Estimate the variance of realized relatedness around pedigree kinship

Pedigree-expected relatedness (this package's existing kinship output, doubled) is only an average – the actual proportion of genome shared identical-by-descent (IBD) between two relatives varies around that expectation because of Mendelian sampling and linkage (finite chromosome number/map length creates covariance in IBD status among nearby loci). This function estimates that variance for Parent-Offspring, Full-Siblings, and Half-Siblings pairs (issue #153, D3a) – the closed-form solution of Hill & Weir (2011), extending the pedigree kinship this package already computes rather than requiring a new population-genetics framework. It lists only those three kinds of pairs; every other pair (grandparent, cousin, avuncular, more distant, unrelated, self) is left out, and convertRelationships lists every pair.

Usage

markerRealizedRelatednessVariance(kmat, ped, nChr, mapLength, ids = NULL)

Arguments

kmat: square kinship matrix, as produced by kinship.
ped: dataframe with (at least) id, sire, and dam columns, as used by convertRelationships. Ids given as numbers or as a factor, here or in ids, are taken as their text.
nChr: chromosome count (e.g. autosome count for the species), meant to be a whole number. Only a single positive numeric value is enforced, so a non-integer such as 20.5 is accepted.
mapLength: numeric; total autosomal genetic map length in Morgans. Must be a single positive value.
ids: character vector of IDs or NULL to which the analysis should be restricted, as in convertRelationships.

Value

A dataframe with one row for each "Parent-Offspring", "Full-Siblings" or "Half-Siblings" pair and columns id1, id2, kinship, relation, R (pedigree relationship, 2 * kinship), varR (the realized-relatedness variance estimate), and sdR (its square root).

Details

Pairs are named by the rules of convertRelationships and listed in its order, but only an animal with each of its parents and two animals sharing a parent are looked at, so the table of every pair is never built. The variance combines nChr chromosomes, each approximated as the average length mapLength / nChr – Hill & Weir (2011) give an exact weighted-sum combination rule (their equation 5) only for lineal descendants; for Full-Sib/Half-Sib pairs they state in prose that this equal-length approximation closely matches a real heterogeneous-length genome, which this package's own PRE-RED research verified numerically against their published human-genome Table 2 (within ~2%; see the test file header for the full derivation).

References

Hill WG, Weir BS. 2011. Variation in actual relationship as a consequence of Mendelian sampling and linkage. Genetics Research 93(1):47-64.

Examples

library(nprcgenekeepr)
ped <- nprcgenekeepr::smallPed
kmat <- kinship(ped$id, ped$sire, ped$dam, ped$gen, sparse = FALSE)
## Rhesus macaque autosome count/map length are used only as an example
## scale -- callers should supply values appropriate to their own species.
markerRealizedRelatednessVariance(kmat, ped, nChr = 20L, mapLength = 28)
#>     id1 id2 kinship         relation    R         varR        sdR
#> 3     A   C   0.250 Parent-Offspring 0.50 0.0000000000 0.00000000
#> 4     A   D   0.250 Parent-Offspring 0.50 0.0000000000 0.00000000
#> 8     A   H   0.250 Parent-Offspring 0.50 0.0000000000 0.00000000
#> 9     A   I   0.250 Parent-Offspring 0.50 0.0000000000 0.00000000
#> 13    A   M   0.250 Parent-Offspring 0.50 0.0000000000 0.00000000
#> 17    A   Q   0.250 Parent-Offspring 0.50 0.0000000000 0.00000000
#> 20    B   C   0.250 Parent-Offspring 0.50 0.0000000000 0.00000000
#> 21    B   D   0.250 Parent-Offspring 0.50 0.0000000000 0.00000000
#> 25    B   H   0.250 Parent-Offspring 0.50 0.0000000000 0.00000000
#> 38    C   D   0.250    Full-Siblings 0.50 0.0018350199 0.04283713
#> 42    C   H   0.250    Full-Siblings 0.50 0.0018350199 0.04283713
#> 43    C   I   0.125    Half-Siblings 0.25 0.0009175099 0.03029043
#> 46    C   L   0.250 Parent-Offspring 0.50 0.0000000000 0.00000000
#> 47    C   M   0.125    Half-Siblings 0.25 0.0009175099 0.03029043
#> 57    D   F   0.250 Parent-Offspring 0.50 0.0000000000 0.00000000
#> 58    D   G   0.250 Parent-Offspring 0.50 0.0000000000 0.00000000
#> 59    D   H   0.250    Full-Siblings 0.50 0.0018350199 0.04283713
#> 60    D   I   0.125    Half-Siblings 0.25 0.0009175099 0.03029043
#> 64    D   M   0.125    Half-Siblings 0.25 0.0009175099 0.03029043
#> 74    E   F   0.250 Parent-Offspring 0.50 0.0000000000 0.00000000
#> 75    E   G   0.250 Parent-Offspring 0.50 0.0000000000 0.00000000
#> 92    F   G   0.250    Full-Siblings 0.50 0.0018350199 0.04283713
#> 128   H   I   0.125    Half-Siblings 0.25 0.0009175099 0.03029043
#> 132   H   M   0.125    Half-Siblings 0.25 0.0009175099 0.03029043
#> 146   I   J   0.250 Parent-Offspring 0.50 0.0000000000 0.00000000
#> 149   I   M   0.125    Half-Siblings 0.25 0.0009175099 0.03029043
#> 182   K   L   0.250 Parent-Offspring 0.50 0.0000000000 0.00000000
#> 218   M   N   0.000 Parent-Offspring 0.00 0.0000000000 0.00000000
#> 220   M   P   0.250 Parent-Offspring 0.50 0.0000000000 0.00000000
#> 254   O   P   0.250 Parent-Offspring 0.50 0.0000000000 0.00000000