We study an extension of the standard framework for pedigree analysis, in which we allow pedigree founders to be inbred. This solves a number of practical challenges in calculating coefficients of relatedness, including condensed identity coefficients. As a consequence we expand considerably the class of pedigrees for which such coefficients may be efficiently computed. An application of this is the modelling of background inbreeding as a continuous effect. We also use inbred founders to shed new light on constructibility of relatedness coefficients, i.e., the problem of finding a genealogy yielding a given set of coefficients. In particular, we show that any theoretically admissible coefficients for a pair of noninbred individuals can be produced by a finite pedigree with inbred founders. Coupled with our computational methods, implemented in the R package ribd, this allows for the first time computer analysis of general constructibility solutions, thus making them accessible for practical use.
While we have focused on autosomal relatedness coefficients in this paper, the ideas presented transfer easily to X-chromosomal coefficients. To our knowledge ribd is the only package with a complete set of functions for computing kinship and identity coefficients both for the autosomes and the X chromosome, as well as a variety of other single-locus and two-locus coefficients. Table 1 shows a comparison with the partially overlapping R packages kinship (Sinnwell et al. 2014), identity (Abney 2009) and XIBD (Henden et al. 2016), and the command-line tool PedKin (Kirkpatrick et al. 2018).
Relatedness Calculator: Part Deux
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It is natural and interesting to seek a further extension of our approach, allowing pedigree founders to be not only inbred, but also related to each other. This would be particularly relevant for pedigrees in isolated populations, where the assumption of unrelatedness between all founders is unrealistic. However, the complexity of multi-person relatedness poses serious challenges for such an extension in full generality. For example, consider the algorithm in Sect. 3.1 for computing identity coefficients. If the founders are allowed to be related, then the boundary conditions (4) cease to hold, and must be replaced with formulas for \(\varphi _ab\), \(\varphi _abc\), \(\varphi _abcd\) and \(\varphi _ab,cd\), expressed by some coefficients describing the founder relationships. One might hope that these formulas only involved coefficients between each pair of related founders. Unfortunately this does not suffice in general, as shown by the following counter-example.
There is one important special case, however, where the (single-locus) inbreeding coefficient in fact captures the complete genetic constituency of the individual, namely when \(f=1\). The choice of mating process used to produce a completely inbred individual, has no bearing on the distribution of IBD alleles among his or her descendants, even at linked loci. In particular, any recursive algorithm for computing multi-locus relatedness coefficients can in principle be modified to allow completely inbred founders.
In this paper we have studied an extension of the conventional approach to pedigree analysis, in which we allow the assignment of inbreeding coefficients to the founders. The motivation is to enable a more compact representation of many pedigrees, while retaining sufficient information for exact computation of relatedness coefficients. This is particularly useful in cases where the true ancestries of certain pedigree members are unknown or unsuitable for computer modelling, such as completely inbred individuals. We believe that our implementation in ribd is the first software capable of computing identity coefficients in such pedigrees, even as simple as that in Fig. 1.
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