Abstract
The variance components (VC) model has been popular for genetic analysis. It has received wide applications in a variety of genetic practices, and been extended to various forms for different settings. However, most of the existing VC models are, explicitly or implicitly, under the assumption of the Hardy–Weinberg and/or linkage equilibria, which is impractical in some realistic settings since more or less deviations from this assumption are common. We propose a new VC model that incorporates both these disequilibria, and includes the existing models as special cases. The corresponding variance components are computed for some commonly used relative pairs conditional on the observed marker identity-by-descent data. Parameters can be estimated by the traditional methods such as the maximum likelihood estimate. Simulation studies suggest that this extended model improves inference significantly over the existing models when deviations of these disequilibria are present.
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Acknowledgements
We appreciate the suggestions/comments from the Editor and the two anonymous reviewers, which greatly improved the quality of this manuscript. The work was supported by the United States Public Service Grant No AG 16996 and the National Center for Research Resources Grant No 2G12RR003048 from the National Institutes of Health. The AADM study was supported by NIH Grants no. 3T37TW00041 from NCMHD and NHGRI. G Chen and Rotimi were also partly supported by the NIGMS/MBRS program.
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Appendices
Appendix A
We first derive (5). When fixed Ak, the events AkAl and AkAm are independent, so by (4) we have,
For (6), we use the method as in Lange16 (pp. 87–89). Since 
and the αk's minimize the squared error 
, take derivative with respect to αk, we get
Sum over k in (A.1) we have
Now (A.1) and (A.2) gives
that is,
Then we have
When i=j, Cov(Gi, Gj)=σG2, Cov(ei, ej)=σe2 and
By (A.1), the above is
Since ∑lpkl=∑lplk=pk, we have
and
so by (A.3) and the above three equations we have
If i≠j, Cov(eiej)=0, by the central limit theorem of Lange22 and assume no dominance, we have approximately Cov(Gi, Gj)=2ΦijσG2 and
By (A.1) and (A.2), the coefficient Δ9ij is zero. By the calculation for E(gi2), the first term above is
the second term is
From (5) it is easy to check that
so by (A.2) the coefficient of Δ8ij is
Since ∑k∑lαkαlpkl=fσa2/2, the above is
By (5) and (A.1), the middle term in the above is
By the same way, the last term in (A.6) is
so the coefficient of Δ8ij is
Now collecting terms we have
Appendix B
When i=j, πii′=2 which is noninformative about trait-marker relationship, so 
, which has the same expression as in (8). When i≠j,
We first derive the conditional probabilities 
in (B.1). Since conditioning on the IBD status, those quantities are independent of relatedness of the pair, only depend on the relationships among the trait and marker alleles through f and ζ, in other words, given IBD status, different alleles in one configuration are independent with those in the other one. We have
Now the two configurations share AkAl in common, if we fix it, the two configurations are independent each other, so we rewrite the above as
Similarly,
and
where p(kl, r)=∑sp(kl, rs). The same reason gives
so we have
Also
where
So
and
so
Also
and
where p(k, rs)=∑lp(kl, rs)=pk(qrs−ζprs+ζ(1−f)psI(r=k)+ζI(r=s=k)), so
Lastly
Now we compute the covariance (B.1) for different values of the π′ij's. If π′ij=0, by (B.2)–(B.4) and Appendix A, we have the same expression of (B.1) as in (8).
If π′ij=1, by (B.5)–(B.7), the coefficient of Δ7ij(πij) and Δ8ij(πij) in (B.1) are the same as that in (A.4) and (A.7); the coefficient of Δ9ij(πij) in (B.1) has four terms corresponding to those in (B.7), the first two terms are zero by the computation in Appendix A, by its symmetry in (k, l, m, n), the last two terms are
since
the first term above is zero. By expanding and check each term using (A.1) and (A.2), the second term above, and hence the coefficient of Δ9ij(πij) is
If π′ij=2, by (B.8), the coefficient of Δ7ij(πij) is the same as before. Now we compute the coefficient of Δ8ij(πij). We expand it in five terms as in (B.9). The first term is (1+f)2(1−ζ)σa2/2 by the computation in Appendix A. Expanding the same way as in Appendix A, the second term is
the last two terms above are zero by (A.1). Since
substitute this into the second and the third term in (B.12), it becomes −(1+f)2∑kαk2pk(qk−ζpk). By expanding the same way, the third term is
The last two terms in (B.14) are zero. Substitute (B.13) into the second and third term in (B.14), it becomes
now combine the second and the third terms gives
the fourth term is
the fifth term is
For the coefficient of Δ9ij(πij), we expand it in four terms according to (B.10), the first two terms are zero by the computation in Appendix A, so it reduces to
the first term above is zero since 
, the second term, and hence the coefficient of Δ9ij(πij) is 
, which is
The first term in the bracket above is
the second term is
the third term is
the fourth term is
Now collect terms, the coefficient of Δ9ij(πij) is
Appendix C
Let ξ=(α, β), ξ0=(α0, β), and define ξ1 and the hat notations for the corresponding estimates. Let Î(ξ) be the empirical Fisher information matrix evaluated at (ξ), by Taylor expansion,
and its is well known that, under H1, as n → ∞,
Also, since the familial structures are homogeneous, so
Thus under H1,
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Yuan, A., Chen, G., Yang, Q. et al. Variance components model with disequilibria. Eur J Hum Genet 14, 941–952 (2006). https://doi.org/10.1038/sj.ejhg.5201645
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DOI: https://doi.org/10.1038/sj.ejhg.5201645
