Building Additive Relationship Matrices From Pedigrees

Why Build an Additive Relationship Matrix?

A pedigree describes who is related to whom through recorded parentage. A matrix makes those relationships available in a consistent form: each row and column corresponds to one individual, and each entry summarizes the additive relationship between a pair. This is useful when a calculation needs relationships for several individuals at once rather than a single kinship coefficient.

The main care is to distinguish off-diagonal entries, which describe relationships between two different individuals, from diagonal entries, which describe an individual's relationship with itself and therefore depend on that individual's inbreeding. Confusing these entries can produce a matrix that looks plausible but encodes the wrong quantities.

Definition: The additive relationship matrix, denoted \(A\), has one row and one column for each individual in a specified order. For two distinct individuals \(i\) and \(j\), its entry is \(A_{ij}=2\phi_{ij}\), where \(\phi_{ij}\) is their kinship coefficient. Its diagonal entry is \(A_{ii}=1+F_i\), where \(F_i\) is individual \(i\)'s inbreeding coefficient. Thus, off-diagonal entries are twice pairwise kinship, while diagonal entries are not obtained by doubling an individual's kinship with itself.

The relationship coefficient \(A_{ij}\) is dimensionless. For example, \(A_{ij}=0.5\) means the pair's additive relationship is one half under this convention; it does not mean that the individuals share exactly one half of their observed alleles at every genetic location. The matrix is symmetric, so \(A_{ij}=A_{ji}\).

Pedigree Assumptions and Recursive Rules

A pedigree calculation needs boundary assumptions for individuals whose parents are not represented. Unless the pedigree specifies otherwise, treat founders as unrelated and non-inbred. Their diagonal entries are 1, and the off-diagonal entries between distinct founders are 0. If founders are known to be related or inbred, these base entries must instead reflect that information.

Arrange individuals so that both parents of each nonfounder appear earlier in the ordering. The recursion then calculates each new individual's relationships using entries already available for the parents and earlier individuals. A parent not recorded in the pedigree is treated as an unrelated, non-inbred founder under the stated boundary convention.

Recursive rules: If individual \(i\) has parents \(s\) and \(d\), then for a previously ordered individual \(j\ne i\), \(A_{ij}=(A_{sj}+A_{dj})/2\). The diagonal entry is \(A_{ii}=1+A_{sd}/2\). The latter rule follows because \(F_i=\phi_{sd}=A_{sd}/2\). For a founder under the unrelated, non-inbred assumption, \(A_{ii}=1\), and distinct founders have \(A_{ij}=0\).

The off-diagonal rule averages the two parental relationships with \(j\): an offspring receives one allele from each parent, so its expected kinship with \(j\) averages the parental kinships. The diagonal rule answers a different question. It uses the relationship between the individual's own parents to determine the individual's inbreeding.

Why the Recursion Works

The kinship coefficient \(\phi_{ij}\) is the probability that a randomly selected allele from individual \(i\) and a randomly selected allele from individual \(j\) are identical by descent. If \(i\) is the offspring of \(s\) and \(d\), its sampled allele came from either parent with probability one half. Therefore, for \(j\) distinct from \(i\),

\phi_{ij}=\frac{\phi_{sj}+\phi_{dj}}{2}.

Multiplying both sides by 2 and using \(A_{ij}=2\phi_{ij}\) gives the matrix recursion:

A_{ij} =2\phi_{ij} =2\left(\frac{\phi_{sj}+\phi_{dj}}{2}\right) =\frac{2\phi_{sj}+2\phi_{dj}}{2} =\frac{A_{sj}+A_{dj}}{2}.

For the diagonal, the standard self-kinship identity is \(\phi_{ii}=(1+F_i)/2\), while the relationship matrix is defined to have \(A_{ii}=1+F_i\). For an offspring, its inbreeding coefficient is the kinship of its parents, \(F_i=\phi_{sd}\). Since \(A_{sd}=2\phi_{sd}\), substitution gives \(A_{ii}=1+F_i=1+\phi_{sd}=1+A_{sd}/2\). This explains why the diagonal uses half the parents' matrix entry, not their full entry.

