Separating Polygenic and QTL Variance

Why Separate Polygenic and QTL Variation?

Relatives tend to resemble one another because they share genetic material. A linkage analysis asks a more specific question: does sharing at a candidate genomic location explain additional trait resemblance beyond the genome-wide resemblance expected from kinship? Answering it requires separating two covariance patterns, not merely adding two labels to the same source of familial similarity.

The genome-wide polygenic component represents aggregate effects distributed across the genome, apart from the candidate locus being evaluated. The QTL component represents the effect associated with that candidate locus. Their variance contributions can be distinguished when their covariance matrices imply different patterns of resemblance among the observed relatives.

Definition: A polygenic variance component is the variance contribution from aggregate genetic effects represented by genome-wide kinship. A QTL variance component is the variance contribution associated with a specified candidate locus, represented through relatives’ identity-by-descent (IBD) sharing at that locus. The residual variance component represents individual-specific variation not accounted for by these genetic components or the mean model.

The Linkage Variance-Component Model

Let \(y\) be the vector of quantitative phenotypes for \(n\) family members, and let \(X\) be a design matrix for fixed effects such as an intercept and specified covariates. Use the model \(y=X\beta+g+q+e\), where \(g\) is the polygenic effect vector, \(q\) is the candidate-locus effect vector, and \(e\) is the residual vector.

For clarity, suppose individuals are not inbred, and scale each genetic variance component as the variance of that component for one individual. Let \(K\) be the genome-wide additive relationship matrix: its diagonal entries are 1 and its off-diagonal entries are twice the kinship coefficients. Let \(L\) be the candidate-locus IBD matrix: \(L_{ij}\) is the expected proportion of the two alleles at the locus that individuals \(i\) and \(j\) share IBD. Its diagonal entries are 1. Thus, for a pair, \(K_{ij}\) describes genome-wide expected relatedness, whereas \(L_{ij}\) describes IBD sharing at this particular locus.

g\sim N(0,\sigma_P^2K),\qquad q\sim N(0,\sigma_Q^2L),\qquad e\sim N(0,\sigma_E^2I)

The model assumes that these three random-effect vectors are mutually independent, and that residuals for different individuals are independent with common variance. Conditional on the fixed effects and covariance matrices, the phenotype vector is multivariate normal. The resulting covariance matrix is

V=\operatorname{Var}(y\mid X)=\sigma_P^2K+\sigma_Q^2L+\sigma_E^2I.

Here \(\sigma_P^2\), \(\sigma_Q^2\), and \(\sigma_E^2\) are nonnegative variance components, measured in squared phenotype units. The fixed effects determine the mean \(X\beta\); they do not add a random covariance component. If covariates are included, the covariance model describes variation around that mean.

Model interpretation: For two distinct relatives \(i\) and \(j\), the model-implied covariance is \(K_{ij}\sigma_P^2+L_{ij}\sigma_Q^2\). The independent residual component contributes to each individual’s variance, but not to covariance between different individuals. On the diagonal, under the stated non-inbreeding convention, the variance is \(\sigma_P^2+\sigma_Q^2+\sigma_E^2\).

Why the Two Genetic Components Have Different Covariance Patterns

The polygenic component follows genome-wide kinship: relatives with greater overall relatedness are expected to have greater polygenic covariance. The candidate-locus component instead follows IBD sharing at the locus. Two pairs with the same pedigree relationship can have the same \(K_{ij}\) but different \(L_{ij}\), because they may inherit different amounts of the same ancestral material at that location. That difference is what can provide information about \(\sigma_Q^2\) separately from \(\sigma_P^2\).

The distinction is clearest if the random effects are independent. Covariance is additive for independent random vectors, so the covariance of \(g+q+e\) is the sum of their covariance matrices. The candidate-locus term must represent a component distinct from the aggregate polygenic term. In practice, this means treating the polygenic component as background variation apart from the locus-specific contribution, rather than counting the candidate locus in both components.

A Short Derivation of the Covariance

Because \(X\beta\) is fixed conditional on the covariates, it has zero conditional variance. For any two phenotype entries, expand the covariance of their random parts. Mutual independence makes all cross-covariances zero.

\begin{aligned} \operatorname{Cov}(y_i,y_j\mid X) &=\operatorname{Cov}(g_i+q_i+e_i,\;g_j+q_j+e_j)\\ &=\operatorname{Cov}(g_i,g_j)+\operatorname{Cov}(q_i,q_j) +\operatorname{Cov}(e_i,e_j)\\ &=\sigma_P^2K_{ij}+\sigma_Q^2L_{ij} +\sigma_E^2 I_{ij}. \end{aligned}

The final equality uses the three specified covariance matrices. For distinct individuals \(I_{ij}=0\), so only the two genetic terms remain. For the same individual \(I_{ii}=1\), and the residual variance contributes to the diagonal. Applying this entry by entry gives \(V=\sigma_P^2K+\sigma_Q^2L+\sigma_E^2I\).

Worked Example: Building a Two-Relative Covariance Matrix

Suppose two non-inbred full siblings have \(K_{12}=0.50\). At a candidate locus, suppose their IBD sharing is \(L_{12}=0.25\). For illustration, take \(\sigma_P^2=0.80\), \(\sigma_Q^2=0.50\), and \(\sigma_E^2=0.70\), all in squared trait units. These are hypothetical model values.

The diagonal variance for either sibling is \(\sigma_P^2+\sigma_Q^2+\sigma_E^2=0.80+0.50+0.70=2.00\) squared trait units. The off-diagonal covariance is \(K_{12}\sigma_P^2+L_{12}\sigma_Q^2=(0.50)(0.80)+(0.25)(0.50)=0.40+0.125=0.525\) squared trait units.

