89
0.15
97
0.030
96
95
0.10
0.020
0.05
0.010
0.00
0.000
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Time t to MRCA
Time t to MRCA
Sample Posteriors for u = 0.002
• By using markers on a non recombining chromosomal
section, we can estimate TMRCA over much, much
longer time scales than with unlinked autosomal markers
• By using the appropriate number of markers we
can get accurate estimates for TMRCA for even
just a few generations. 2050 markers will do.
• Hence, we have a fairly large window of resolution
for TMRCA when using a modest number of completely
linked markers.

]
[



Extensions I: Different Mutation RatesLet marker locus k have mutation rate uk.
Code the observations as xk = 1 if a match,
otherwise code xk = 0
The posterior becomes:
Mutation 2
Under IAM, score as a match, and hence no mutations
In reality, there are two mutations
Stepwise Mutation Model (SMM)Microsatellite allelic variants are scored by their number
of repeat units. Hence, two “matching” alleles can actually
hide multiple mutations (and hence more time to the MRCA)
The Infinite alleles model (IAM) is not especially
realistic with microsatellite data, unless the fraction
of matches is very high.
The SMM model is an attempt to correct for
multiple hits by fully accounting for the mutational
structure.
Good fit to array sizes in natural populations when
assuming the symmetric singlestep model
• Equal probability of a onestep move up or down
In direct studies of (Y chromosome) microsatellites
35 of 37 dectected mutations in pedigrees were
single step, other 2 were twostep
The simplest implementation of the SMM model is
to simply replace the match probabilities q(t) under
the IAM model with those for the SMM model.
This simply codes the marker loci as match / no match
We refer to this as the SMMO model

>

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Formally, the SMM model assumes the following
transition probabilities
Note that two alleles can match only if they have
experienced an even number of mutations in total between
them. In such cases, the match probabilities become
Under the SMM model, the prior hyperparameter
can now become important.
This is the case when the number n of markers is
small and/or k/n is not very close to one
Why? Under the prior, TMRCA is forced by a
geometric with 1/Ne. Under the IAM model for
most values this is still much more time than
the likelihood function predicts from the marker data
Under the SMM model, the likelihood alone predicts
a much longer value so that the forcing effect of the
initial geometric can come into play
IAM, both flat prior and Ne = 5000
SSMO, Ne = 5000
SSMO, flat prior
Pr(TMRCA < t)
Prior with Ne =5000
Time, t

The jthorder modifed
Type I Bessel Function
…
…
An Exact Treatment: SMMEWith a little work we can show that the probability
two sites differ by j steps is just
The resulting likelihood thus becomes
Where nj is the number of sites that differ
by k (observed) steps


…

Number of exact matches
Number of k steps differences
With this likelihood, the resulting prior becomes
This rearranges to give the general posterior
under the Exact SMM model (SMME) as
Consider comparing haplotypes 1 and 3 from Thomas
et al.’s (2000) study on the Lemba and Cohen Y
chromosome modal haplotypes. Here six markers used,
four exactly match, one differs by one repeat, the other
by two repeats
Hence, n = 6, k = 4 for IAM and SMM0 models
n0 = 4, n1 = 1, n2 = 1, n = 6 under SMME model
Assume Hammer’s value of Ne=5000 for the prior
The expressions developed above have direct
commercial applications
Family Tree DNA (ftDNA)  provides Y
chromosome marker kits for genealogical studies
So far, ftDNA has processed over 10,000 such
kits
This amounts to a rough gross of around 3 million
dollars.
NRC 1996 recommendations (autosomal loci)
Product rule across markers
Population substructure correction within markers
Prob(Y match)*Prob(autosomal match)
Problem: Y markers may provide information
about population substructure membership
For example, a particular haplotype may be restricted
to a certain subpopulation, e.g., Native Americans
Let y denote the observed Y haplotype
A the multilocus autosomal marker genotype
P(y,A) = P(A  y)*P(y)
Simple approach: knowledge of y indicates membership
in a particular subpopulation, P(A) computed using
allele frequencies for that subpopulation.
Suggestion: Multiply freq(y)* max Freq(A over subgroups)
Suppose two individuals share the same y haplotype.
What is there average coancestry, q?
Balding and Nichols give expressions for autosomal
singlelocus genotype frequencies given that the
population shows structure with coancestry q.
Second approach: Compute q from haplotype matching.
Using this q value in Balding  Nichols expressions
to compute (singlelocus) autosomal frequencies.
>
>
Posterior Distribution for a match at all
n markers with a prior of l = 1/Ne
With a MRCA of t generations, q = (1/2)2t+1
Typical situation is where we can exclude
fatherson and paternal halfsibs, k > 2
Typical values, n = 11, m = 1/500
• For l = 1/5000, E [ q ] = 0.00186
• For l = 1/500, E [ q ] = 0.00194
• For l = 1/50, E [ q ] = 0.00272
For these values, unless pi < 0.01, BaldingNichols
expression are essentially HW.
Estimate P(y) from a database (counting methods,
Bayesian estimators)
Compute mutlilocus autosomal frequencies by
each major ethnic group using the product of
the singlelocus genotypes computed using
groupspecific allele frequencies and q = 0.002
correction.
Conservative P(y,A) = P(y)*max P(A)