Constructing the Matrix Step by Step

A reliable calculation proceeds in pedigree order. First set the founder entries under the chosen boundary assumptions. Then calculate the relationships of each new individual with all earlier individuals. Finally calculate that individual's diagonal from the relationship between its parents. Fill the symmetric position at the same time: once \(A_{ij}\) is known, set \(A_{ji}=A_{ij}\).

Set the base.
Assign founder diagonals and founder-to-founder entries using the pedigree boundary assumptions.

Add an offspring's off-diagonal entries.
For each earlier individual \(j\), average the entries between \(j\) and the offspring's two parents.

Add the offspring's diagonal.
Use \(1+A_{sd}/2\), where \(s\) and \(d\) are its parents.

Check symmetry and interpretation.
Confirm that \(A_{ij}=A_{ji}\), and that the diagonal reflects inbreeding rather than a pairwise relationship between distinct individuals.

Worked Example: Two Unrelated Founders and Their Child

Suppose a hypothetical pedigree has two founders, \(P\) and \(Q\), and their child \(C\). The founders are assumed unrelated and non-inbred. Use the order \(P,Q,C\) to build \(A\).

The founder entries are \(A_{PP}=A_{QQ}=1\) and \(A_{PQ}=A_{QP}=0\). For \(C\) and \(P\), the recursion gives \(A_{CP}=(A_{PP}+A_{QP})/2=(1+0)/2=0.5\). For \(C\) and \(Q\), it gives \(A_{CQ}=(A_{PQ}+A_{QQ})/2=(0+1)/2=0.5\).

The parents are unrelated, so \(A_{PQ}=0\). Thus \(A_{CC}=1+A_{PQ}/2=1+0/2=1\). Filling the symmetric entries gives

A= \begin{pmatrix} 1 & 0 & 0.5\\ 0 & 1 & 0.5\\ 0.5 & 0.5 & 1 \end{pmatrix}.

The parent–offspring entries are 0.5, while the child's diagonal is 1 because the child's parents are unrelated and the child is not inbred under this pedigree.

Worked Example: Two Full Siblings

Suppose unrelated, non-inbred founders \(A\) and \(B\) have two children, \(C\) and \(D\). Use the order \(A,B,C,D\). The founders have \(A_{AA}=A_{BB}=1\) and \(A_{AB}=0\).

Each child has relationship 0.5 with each parent. For instance, \(A_{CA}=(A_{AA}+A_{BA})/2=(1+0)/2=0.5\), and \(A_{CB}=(A_{AB}+A_{BB})/2=(0+1)/2=0.5\). The same calculations apply to \(D\).

For the siblings, use \(C\)'s parents to calculate its relationship with \(D\): \(A_{CD}=(A_{AD}+A_{BD})/2=(0.5+0.5)/2=0.5\). Each child's parents are unrelated, so \(A_{CC}=A_{DD}=1+0/2=1\). The resulting matrix is

A= \begin{pmatrix} 1 & 0 & 0.5 & 0.5\\ 0 & 1 & 0.5 & 0.5\\ 0.5 & 0.5 & 1 & 0.5\\ 0.5 & 0.5 & 0.5 & 1 \end{pmatrix}.

The entry \(A_{CD}=0.5\) is an off-diagonal relationship between two different individuals. The entries \(A_{CC}=A_{DD}=1\) are diagonals, which would be larger than 1 if either sibling were inbred.

Worked Example: A Child of Related Parents

Consider a hypothetical pedigree in which founders \(U\) and \(V\) have children \(C\) and \(D\). Each of \(C\) and \(D\) has a child with a different unrelated, non-inbred founder: \(X\) is the child of \(C\) and \(E\), and \(Y\) is the child of \(D\) and \(F\). Here \(E\) and \(F\) are unrelated to the other founders and to each other. Finally, \(X\) and \(Y\) have a child \(Z\). We will calculate the entries for \(X,Y,Z\), retaining the pedigree information needed to obtain their relationships.