V= \begin{pmatrix} 2.00 & 0.525\\ 0.525 & 2.00 \end{pmatrix} \quad\text{(squared trait units).}

The polygenic term contributes \(0.40\) to the siblings’ covariance, and the candidate-locus term contributes \(0.125\). The residual term contributes \(0.70\) to each diagonal variance and nothing to their covariance, given the independent-residual assumption.

When Can the Components Be Distinguished?

A model can be written down without its components being well determined by the available data. Separation depends on the covariance matrices providing distinct patterns after the mean structure is accounted for. If one component matrix can be expressed as a linear combination of the others in the information available from the data, different component values can imply the same covariance. Then the components are not separately identifiable from that information.

For example, in a set of full-sibling pairs, \(K_{ij}=0.50\) for every pair. If all pairs also have the same candidate-locus sharing value, the off-diagonal covariances across those pairs follow the same single combination of the two genetic variances. Variation in locus-specific sharing across pairs adds a second pattern: polygenic covariance stays at the value determined by \(K_{ij}\), while QTL covariance changes with \(L_{ij}\). A well-designed analysis also needs enough independent family information to estimate those patterns in the presence of residual variation.

Worked Example: Checking Separation Across Two Pair Types

Suppose hypothetical data provide estimates of within-pair covariance for two groups of full-sibling pairs. The pairs have \(K_{ij}=0.50\) in both groups, but their candidate-locus sharing values are \(0.25\) and \(0.75\). Suppose the corresponding estimated covariances are \(0.55\) and \(0.80\) squared trait units. To illustrate how the two genetic contributions are separated, use the model equations for pair covariance, excluding the residual term because the individuals in each pair are distinct.

For the first group, \(0.50\sigma_P^2+0.25\sigma_Q^2=0.55\). For the second group, \(0.50\sigma_P^2+0.75\sigma_Q^2=0.80\). Subtracting the first equation from the second gives \(0.50\sigma_Q^2=0.25\), so \(\sigma_Q^2=0.25/0.50=0.50\) squared trait units. Substitution into the first equation gives \(0.50\sigma_P^2+(0.25)(0.50)=0.55\), hence \(0.50\sigma_P^2=0.425\) and \(\sigma_P^2=0.425/0.50=0.85\) squared trait units.

The coefficient matrix for these two equations has determinant \((0.50)(0.75)-(0.50)(0.25)=0.25\), which is nonzero, so the two equations have a unique solution. If the two groups had identical IBD sharing, the rows would be identical and the determinant would be zero: these pair covariances alone could not separate the two genetic components. These calculations illustrate the covariance structure; actual estimates require appropriate data and an estimation method.

Worked Example: Completing the Diagonal Variance

Continue with the hypothetical estimates \(\sigma_P^2=0.85\) and \(\sigma_Q^2=0.50\) squared trait units. Suppose the estimated total within-individual variance is \(2.00\) squared trait units. Under the stated non-inbreeding model, the diagonal variance is the sum of all three components: \(2.00=\sigma_P^2+\sigma_Q^2+\sigma_E^2\).

Solving gives \(\sigma_E^2=2.00-0.85-0.50=0.65\) squared trait units. The model therefore attributes \(0.85\) to polygenic variation, \(0.50\) to candidate-locus variation, and \(0.65\) to residual variation. The component values sum to \(0.85+0.50+0.65=2.00\), matching the stipulated diagonal variance. This arithmetic checks the decomposition; it does not establish that the stipulated component estimates are statistically precise.

Common Mistakes and Practical Interpretation

Treating kinship and locus sharing as interchangeable. Kinship describes genome-wide relatedness; \(L_{ij}\) describes IBD sharing at the candidate locus. Even when both concern relatives, they need not have the same pairwise values.
Assigning residual variance to pair covariance. Under independent residuals, \(\sigma_E^2\) contributes to diagonal entries, not to covariance between different family members. A model with shared household or other correlated environmental effects would need to represent that dependence separately.
Assuming distinct labels guarantee separate estimates. If the available \(K\) and \(L\) patterns are too similar, the data may provide little information to tell the components apart. Different notation cannot fix a lack of distinguishable covariance patterns.
Counting the candidate locus twice. The intended decomposition treats the candidate-locus contribution as distinct from the background polygenic component. Including the same locus in both terms changes the interpretation and can inflate or obscure the component estimates.
Interpreting variance as an individual effect. A variance component is a population-level contribution to variation under the model, in squared trait units. It is not a predicted trait change for an individual, nor does it by itself establish a causal effect.

Key takeaway: The model separates polygenic and QTL variance through covariance structure: genome-wide kinship weights \(\sigma_P^2\), candidate-locus IBD sharing weights \(\sigma_Q^2\), and independent individual-specific residuals contribute \(\sigma_E^2I\). Separation requires enough distinct information in those patterns.

Check Your Understanding

Answer each question using the model and assumptions in this tutorial.

For two distinct relatives, write the model-implied covariance and identify which component contributes nothing under independent residuals.
Explain why two full-sibling pairs with the same kinship but different candidate-locus IBD sharing can help distinguish polygenic from QTL variance.
If two pair groups have the same \(K_{ij}\) and the same \(L_{ij}\), what limitation does that create for separating their genetic covariance contributions?
A researcher includes the candidate locus in both the background polygenic component and the locus-specific component. What interpretive problem can result?
Under the non-inbreeding model, calculate the residual variance if the total variance is \(1.80\), \(\sigma_P^2=0.60\), and \(\sigma_Q^2=0.45\), all in squared trait units.