Building Pedigree Covariance Matrices

Why a Pedigree Covariance Matrix Has Several Parts

For family members, a pair of similar trait values may reflect inherited effects, a genetic effect linked to a particular locus, or an environment they share. Individuals may also differ for reasons that are not shared with relatives. A pedigree covariance matrix keeps these sources distinct while expressing their combined implication: how variable each person’s trait is and how much two people’s traits covary.

The matrix is built in a specified order of individuals. Each diagonal entry is the variance of one person’s trait; each off-diagonal entry is the covariance between a pair. The entries depend on both the assumed variance components and matrices that describe genetic relatedness, identity-by-descent (IBD) sharing at the locus of interest, and shared environment.

Definition: A pedigree trait covariance matrix is the covariance matrix $V$ of the trait values for the individuals in a pedigree, under a specified model. It combines the covariance contributions of the model’s random components. Its rows and columns must use the same individual ordering.

Four Covariance Components

Let $y_i$ be the trait value for individual $i$, and let $y$ be the vector of trait values in the chosen order. A convenient model separates the trait into additive polygenic, locus-specific, shared-environment, and individual residual components. Any fixed mean terms are not part of the covariance matrix.

Model: Write $y=X\beta+a+q+c+e$. Here $X\beta$ represents fixed mean terms; $a$ is the additive polygenic effect; $q$ is the effect associated with the locus being considered; $c$ is the shared-environment effect; and $e$ is the individual residual. Assume these four random components have mean zero and are mutually uncorrelated. Assume the residuals for different individuals are uncorrelated.

The additive relationship matrix $A$ describes pedigree relatedness for additive genetic effects. Let $\phi_{ij}$ be the kinship coefficient: the probability that a randomly selected allele from individual $i$ and a randomly selected allele at the same autosomal locus from individual $j$ are identical by descent. Define $A_{ij}=2\phi_{ij}$. Thus, for non-inbred individuals, $A_{ii}=1$; a parent and child, or a pair of full siblings, have $A_{ij}=0.5$. The additive covariance is $\sigma_A^2 A$, where $\sigma_A^2$ is the additive variance component.

The locus-specific matrix $\Pi$ describes IBD sharing at the locus under study. Its entry $\pi_{ij}$ is the expected proportion of the two alleles at that locus that individual $i$ and individual $j$ share IBD. Set $\pi_{ii}=1$. For a pair of full siblings, $\pi_{ij}$ is often an estimated or conditional expected sharing value, not necessarily the pedigree-wide expectation. The locus-specific covariance is $\sigma_Q^2\Pi$, where $\sigma_Q^2$ is the variance component for the locus effect.

The matrix $C$ encodes which individuals share the relevant environment and to what degree. In a simple household model, $C_{ij}=1$ when $i$ and $j$ share a household, including $C_{ii}=1$, and $C_{ij}=0$ otherwise. Its covariance contribution is $\sigma_C^2 C$. This is a model choice: a different environmental structure requires a correspondingly different, valid covariance matrix.

Finally, $I$ is the identity matrix, and the independent residual contribution is $\sigma_E^2 I$. This puts residual variance $\sigma_E^2$ on each diagonal and zero residual covariance between distinct individuals.

Key result: Under the stated model, $$ V=\sigma_A^2 A+\sigma_Q^2\Pi+\sigma_C^2 C+\sigma_E^2 I. $$ Each variance component is nonnegative. The matrices $A$, $\Pi$, and $C$ must represent valid covariance structures; in particular, they should be positive semidefinite. The identity matrix $I$ has the same number of rows and columns as the other component matrices.

Building the Matrix in a Consistent Order

The formula is straightforward, but its entries are only meaningful if every component matrix refers to the same people in the same order. Constructing each matrix separately and then adding them without checking the ordering is a serious error: the resulting entries can describe the wrong pairs.

Fix the individual order.
Write down the exact sequence of phenotyped individuals. Use that sequence for every row and column of every matrix.

Build the pedigree component.
Use pedigree kinship coefficients to set $A_{ij}=2\phi_{ij}$, accounting for inbreeding where relevant.

Build the locus and environment components.
Use locus-specific IBD sharing values for $\Pi$, and the stated environmental-sharing rule for $C$. Confirm that the diagonal convention is compatible with each variance component.

Scale and add.
Multiply each component matrix by its variance component, add $\sigma_E^2$ to each diagonal through $\sigma_E^2I$, and sum the matrices entry by entry.

Check the result.
Verify symmetry, dimensions, diagonal variances, and the interpretation of representative off-diagonal covariances.

Worked Examples

Worked Example: Converting Kinship to Additive Relationships

Suppose three non-inbred individuals are ordered as a parent, that parent’s child, and an unrelated individual. The kinship coefficient for the parent and child is $1/4$. The unrelated individual has kinship coefficient zero with each of them. Construct $A$.

For each diagonal, non-inbreeding gives $\phi_{ii}=1/2$, so $A_{ii}=2(1/2)=1$. For the parent and child, $A_{12}=A_{21}=2(1/4)=0.5$. The unrelated pairs have $A_{13}=A_{31}=A_{23}=A_{32}=0$. Therefore, $$ A= \begin{pmatrix} 1 & 0.5 & 0\\ 0.5 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}. $$

The entry 0.5 is an additive relationship coefficient, not a trait covariance by itself. For example, if $\sigma_A^2=4$ trait-units squared, the additive covariance between parent and child is $4(0.5)=2$ trait-units squared. Their total trait covariance could be different after the other components are added.

Worked Example: Interpreting Sibling IBD Sharing

For two full siblings, suppose the probabilities of sharing zero, one, or two alleles IBD at a locus are $1/4$, $1/2$, and $1/4$, respectively. Calculate the expected proportion of alleles shared IBD under this distribution.