First, \(C\) and \(D\) are full siblings, so \(A_{CD}=0.5\). For \(C\) and \(Y\), the parents of \(Y\) are \(D\) and \(F\), giving \(A_{CY}=(A_{CD}+A_{CF})/2=(0.5+0)/2=0.25\). Also, \(A_{EY}=(A_{ED}+A_{EF})/2=(0+0)/2=0\), because \(E\) is unrelated to \(D\) and \(F\). Since \(X\) has parents \(C\) and \(E\), \(A_{XY}=(A_{CY}+A_{EY})/2=(0.25+0)/2=0.125\).

The parents of \(Z\) are \(X\) and \(Y\), so \(A_{ZZ}=1+A_{XY}/2=1+0.125/2=1.0625\). Equivalently, \(F_Z=A_{XY}/2=0.0625\). The relationships between \(Z\) and each parent are \(A_{ZX}=(A_{XX}+A_{YX})/2=(1+0.125)/2=0.5625\) and \(A_{ZY}=(A_{XY}+A_{YY})/2=(0.125+1)/2=0.5625\). Neither \(X\) nor \(Y\) is inbred, so \(A_{XX}=A_{YY}=1\). The submatrix for \(X,Y,Z\) is therefore

A_{\{X,Y,Z\}}= \begin{pmatrix} 1 & 0.125 & 0.5625\\ 0.125 & 1 & 0.5625\\ 0.5625 & 0.5625 & 1.0625 \end{pmatrix}.

The diagonal above 1 records \(Z\)'s inbreeding. The off-diagonal value \(A_{XY}=0.125\) records the relationship between \(Z\)'s parents; it is that value, halved, that determines \(Z\)'s inbreeding coefficient.

Common Mistake: Treating Every Entry the Same Way

A tempting but incorrect approach is to use \(2\phi_{ij}\) for every matrix entry, including the diagonal, or to assign every diagonal entry the value 1. Both shortcuts ignore the distinct definition of a diagonal entry. For a non-inbred individual, \(A_{ii}=1\); for an inbred individual, \(A_{ii}=1+F_i\), which is greater than 1 when \(F_i>0\).

For example, in the preceding pedigree, using \(1\) for \(A_{ZZ}\) would discard the fact that \(Z\)'s parents are related. The correct calculation is \(A_{ZZ}=1+A_{XY}/2=1.0625\). Conversely, using \(A_{XY}\) itself as the inbreeding coefficient would give 0.125, twice the correct value \(F_Z=0.0625\). Keep the two operations separate: off-diagonal entries are twice kinship, while an offspring's diagonal is 1 plus parental kinship.

A second practical error is to begin the recursion before establishing the founder assumptions. The zero relationship between two founders is not a universal fact; it is a modeling boundary assumption. If known pedigree information makes founders related, that information must be used rather than silently replaced with zeros.

Key takeaway: For distinct individuals, use \(A_{ij}=2\phi_{ij}\), and for an individual's diagonal use \(A_{ii}=1+F_i\). In a pedigree recursion, calculate a child's off-diagonal entries by averaging its parents' entries and its diagonal by adding half the parents' matrix entry to 1.

Check Your Understanding

Use the definitions and recursive rules above to answer these questions.

Under the unrelated, non-inbred founder assumption, what are the diagonal and off-diagonal entries for two founders?
Two distinct individuals have kinship coefficient \(0.125\). What is their off-diagonal additive relationship entry?
An offspring's parents have matrix entry \(A_{sd}=0.25\). What are the offspring's inbreeding coefficient and diagonal entry?
Why is the diagonal rule \(A_{ii}=1+A_{sd}/2\), rather than \(1+A_{sd}\)?
When would it be inappropriate to set the relationship between two founders to zero?