Variation in actual relationship as a consequence of Mendelian sampling and linkage

Abstract

Although the expected relationship or proportion of genome shared by pairs of relatives can be obtained from their pedigrees, the actual quantities deviate as a consequence of Mendelian sampling and depend on the number of chromosomes and map length. Formulae have been published previously for the variance of actual relationship for a number of specific types of relatives but no general formula for non-inbred individuals is available. We provide here a unified framework that enables the variances for distant relatives to be easily computed, showing, for example, how the variance of sharing for great grandparent-great grandchild, great uncle-great nephew, half uncle-nephew and first cousins differ, even though they have the same expected relationship. Results are extended in order to include differences in map length between sexes, no recombination in males and sex linkage. We derive the magnitude of skew in the proportion shared, showing the skew becomes increasingly large the more distant the relationship. The results obtained for variation in actual relationship apply directly to the variation in actual inbreeding as both are functions of genomic coancestry, and we show how to partition the variation in actual inbreeding between and within families. Although the variance of actual relationship falls as individuals become more distant, its coefficient of variation rises, and so, exacerbated by the skewness, it becomes increasingly difficult to distinguish different pedigree relationships from the actual fraction of the genome shared.

Fig. 1

Examples of relationship

Fig. 1

Examples of relationship

Fig. 2

(a) SD and (b) CV of actual relationship (proportion of genome shared, $\stackrel{⌣}{R}={\stackrel{⌣}{k}}_2+\frac{1}{2}{\stackrel{⌣}{k}}_1$), for a single chromosome as a function of map length and relationship for full sibs (FS) and their descendants: uncle nephew (UN), cousins (C), cousins once removed (C1R), second cousins (2C), second cousins once removed (2C1R) and third cousins (3C).

Fig. 3

SD of actual relationship (proportion of genome shared, $\stackrel{⌣}{R}={\stackrel{⌣}{k}}_2+\frac{1}{2}{\stackrel{⌣}{k}}_1$), for a single chromosome as a function of map length and relationship for three different pedigrees for two different pedigree relationships: R = 0·125: great grandparent–great grandoffspring (GGPGGO), half-uncle–nephew (HUN), great uncle–great nephew (GUGN), cousins (C); and R = 0·03125: greatgreatgreat grandparent–GGGGoffspring (G4PG4O), half–cousins once removed (HC1R) and second cousins (2C).

Fig. 4

Skewness of actual relationship (proportion of genome shared) for a single chromosome as a function of map length and relationship for (a) descendants of full sibs (as Fig. 2), and (b) for different pedigrees for two different degrees of relationships (as Fig. 3). For full sibs and uncle–nephew there is no skew.

Fig. 5

Distribution of actual genome sharing (k̆₁) for samples of ‘human’ genomes for different degrees of pedigree relationship of descendants of full sibs (as Fig. 2) (10 000 replicates each).

Fig. 6

Estimated ibd coefficients, k̂₀ and k̂₁, from SNP data for individuals with known pedigree relationship (PO denotes parent-offspring, DFC double first cousins, other symbols as Figs 2 and 3), together with predicted ‘error bars’ of two SD about expectation. Bars are offset from k₀ + k₁ = 1 if k₂ = 0.