Pedigree-Based Covariance for Several Relatives

From Pairwise Kinship to a Family-Wide Covariance Matrix

A pedigree can tell us how genetically related two people are, but analyses often need to account for several relatives at once. The natural summary is a matrix: each row and column represents one individual, and each entry records the additive genetic covariance for that pair. Constructing it carefully requires keeping track of two linked quantities: the kinship coefficient and the additive relationship.

This tutorial works through a small hypothetical family. All four founders are assumed to be unrelated to one another and noninbred, and there are no pedigree links outside the family shown. These assumptions make the arithmetic transparent; when founders are related or descendants are inbred, the same matrix principles apply, but the relevant kinship and diagonal values change.

For an individual \(i\), let \(F_i\) denote the inbreeding coefficient, which is the probability that the individual's two alleles at a locus are identical by descent. The kinship of an individual with themself is \(\phi_{ii}=(1+F_i)/2\). Thus the diagonal relationship is \(A_{ii}=1+F_i\), and the diagonal additive genetic covariance is \((1+F_i)\sigma_A^2\). An outbred individual has \(F_i=0\), so their diagonal relationship is 1.

A Small Pedigree and the Recursion

Let P and Q be unrelated founders who have two children, A and B. A has a child, C, with founder R; B has a child, D, with founder S. P, Q, R, and S are mutually unrelated and noninbred. A and B are full siblings; C and D are first cousins. Every individual in the pedigree is outbred.

For a nonfounder \(i\) with parents \(s_i\) and \(d_i\), kinship with a different individual \(j\) can be calculated by averaging the kinships of the two parents with \(j\): \(\phi_{ij}=\{\phi_{s_i j}+\phi_{d_i j}\}/2\). The parents transmit one allele each, so a randomly sampled allele from \(i\) comes from either parent with probability one-half. This recursion is applied in pedigree order, beginning with founders and previously calculated relatives.

The founders' unrelatedness gives \(\phi_{PQ}=\phi_{PR}=\phi_{PS}=\phi_{QR}=\phi_{QS}=\phi_{RS}=0\), and each founder has self-kinship \(1/2\). Because A and B share parents P and Q, their kinship is \(1/4\). They are noninbred, so \(\phi_{AA}=\phi_{BB}=1/2\).

Assembling the Relationship Matrix

Choose and state an individual order before filling a matrix. Here the order is A, B, C, D, used for both rows and columns. Each diagonal entry is \(1+F_i=1\), because all four individuals are noninbred. Each off-diagonal entry is twice the corresponding kinship coefficient. For example, \(A_{AC}=2\phi_{AC}\), and \(A_{CD}=2\phi_{CD}\).

Scaling to Obtain Additive Genetic Covariance

The relationship matrix is dimensionless. To obtain additive genetic covariances in the units of genetic value squared, multiply every entry by the reference variance \(\sigma_A^2\). This relies on the standard additive pedigree model: alleles are transmitted according to Mendelian inheritance, the pedigree correctly captures the relevant identity by descent, and the covariance attributed to relatedness is the additive genetic covariance. The formula does not, by itself, include shared environmental effects or other sources of covariance.

In matrix notation, if \(\mathbf{G}=(G_A,G_B,G_C,G_D)^\mathsf{T}\), then:

In particular, \(\operatorname{Var}(G_i)=(1+F_i)\sigma_A^2\). For two different individuals, \(\operatorname{Cov}(G_i,G_j)=2\phi_{ij}\sigma_A^2\). A positive covariance indicates that the additive genetic values tend to vary in the same direction under this model; it does not imply that one individual's genes cause the other's outcome.

Using the Matrix for a Combined Genetic Value

A matrix is useful because it supports calculations involving more than one relative. For a vector of weights \(\mathbf{w}\), the variance of the weighted sum \(\mathbf{w}^\mathsf{T}\mathbf{G}\) is \(\mathbf{w}^\mathsf{T}\operatorname{Cov}(\mathbf{G})\mathbf{w}\). The weights might represent an average, a difference, or another specified combination of genetic values. This formula uses both diagonal variances and off-diagonal covariances; ignoring the latter generally gives the wrong variance for related individuals.

Why the Covariance Matrix Matters

Many statistical models for related individuals use a covariance matrix to represent dependence due to shared ancestry. A separate pairwise calculation can be enough for one comparison, but a matrix gives a consistent account of all selected individuals at once. It also makes it possible to calculate uncertainty for weighted combinations and to specify the genetic component of a model without treating relatives as genetically independent.

The matrix entries describe additive genetic covariance, not necessarily the total covariance of measured traits. If relatives also share an environment, their observed traits may covary for that reason as well. That additional source must be represented separately in a model; it should not be silently added to the pedigree-based genetic covariance.

Common Mistakes

• Using kinship as if it were the additive relationship. The cousin kinship here is \(1/16\), but their additive relationship is \(2\phi_{CD}=1/8\). The covariance formula uses the doubled value.
• Putting the kinship coefficient directly on the diagonal. A noninbred individual's self-kinship is \(1/2\); their diagonal additive relationship is \(2\phi_{ii}=1\), not \(1/2\).
• Changing row order without changing column order. Every row and column must refer to the same declared individual order, or the entries will describe the wrong pairs.
• Ignoring covariance when combining relatives. The variance of a sum includes twice the pairwise covariance. Treating related individuals as independent drops that term.
• Calling the genetic covariance the total trait covariance. Pedigree-based additive covariance does not automatically include shared environment or other causes of dependence.