6 Identity by Descent in the Mapping of Genetic Traits
153
Glazner and Thompson (2015) proposed an alternate approach, in which a
joint IBD state among multiple individuals is built up successively from pairwise
inferences. The 15-state HMM is run on pairs of individuals as in Brown et al.
(2012). However, in adding individuals to a joint configuration, the IBD trajectories
across the chromosome are constrained by previously realized IBD. Using this
method in a simulated example, Glazner and Thompson (2015) showed that joint
IBD realized in the absence of an assumed pedigree structure can be used to
recover a likelihood across genome locations j using Equation (6.13). The “gold
standards” are likelihoods that would be obtained if IBD were perfectly imputed
from marker data X: Pr(Y | Z j ; Y ) at multiple locations j across a chromosome.
For IBD realized using haplotypic marker data the approximation is very good,
but for genotypic marker data, X, it is less so. Additionally, the method becomes
computationally intensive, and performance degrades, as larger sets of related
individuals have observed phenotypes, Y. There are additional problems also in the
use of Equation (6.13) for likelihood-based mapping in the absence of a pedigree.
First, the usual baseline probability Pr(Y; Y ) is not available; without a pedigree
there is no basis to compute it. Second, no constant baseline works well: the
likelihood (6.13) is affected by the inferred level of IBD across the chromosome,
and for IBD estimated without the constraints of a pedigree structure, this can vary
widely.
Because of unsolved problems in computationally feasible and effective ways to
realize joint IBD in the absence of a pedigree structure, we return to the pairwise
model of Equation (6.10) for our final example of IBD-based mapping in the
absence of an assumed pedigree. In this model, the local kinships j are now
estimated using the methods of Sect. 6.3.4, and the genome-wide kinship is
estimated by averaging the local kinships j across the genome. This approach was
first taken by Day-Williams et al. (2011). They used the estimator of local pairwise
kinship j outlined in Sect. 6.3.3 and denoted DW. The resulting estimators,
smoothed across the chromosome, may additionally be constrained so that each
j for each pair of individuals takes the values 0, 1/4, 1/2, or 1 (Table 6.2). By
contrast, the joint realizations of IBD among individuals produced by Glazner and
Thompson (2015) can be immediately reduced to a set of pairwise realizations of the
15 states of Table 6.2 and hence to local kinships j ; these estimates are denoted
GT. At each j , and for each pair of individuals, the averages across realizations can
also be constrained to the values 0, 1/4, 1/2, and 1.
The two estimation methods GT and DW were compared on a simulated example
using the variance component log-likelihood ratio (6.11) computed at locations j
across the chromosome. Since there was no evidence of a genome-wide genetic
effect, for simplicity it was assumed that σ 2
a = 0. The two sets of local estimates
were each used constrained and unconstrained. Each of the four log-likelihood
curves were compared with the curve that would be obtained if the actual Z and
hence the realized j were known at each location j . Details may be found in
Glazner and Thompson (2015), but generally the GT estimators performed better
than the DW estimators. Also, whereas for GT there was little difference in the
results between the constrained and unconstrained versions, for DW the constraint
153
Glazner and Thompson (2015) proposed an alternate approach, in which a
joint IBD state among multiple individuals is built up successively from pairwise
inferences. The 15-state HMM is run on pairs of individuals as in Brown et al.
(2012). However, in adding individuals to a joint configuration, the IBD trajectories
across the chromosome are constrained by previously realized IBD. Using this
method in a simulated example, Glazner and Thompson (2015) showed that joint
IBD realized in the absence of an assumed pedigree structure can be used to
recover a likelihood across genome locations j using Equation (6.13). The “gold
standards” are likelihoods that would be obtained if IBD were perfectly imputed
from marker data X: Pr(Y | Z j ; Y ) at multiple locations j across a chromosome.
For IBD realized using haplotypic marker data the approximation is very good,
but for genotypic marker data, X, it is less so. Additionally, the method becomes
computationally intensive, and performance degrades, as larger sets of related
individuals have observed phenotypes, Y. There are additional problems also in the
use of Equation (6.13) for likelihood-based mapping in the absence of a pedigree.
First, the usual baseline probability Pr(Y; Y ) is not available; without a pedigree
there is no basis to compute it. Second, no constant baseline works well: the
likelihood (6.13) is affected by the inferred level of IBD across the chromosome,
and for IBD estimated without the constraints of a pedigree structure, this can vary
widely.
Because of unsolved problems in computationally feasible and effective ways to
realize joint IBD in the absence of a pedigree structure, we return to the pairwise
model of Equation (6.10) for our final example of IBD-based mapping in the
absence of an assumed pedigree. In this model, the local kinships j are now
estimated using the methods of Sect. 6.3.4, and the genome-wide kinship is
estimated by averaging the local kinships j across the genome. This approach was
first taken by Day-Williams et al. (2011). They used the estimator of local pairwise
kinship j outlined in Sect. 6.3.3 and denoted DW. The resulting estimators,
smoothed across the chromosome, may additionally be constrained so that each
j for each pair of individuals takes the values 0, 1/4, 1/2, or 1 (Table 6.2). By
contrast, the joint realizations of IBD among individuals produced by Glazner and
Thompson (2015) can be immediately reduced to a set of pairwise realizations of the
15 states of Table 6.2 and hence to local kinships j ; these estimates are denoted
GT. At each j , and for each pair of individuals, the averages across realizations can
also be constrained to the values 0, 1/4, 1/2, and 1.
The two estimation methods GT and DW were compared on a simulated example
using the variance component log-likelihood ratio (6.11) computed at locations j
across the chromosome. Since there was no evidence of a genome-wide genetic
effect, for simplicity it was assumed that σ 2
a = 0. The two sets of local estimates
were each used constrained and unconstrained. Each of the four log-likelihood
curves were compared with the curve that would be obtained if the actual Z and
hence the realized j were known at each location j . Details may be found in
Glazner and Thompson (2015), but generally the GT estimators performed better
than the DW estimators. Also, whereas for GT there was little difference in the
results between the constrained and unconstrained versions, for DW the constraint