Worked Pedigree From Diagram to Kinship and Covariance

From Pedigree to Covariance

A pedigree is more than a family diagram: under stated assumptions, it gives the expected sharing of alleles identical by descent between individuals. Those expectations can be organized as kinship coefficients, converted to additive relationship coefficients, and then scaled by an additive genetic variance to obtain genetic covariances. A reliable calculation therefore begins with the pedigree’s assumptions and proceeds in a fixed order.

The worked pedigree below includes full siblings whose descendants mate, creating an inbred individual, and then a parent–offspring mating that creates a further inbred descendant. It is hypothetical. The calculations concern expected genetic relationships under the pedigree model, not observed genotypes at a particular variant.

The pedigree calculations assume autosomal inheritance, correct parentage, and Mendelian transmission: each offspring receives one allele at a locus from each parent, with either parental allele having probability one-half. Unless specified otherwise, founders are assumed unrelated to one another and noninbred. Here, “unrelated” means no identity by descent within the pedigree being modeled; it is a modeling assumption, not a claim that the founders could not share any distant ancestors in reality. The additive covariance calculations additionally assume a common additive genetic variance, denoted \(\sigma_A^2\), and include no shared environmental covariance.

The Hypothetical Pedigree

There are five founders: \(A\), \(B\), \(E\), \(G\), and \(K\). They are assumed mutually unrelated and noninbred. The other individuals and their parents are as follows.

In generation order, \(A\) and \(B\) have children \(C\) and \(D\). Individual \(C\) has a child \(F\) with founder \(E\), while \(D\) has a child \(H\) with founder \(G\). The first cousins \(F\) and \(H\) have child \(I\). Individual \(I\) has child \(L\) with founder \(K\), and then \(I\) and \(L\) have child \(M\). This ordering makes the assumptions visible and ensures each nonfounder’s parents are identified before the individual’s coefficients are calculated.

The off-diagonal recursion follows because a randomly sampled allele from \(i\) came from either parent with probability one-half. Conditional on its parental source, the chance of identity by descent with an allele sampled from \(j\) is the corresponding parental kinship. For the diagonal, two alleles sampled from one individual are the same homolog with probability one-half; otherwise they are the individual’s two distinct alleles, which are identical by descent with probability \(F_i\). Thus \(\phi_{ii}=(1+F_i)/2\). These rules also make clear why an inbreeding coefficient is not itself a kinship coefficient between two different people.

Worked Example: Kinship of the First Cousins

Inbreeding and Kinship Along the Pedigree

The inbreeding coefficient of a nonfounder is the kinship of that individual’s parents. Since \(I\)’s parents are \(F\) and \(H\), the cousin calculation gives \(F_I=\phi_{FH}=1/16\). By contrast, \(F\) and \(H\) themselves have \(F_F=F_H=0\) under the assumed unrelated-parent pairings. Their child \(I\) is inbred because the parents are related; neither parent needs to be inbred for their child to be inbred.

The same recursion can fill selected entries of an additive relationship matrix. For example, the relationship between \(I\) and \(F\) can be found from \(I\)’s parents \(F\) and \(H\). Since \(F\) is noninbred, \(\phi_{FF}=1/2\), and the first worked example gives \(\phi_{HF}=1/16\). Therefore,

For \(M\), its parents are \(I\) and \(L\). We have \(\phi_{II}=17/32\) and \(\phi_{LI}=17/64\), so \(\phi_{MI}=(17/32+17/64)/2=51/128\), and \(A_{MI}=51/64\). The relationship is larger than the ordinary parent–offspring value for an unrelated mating because \(M\)’s other parent, \(L\), is also related to \(I\).

Worked Example: From Relationships to Genetic Covariances

A Compact Matrix Check

A small matrix is a useful way to check that the pedigree calculations use one convention consistently. For individuals \(I,L,M\), the diagonal entries are \(1+F_i\), and the off-diagonal entries are twice the corresponding kinship coefficients. To obtain the remaining entry, use \(M\)’s parents \(I,L\): \(\phi_{ML}=(\phi_{IL}+\phi_{LL})/2=(17/64+1/2)/2=49/128\), since \(L\) is noninbred. Thus \(A_{LM}=49/64\).

The matrix is symmetric, as a covariance or relationship matrix must be. Its diagonal is not all one: the inbred individuals \(I\) and \(M\) have diagonal entries greater than one. Multiplying the whole matrix by \(\sigma_A^2=4\) gives the additive genetic covariance matrix in trait-units squared. This matrix check is not a substitute for tracing parentage carefully, but it can reveal a misplaced entry or a kinship-versus-relationship factor-of-two error.

Common Mistakes and Interpretation

• Leaving the founder assumption implicit. A founder is not automatically unrelated to every other founder. State the assumption; if founders are related, their kinship entries must reflect that information.
• Using kinship where a relationship coefficient is required. For distinct individuals, \(A_{ij}=2\phi_{ij}\). Multiplying covariance by \(\phi_{ij}\) alone understates it by a factor of two under the stated additive model.
• Setting every diagonal relationship to one. The diagonal is \(1+F_i\), so an inbred individual has a larger additive genetic variance than a noninbred individual when the same \(\sigma_A^2\) is used.
• Confusing an individual’s inbreeding with the parents’ relationship scale. The offspring’s \(F\) equals the parents’ kinship \(\phi\), not twice that value. Twice the kinship is the additive relationship between distinct parents.
• Reading a genetic covariance as a total trait covariance. The calculation \(\operatorname{Cov}(g_i,g_j)=A_{ij}\sigma_A^2\) is for additive genetic values. A total phenotypic covariance may also involve environmental or other modeled components.