kappaIBD {ribd}

IBD (kappa) coefficients

Description

Computes the three IBD coefficients summarising the relationship between two non-inbred individuals. Both autosomal and X chromosomal versions are implemented. The pedigree founders (other than the individuals in question) are allowed to be inbred; see pedtools::founderInbreeding() for how to set this up, and see Examples below.

Usage

kappaIBD(x, ids = labels(x), inbredAction = 1, simplify = TRUE, Xchrom = FALSE)

Arguments

x

A pedigree in the form of a ped object (or a list of such).

ids

A character (or coercible to character) containing ID labels of two or more pedigree members.

inbredAction

An integer telling the program what to do if either of the ids individuals are inbred. Possible values are: 0 = do nothing; 1 = print a warning message (default); 2 = raise an error. In the first two cases the coefficients are reported as NA.

simplify

Simplify the output (to a numeric of length 3) if ids has length 2. Default: TRUE.

Xchrom

A logical, indicating if the autosomal (default) or X-chromosomal kappa coefficients should be computed.

Details

For non-inbred individuals a and b, their autosomal IBD coefficients (\kappa_0, \kappa_1, \kappa_2) are defined as follows:

\kappa_i = P(\text{a and b share exactly i alleles IBD at a random autosomal locus})

The autosomal kappa coefficients are computed from the kinship coefficients. When a and b are both nonfounders, the following formulas hold:

• \kappa_2 = \varphi_{MM} \cdot \varphi_{FF} + \varphi_{MF} \cdot\varphi_{FM}

• \kappa_1 = 4 \varphi_{ab} - 2 \kappa_2

• \kappa_0 = 1 - \kappa_1 - \kappa_2

Here \varphi_{MF} denotes the kinship coefficient between the mother of a and the father of b, etc. If either a or b is a founder, then \kappa_2 = 0, while the other two formulas remain as before.

The X-chromosomal IBD coefficients are defined similarly to the autosomal case. Here \kappa_2 is undefined when one or both individuals are male, which greatly simplifies the calculations when males are involved. The formulas are (with \varphi_{ab} now referring to the X-chromosomal kinship coefficient):

• Both male: (\kappa_0, \kappa_1, \kappa_2) = (1-\varphi_{ab}, \varphi_{ab}, \text{NA})

• One male, one female: (\kappa_0, \kappa_1, \kappa_2) = (1-2 \varphi_{ab}, 2 \varphi_{ab}, \text{NA})

• Two females: Similar formulas as in the autosomal case.

Value

If ids has length 2 and simplify = TRUE: A numeric vector of length 3: (\kappa_0, \kappa_1, \kappa_2).

Otherwise: A data frame with one row for each pair of individuals, and 5 columns. The first two columns contain the ID labels, and columns 3-5 contain the IBD coefficients.

Kappa coefficients of inbred individuals (meaning X-inbred females if Xchrom = T) are reported as NA, unless inbredAction = 2.

The X-chromosomal \kappa_2 is NA whenever at least one of the two individuals is male.

See Also

kinship(), identityCoefs() for coefficients allowing inbreeding, showInTriangle() for plotting kappa coefficients in the IBD triangle.

Examples

### Siblings
x = nuclearPed(2)
kappaIBD(x)

k = kappaIBD(x, 3:4)
stopifnot(identical(k, c(.25, .5, .25)))

### Quad half first cousins
x = quadHalfFirstCousins()
k = kappaIBD(x, ids = leaves(x))
stopifnot(identical(k, c(17/32, 14/32, 1/32)))

### Paternal half brothers with 100% inbred father
# Genetically indistinguishable from an (outbred) father-son relationship
x = halfSibPed() |> setFounderInbreeding(ids = 2, value = 1)
plot(x, hatched = 4:5)

k = kappaIBD(x, 4:5)
stopifnot(identical(k, c(0, 1, 0)))

### X-chromosomal kappa
y = nuclearPed(2, sex = 2)
kappaIBD(y, Xchrom = TRUE)