140
E. A. Thompson
Combining these we have
Pr(X 1 , X 2 , X 3 , Z 2 ) ∝ (2.490, 7.524, 0.304) × 10
−5 for Z 2 = 0, 1, 2.
and normalizing these gives the probability of Z 2 given X 1 , X 2 , X 3 as approximately (0.24, 0.73, 0.03) for Z 2 = 0, 1, 2. Note that whereas the data at locus-2
increased the relative probability that Z 2 = 2, incorporating the data at loci 1 and
3 greatly decreases the probability since the state of 2 IBD is impossible at each of
these flanking loci. For Z 2 = 2, a recombination would be required both between
locus-1 and locus-2 and between locus-2 and locus-3.
6.3.2 Monte Carlo Realization of IBD in Defined Pedigrees
The principles that underlie the computations of Sect. 6.3.1 are that the data at
each marker locus depends only on the latent IBD state at that locus and that the
transitions in latent IBD state follow a Markov process across the chromosome.
This is the classic framework for a hidden Markov model (HMM) which enables
many computations to be made. In fact, IBD is in general not Markovian, since
many different patterns of inheritance may give rise to the same IBD state
among observed individuals. However, in the absence of genetic interference, the
indicators of maternal or paternal transmission in a meiosis are indeed Markov (see
Equation (6.1)). This Markov nature of inheritance vectors (Sect. 6.2.1) has been
used by many to enable computations on pedigrees. For example, as implemented in
the MERLIN software (Abecasis et al., 2002), probabilities of IBD pairwise among
the members of a small pedigree may be computed at each marker locus, conditional
on the joint marker data on all pedigree members and on marker genotypes at
all loci. More generally, as seen in Sect. 6.2.1, the inheritance vector at a locus
determines the IBD state at that locus, among all members of the pedigree.
While location-specific probabilities of IBD are useful, there are good reasons
to prefer Monte Carlo realizations of latent inheritance vectors or IBD. First, it is
not feasible to consider probabilities of IBD jointly across multiple loci, but a set
of realizations directly display the segmental nature of inheritance and can identify
locations of recombination events that change the IBD among observed individuals.
Second, the variation in a set of realizations provides a measure of uncertainty in
the IBD information which cannot be captured in a single probability. In using IBD
inferred from marker data in subsequent genetic analyses, it is important to have
some measure of this uncertainty.
The same HMM methods that allow computation of IBD probabilities also allow
Monte Carlo realization of IBD conditional on genetic marker data (Thompson,
2000). On small pedigrees, where exact computation is feasible, independent
realizations may be generated. On larger pedigrees, where the space of inheritance
vectors at each locus is too large for exact computation, Markov chain Monte Carlo
(MCMC) methods must instead be used (Sobel and Lange, 1996; Thompson, 2000).
However, the same principles apply. The Markov dependence of inheritance vectors
E. A. Thompson
Combining these we have
Pr(X 1 , X 2 , X 3 , Z 2 ) ∝ (2.490, 7.524, 0.304) × 10
−5 for Z 2 = 0, 1, 2.
and normalizing these gives the probability of Z 2 given X 1 , X 2 , X 3 as approximately (0.24, 0.73, 0.03) for Z 2 = 0, 1, 2. Note that whereas the data at locus-2
increased the relative probability that Z 2 = 2, incorporating the data at loci 1 and
3 greatly decreases the probability since the state of 2 IBD is impossible at each of
these flanking loci. For Z 2 = 2, a recombination would be required both between
locus-1 and locus-2 and between locus-2 and locus-3.
6.3.2 Monte Carlo Realization of IBD in Defined Pedigrees
The principles that underlie the computations of Sect. 6.3.1 are that the data at
each marker locus depends only on the latent IBD state at that locus and that the
transitions in latent IBD state follow a Markov process across the chromosome.
This is the classic framework for a hidden Markov model (HMM) which enables
many computations to be made. In fact, IBD is in general not Markovian, since
many different patterns of inheritance may give rise to the same IBD state
among observed individuals. However, in the absence of genetic interference, the
indicators of maternal or paternal transmission in a meiosis are indeed Markov (see
Equation (6.1)). This Markov nature of inheritance vectors (Sect. 6.2.1) has been
used by many to enable computations on pedigrees. For example, as implemented in
the MERLIN software (Abecasis et al., 2002), probabilities of IBD pairwise among
the members of a small pedigree may be computed at each marker locus, conditional
on the joint marker data on all pedigree members and on marker genotypes at
all loci. More generally, as seen in Sect. 6.2.1, the inheritance vector at a locus
determines the IBD state at that locus, among all members of the pedigree.
While location-specific probabilities of IBD are useful, there are good reasons
to prefer Monte Carlo realizations of latent inheritance vectors or IBD. First, it is
not feasible to consider probabilities of IBD jointly across multiple loci, but a set
of realizations directly display the segmental nature of inheritance and can identify
locations of recombination events that change the IBD among observed individuals.
Second, the variation in a set of realizations provides a measure of uncertainty in
the IBD information which cannot be captured in a single probability. In using IBD
inferred from marker data in subsequent genetic analyses, it is important to have
some measure of this uncertainty.
The same HMM methods that allow computation of IBD probabilities also allow
Monte Carlo realization of IBD conditional on genetic marker data (Thompson,
2000). On small pedigrees, where exact computation is feasible, independent
realizations may be generated. On larger pedigrees, where the space of inheritance
vectors at each locus is too large for exact computation, Markov chain Monte Carlo
(MCMC) methods must instead be used (Sobel and Lange, 1996; Thompson, 2000).
However, the same principles apply. The Markov dependence of inheritance vectors