realisedIbdVariance {ribd}

Variance of realised relatedness coefficients

Description

Compute the variance of realised relatedness coefficients, by doubly integrating the corresponding two-locus coefficients.

Usage

realisedIbdVariance(x, ids = leaves(x), coeff, L = 1)

Arguments

x

A ped object.

ids

A vector naming two members of x.

coeff

A string naming a coefficient for which the variance is to be computed. See Details for legal values.

L

A positive number; the chromosome length in Morgan.

Details

The double integral method was used by Guo to compute the variation in proportion of the genome shared IBD (Guo 1995, see also Thompson 2013). The method extends directly to other coefficients. The implementation here supports Cotterman's kappa coefficients (of noninbred individuals), and Jacquard's condensed identity coefficients.

This function is a bare-bones implementation of the double integral method, based on stats::integrate, and can probably be optimised in various ways.

The coeff parameter must be either a character naming the coefficient to compute, or a function. If a character, it must be one of the following names:

• "inb" (inbreeding coefficient)

• "kinship", "phi" (synonyms for the kinship coefficient)

• "k0", "k1", "k2" (kappa coefficients of noninbred individuals)

• "D1", "D2", ... "D9" (condensed identity coefficients)

Value

A positive number.

References

• Guo (1995) Proportion of genome shared identical by descent by relatives: concept, computation, and applications. Am J Hum Genet.

• Hill & Weir (2011). Variation in actual relationship as a consequence of Mendelian sampling and linkage. Genet Res.

• Thompson (2013). Identity by Descent: Variation in Meiosis, Across Genomes, and in Populations. Genetics.

Examples

###################################
### Box 1 of Hill & Weir (2011) ###
###################################

# Eq. 4b of Hill & Weir
phi = function(n, l) {
 1/(2*l^2) * (1/4)^n * sum(sapply(1:n, function(r)
   choose(n, r) * (2*r*l - 1 + exp(-2*r*l))/r^2))
}

# Chromosome of 1 Morgan
L = 1

### Full sibs ###

## Not run: 
x = nuclearPed(2)
realisedIbdVariance(x, ids = 3:4, coeff = "k2", L = L)

# Hill & Weir (Box 1)
16*phi(4,L) - 16*phi(3,L) + 8*phi(2,L) - 2*phi(1,L)

## End(Not run)

### Double first cousins ###

## Not run: 
dfc = doubleFirstCousins()

# Runtime ~1 min
realisedIbdVariance(dfc, coeff = "k0", L = L)
realisedIbdVariance(dfc, coeff = "k1", L = L)
realisedIbdVariance(dfc, coeff = "k2", L = L)

# Hill & Weir, Box 1
var_k2 = 64*phi(8,L) - 64*phi(7,L) + 40*phi(6,L) - 20*phi(5,L) +
  33/4*phi(4,L) - 5/2*phi(3,L) + 5/8*phi(2,L)-1/8*phi(1,L)
var_k1 = 4*var_k2
var_k0 = var_k2 + 2 * (4*phi(4,L) - 2*phi(3,L) + 3/4*phi(2,L) - 1/4*phi(1,L))

var_k0
var_k1
var_k2

## End(Not run)