The proportions shared in the three states are 0, $1/2$, and 1. Thus the expected sharing is $$ \pi_{12}=(1/4)(0)+(1/2)(1/2)+(1/4)(1)=0+0.25+0.25=0.50. $$

For this pair, the locus-specific matrix has diagonal entries 1 and off-diagonal entries 0.5: $$ \Pi= \begin{pmatrix} 1 & 0.5\\ 0.5 & 1 \end{pmatrix}. $$ This value is a pedigree-wide expectation under the stated probabilities. In an analysis, locus-specific IBD information can give a different expected sharing value for a particular pair. The matrix $\Pi$ represents sharing at the locus, whereas $A$ represents the pedigree’s additive relationship structure.

Worked Example: Combining Components for Two Siblings

Consider two full siblings in the same household, ordered as individual 1 then individual 2. Suppose $A_{12}=0.5$, the locus-specific expected sharing is $\pi_{12}=0.25$, and both siblings share the modeled household environment. Let the hypothetical variance components be $\sigma_A^2=4$, $\sigma_Q^2=2$, $\sigma_C^2=1$, and $\sigma_E^2=3$, all in trait-units squared.

The component matrices are $$ A=\begin{pmatrix}1&0.5\\0.5&1\end{pmatrix},\quad \Pi=\begin{pmatrix}1&0.25\\0.25&1\end{pmatrix},\quad C=\begin{pmatrix}1&1\\1&1\end{pmatrix},\quad I=\begin{pmatrix}1&0\\0&1\end{pmatrix}. $$

The diagonal variance for either sibling is $$ 4(1)+2(1)+1(1)+3(1)=10 $$ trait-units squared. The covariance between the siblings is $$ 4(0.5)+2(0.25)+1(1)+3(0)=2+0.5+1+0=3.5 $$ trait-units squared. Therefore, $$ V= \begin{pmatrix} 10&3.5\\ 3.5&10 \end{pmatrix}. $$

The residual variance increases each person’s diagonal variance but contributes nothing to the off-diagonal covariance under the independent-residual assumption. The shared environment contributes to both the diagonal and sibling covariance in this household model.

Why the Sum Formula Works

The component matrices can be added because covariance is additive across mutually uncorrelated random components. For any two individuals $i$ and $j$, expand the covariance of their modeled trait values. All cross-component covariances are zero under the model assumptions, leaving the sum of the four within-component covariances.

\begin{aligned} \operatorname{Cov}(y_i,y_j) &=\operatorname{Cov}(a_i+q_i+c_i+e_i,\ a_j+q_j+c_j+e_j)\\ &=\operatorname{Cov}(a_i,a_j)+\operatorname{Cov}(q_i,q_j) +\operatorname{Cov}(c_i,c_j)+\operatorname{Cov}(e_i,e_j)\\ &=\sigma_A^2 A_{ij}+\sigma_Q^2\pi_{ij} +\sigma_C^2 C_{ij}+\sigma_E^2 I_{ij}. \end{aligned}

The last equality uses the definitions of the component covariance structures. In matrix form, the same entry-by-entry result is exactly the sum defining $V$. If two random components are correlated, their cross-covariance terms do not vanish and this formula must be extended.

Checks and Common Pitfalls

A valid construction should be symmetric: a covariance between $i$ and $j$ equals the covariance between $j$ and $i$. Its diagonal entries are variances and must be nonnegative. Under the model, the diagonal for individual $i$ is $$ V_{ii}=\sigma_A^2A_{ii}+\sigma_Q^2\pi_{ii}+\sigma_C^2C_{ii}+\sigma_E^2. $$ For a non-inbred individual with $A_{ii}=1$, and the usual conventions $\pi_{ii}=C_{ii}=1$, this is the sum of the four variance components. If an individual is inbred, $A_{ii}=1+F_i$, where $F_i$ is the inbreeding coefficient, so the additive diagonal contribution is adjusted accordingly.

A covariance matrix must also be positive semidefinite: for every vector $u$, $u^\mathsf{T}Vu\geq 0$. If each component matrix is positive semidefinite and each variance component is nonnegative, their weighted sum is positive semidefinite. This follows because $u^\mathsf{T}Vu$ is the sum of the corresponding nonnegative quadratic forms. Pairwise entries that look individually plausible do not by themselves guarantee a valid matrix; the full matrix structure matters.

Wrong: “Full siblings have additive covariance 0.5.” Correction: 0.5 is their additive relationship coefficient. Their additive trait covariance is $0.5\sigma_A^2$.
Wrong: “The pedigree relationship and locus-specific IBD matrices are interchangeable.” Correction: $A$ describes pedigree-based additive relatedness, while $\Pi$ represents sharing at the locus under study.
Wrong: “The residual variance should be added to every matrix entry.” Correction: Under independent individual residuals, it is added only to the diagonal through $\sigma_E^2I$.
Wrong: “A shared-environment effect is the same as genetic relatedness.” Correction: The environmental matrix encodes the chosen exposure-sharing rule; it can connect people who are not genetically related.

Key takeaway: A pedigree covariance matrix is assembled by aligning the individual order, scaling each valid component matrix by its variance component, and adding the contributions. The interpretation of every entry depends on the definitions and assumptions behind those matrices.

Check Your Understanding

Answer these questions using the model and conventions above.

For two non-inbred full siblings, what is their additive relationship coefficient, and what is their additive trait covariance in terms of $\sigma_A^2$?
Why can the locus-specific entry $\pi_{ij}$ differ from the pedigree-based entry $A_{ij}$?
In a household model, what entries of $C$ are 1 for two individuals in the same household, including the diagonal?
Which entries of $V$ receive a contribution from independent residual variance?
State one condition needed for the sum of component matrices to be a valid covariance matrix.