# ribd Release 1.2.0

Pedigree-based Relatedness Coefficients

License
GPL-3.0

# ribd

## Overview

The goal of ribd is to compute various coefficients of relatedness and identity-by-descent (IBD) between pedigree members. It extends the pedtools package which provides useful utilities for pedigree construction and manipulation.

The main functions in ribd are the following:

• kinship(), kinshipX() : Kinship coefficients
• inbreeding(), inbreedingX() : Inbreeding coefficients
• kappaIBD(), kappaIBDX() : IBD coefficients ((\kappa_0, \kappa_1, \kappa_2)) (noninbred individuals only)
• condensedIdentity(), condensedIdentityX() : Jacquard’s condensed identity coefficients

A unique feature of ribd is the ability to handle pedigrees with inbred founders in all of the above calculations. More about this below!

The package also computes a variety of lesser-known pedigree coefficients:

• generalisedKinship() : Generalised kinship coefficients, as defined by Weeks and Lange (1988)
• multiPersonIBD() : Multi-person IBD coefficients (noninbred individuals only)
• twoLocusKinship() : Two-locus kinship coefficients, as defined by Thompson (1988)
• twoLocusIBD() : Two-locus IBD coefficients (noninbred pair of individuals)
• twoLocusIdentity() : Two-locus condensed identity coefficients (any pair of individuals)
• twoLocusGeneralisedKinship() : Generalised two-locus kinship coefficients (not exported)

## Installation

To get the current official version of ribd, install from CRAN as follows:

install.packages("ribd")

Alternatively, you can obtain the latest development version from GitHub:

# install.packages("devtools") # install devtools if needed
devtools::install_github("magnusdv/ribd")

## Getting started

library(ribd)

To illustrate the use of ribd we compute the condensed identity coefficients after one generation of full sib mating. This is a suitable example because the answer is well known, and it is one of the simplest in which all 9 coefficients are non-zero.

We create the pedigree with pedtools as follows:

x = fullSibMating(1)
plot(x)

The identity coefficients of the children are computed with condensedIdentity()

condensedIdentity(x, ids = 5:6)
#> [1] 0.06250 0.03125 0.12500 0.03125 0.12500 0.03125 0.21875 0.31250 0.06250

## Inbred founders

How would the above result would change if individual 1 was himself inbred, say, as a child of half siblings? A possible, but cumbersome, approach to answer this question would be to expand the pedigree to include the complete family history. Here is one way to do this, using pedtools::mergePed() to merge x with a suitably labelled half-sib pedigree:

y = halfSibPed(sex1 = 1, sex2 = 2)
y = addChildren(y, father = 4, mother = 5, nch = 1)
y = relabel(y, c(101:105, 1)) # prepare merge by relabeling
z = mergePed(y, x)
plot(z)

Now that we have the complete pedigree we could answer the original question by running condensedIdentity() on z.

condensedIdentity(z, ids = 5:6)
#> [1] 0.06640625 0.03515625 0.13281250 0.03125000 0.13281250 0.03125000 0.21875000
#> [8] 0.29687500 0.05468750

Although the above strategy worked nicely in this case, it quickly gets awkward or impossible to model founder inbreeding by creating the complete pedigree. For example, inbreeding coefficients close to zero require enormous pedigrees! And even worse: What if individual 1 was 100% inbred? This cannot be modelled in this way, as it calls for an infinite pedigree.

A much easier approach is to use the founderInbreeding() feature offered by pedtools: We simply specify the inbreeding level of individual 1 (in the original x) to be that of a child of half siblings, i.e. (1/8).

founderInbreeding(x, ids = 1) = 1/8

When we now run condensedIdentity() on x, this inbreeding is taken into account, giving the same answer as for z above.

condensedIdentity(x, ids = 5:6)
#> [1] 0.06640625 0.03515625 0.13281250 0.03125000 0.13281250 0.03125000 0.21875000
#> [8] 0.29687500 0.05468750

## The pairwise condensed identity states

The following figure shows the 9 condensed identity states of two individuals a and b. Each state shows the pattern of IBD between the 4 homologous alleles at an autosomal locus. The states are shown in the ordering used by Jacquard and most subsequent authors. This is also the order of the coefficients output by condensedIdentity().

### Identity states on X

The X chromosomal version of condensedIdentity() is called condensedIdentityX(). What this function computes requires some explanation, which we offer here.

The point of condensedIdentityX() is to compute the coefficients (i.e., the expected proportions) of the pairwise identity states for a locus on the X chromosome. What these states are depends on the sexes of the involved individuals: either female-female, female-male, male-female or male-male. In some sense the first case is the easiest: When both are female the states are the same as in the autosomal case.

Males, being hemizygous, have only 1 allele of a locus on X. Hence when males are involved the total number of alleles is less than 4, rendering the autosomal states pictured above meaningless. However, to avoid drawing (and learning the ordering of) new states for each sex combination, we can re-use the autosomal state pictograms by invoking the following simple rule: Replace the single allele of any male, with a pair of autozygous alleles. This gives a one-to-one map from the X states to the autosomal states.

For simplicity the output always contains 9 coefficients, but with NA’s in the positions of undefined states (depending on the sex combination). Hopefully this should all be clear from the following table: