Decomposing Phenotypic Covariance Into Genetic And Residual Components

Why a Phenotypic Relationship Has More Than One Source

Two traits can vary together because their genetic components vary together, because their residual influences vary together, or because both mechanisms contribute. The observed relationship between the traits combines these sources. Consequently, a phenotypic correlation, considered by itself, does not tell us how much of the relationship is genetic.

This tutorial develops the covariance decomposition for two traits. Its purpose is to clarify what the components mean and how they combine, not to estimate them from genomic data. A covariance is scale-dependent, so every component must be interpreted for the particular traits, population, and measurement scales under consideration.

Definition: For two centered traits, write \(Y_1=G_1+E_1\) and \(Y_2=G_2+E_2\). Here \(Y_j\) is the phenotypic value for trait \(j\), \(G_j\) is its genetic component as defined by the target estimand, and \(E_j\) is the residual component. The residual includes variation not represented by that genetic component; it can include environmental influences and measurement error. The genetic covariance is \(\operatorname{Cov}(G_1,G_2)\), the residual covariance is \(\operatorname{Cov}(E_1,E_2)\), and the phenotypic covariance is \(\operatorname{Cov}(Y_1,Y_2)\).

Centering means subtracting the relevant population means, or equivalently using covariance, which is invariant to adding constants. In applications, traits may also be adjusted for specified covariates; the decomposition then concerns the adjusted traits and the scales on which they are analyzed. The genetic component must be defined consistently with the scientific estimand. For example, a covariance for an additive genetic component is not automatically a covariance for every possible genetic contribution.

The Decomposition and Its Assumptions

Expanding the covariance of the two sums gives the general identity below. It does not require genetic and residual components to be independent.

\begin{aligned} \operatorname{Cov}(Y_1,Y_2) &=\operatorname{Cov}(G_1+E_1,G_2+E_2)\\ &=\operatorname{Cov}(G_1,G_2)+\operatorname{Cov}(G_1,E_2)\\ &\quad+\operatorname{Cov}(E_1,G_2)+\operatorname{Cov}(E_1,E_2). \end{aligned}

The familiar two-component decomposition follows when the cross-component covariances vanish: \(\operatorname{Cov}(G_1,E_2)=0\) and \(\operatorname{Cov}(E_1,G_2)=0\). Under this assumption, genetic and residual components may still be correlated across traits—that is precisely what the genetic and residual covariances measure. The assumption rules out covariance between one trait's genetic component and the other trait's residual component; it does not say that all residual influences are unrelated.

Theorem: If \(Y_j=G_j+E_j\) for \(j=1,2\), and \(\operatorname{Cov}(G_1,E_2)=\operatorname{Cov}(E_1,G_2)=0\), then \(\operatorname{Cov}(Y_1,Y_2)=\operatorname{Cov}(G_1,G_2)+\operatorname{Cov}(E_1,E_2)\). For each trait, if \(\operatorname{Cov}(G_j,E_j)=0\), then \(\operatorname{Var}(Y_j)=\operatorname{Var}(G_j)+\operatorname{Var}(E_j)\).

The variance result is obtained by the same expansion: \(\operatorname{Var}(G_j+E_j)=\operatorname{Var}(G_j)+\operatorname{Var}(E_j)+2\operatorname{Cov}(G_j,E_j)\). Thus it requires its own within-trait zero-covariance assumption. When both the cross-trait and within-trait conditions hold, write \(V_{Gj}=\operatorname{Var}(G_j)\), \(V_{Ej}=\operatorname{Var}(E_j)\), and \(V_{Pj}=\operatorname{Var}(Y_j)\), so \(V_{Pj}=V_{Gj}+V_{Ej}\).

A covariance has squared units if the two traits have the same units, and the product of their units if the traits have different units. A correlation is dimensionless. Provided the relevant variances are positive, define the genetic and residual correlations by \(\rho_G=\operatorname{Cov}(G_1,G_2)/\sqrt{V_{G1}V_{G2}}\) and \(\rho_E=\operatorname{Cov}(E_1,E_2)/\sqrt{V_{E1}V_{E2}}\). The phenotypic correlation is \(\rho_P=\operatorname{Cov}(Y_1,Y_2)/\sqrt{V_{P1}V_{P2}}\). Substituting the component covariances yields:

\rho_P= \frac{\rho_G\sqrt{V_{G1}V_{G2}}+ \rho_E\sqrt{V_{E1}V_{E2}}} {\sqrt{(V_{G1}+V_{E1})(V_{G2}+V_{E2})}}.

This expression shows why the three correlations need not be equal. The phenotypic correlation is a standardized sum of component covariances, not generally an average of the component correlations. The components' variances determine how much each covariance contributes relative to total phenotypic variation.

Working Through the Components

Worked Example: Combining Genetic and Residual Covariance

Suppose a hypothetical analysis defines genetic components for two traits measured in the same units. It gives \(V_{G1}=4\), \(V_{G2}=9\), and \(\operatorname{Cov}(G_1,G_2)=3\); the residual components have \(V_{E1}=6\), \(V_{E2}=7\), and \(\operatorname{Cov}(E_1,E_2)=-1\). Assume the genetic-residual cross-covariances vanish, including the within-trait covariances. Find the phenotypic covariance and compare the correlations.

The assumed variance decompositions give \(V_{P1}=4+6=10\) and \(V_{P2}=9+7=16\). The covariance decomposition gives:

\operatorname{Cov}(Y_1,Y_2)=3+(-1)=2.

Therefore, the phenotypic correlation is \(2/\sqrt{10\cdot16}=2/\sqrt{160}\approx0.1581\). The genetic correlation is \(3/\sqrt{4\cdot9}=3/6=0.5000\). The residual correlation is \(-1/\sqrt{6\cdot7}=-1/\sqrt{42}\approx-0.1543\).

In this hypothetical example, the positive genetic covariance and negative residual covariance partly offset each other. All the specified covariance matrices are admissible: \(3\leq\sqrt{4\cdot9}=6\), and \(|-1|\leq\sqrt{6\cdot7}=\sqrt{42}\). The observed phenotypic correlation is positive but much smaller than the genetic correlation.

Why Phenotypic Correlation Alone Is Not Enough

The decomposition is one equation for two component covariances. Even if both phenotypic variances are known, their sum does not determine the separate genetic and residual covariances. Additional information and an analysis that targets the components are needed; the phenotypic correlation alone cannot supply that separation.

Worked Example: The Same Phenotypic Correlation, Different Genetic Covariances

Consider two hypothetical component structures for traits whose total variance is 1 each. In both structures, each trait has genetic variance \(0.5\) and residual variance \(0.5\), with zero genetic-residual cross-covariances. In Structure A, let the genetic covariance be \(0.3\) and residual covariance be \(0.1\). In Structure B, let the genetic covariance be \(0\) and residual covariance be \(0.4\).

For Structure A, the component covariances sum to \(0.3+0.1=0.4\), and the phenotypic variances are \(0.5+0.5=1\) for both traits. Hence \(\rho_P=0.4/\sqrt{1\cdot1}=0.4\). Its genetic correlation is \(0.3/\sqrt{0.5\cdot0.5}=0.6\), while its residual correlation is \(0.1/0.5=0.2\).

For Structure B, the covariance sum is \(0+0.4=0.4\), with the same phenotypic variances of 1. Thus \(\rho_P=0.4\) again. But its genetic correlation is \(0/0.5=0\), and its residual correlation is \(0.4/0.5=0.8\).

Both structures are valid: each component covariance is no larger in absolute value than its component standard-deviation product of \(0.5\). The same phenotypic correlation, \(0.4\), is therefore compatible with a genetic correlation of \(0.6\) or with no genetic covariance at all. This is a concrete demonstration of non-identification from phenotypic correlation alone.

The residual component should not be read as a single environmental cause. It is defined relative to the chosen genetic component and can collect many sources of variation. If two traits share environmental conditions, measurement processes, or other non-genetic influences, their residual covariance can be nonzero. That covariance can reinforce or oppose genetic covariance in the observed phenotype.

Worked Example: Positive Genetic Correlation but Negative Phenotypic Correlation

Suppose a third hypothetical pair of traits has \(V_{G1}=4\), \(V_{G2}=1\), and genetic covariance \(1\). Its residual variances are \(V_{E1}=1\), \(V_{E2}=9\), with residual covariance \(-1.5\). Assume the genetic-residual cross-covariances vanish. Calculate the component correlations and phenotypic correlation.

The genetic correlation is \(1/\sqrt{4\cdot1}=0.5\). The residual correlation is \(-1.5/\sqrt{1\cdot9}=-1.5/3=-0.5\). The total variances are \(V_{P1}=4+1=5\) and \(V_{P2}=1+9=10\); the total covariance is \(1+(-1.5)=-0.5\). Therefore:

\rho_P=\frac{-0.5}{\sqrt{5\cdot10}} =\frac{-0.5}{\sqrt{50}} \approx -0.07071.

The component covariance matrices satisfy the required bounds: \(|1|\leq\sqrt{4\cdot1}=2\), and \(|-1.5|\leq\sqrt{1\cdot9}=3\). Despite a positive genetic correlation of \(0.5\), the phenotypic correlation is slightly negative because the negative residual covariance more than offsets the positive genetic covariance. The example also shows that component correlations alone do not determine the phenotypic correlation: component variances affect the covariance contributions and the total standardization.

Interpretation and a Common Pitfall

A genetic covariance describes how the specified genetic components of two traits vary together in the population and on the traits' measurement scales. It is not, by itself, evidence that one trait causes the other, nor does it identify a particular biological mechanism. Likewise, a residual covariance is a statistical component, not proof of one shared environmental cause. Interpretation depends on the genetic estimand, the trait definitions, and the population to which the decomposition refers.

Common mistake: “The phenotypic correlation is \(0.4\), so the genetic correlation must be \(0.4\).” This reasoning treats the observed correlation as though it were a direct measure of genetic correlation. The correction is to distinguish covariance from correlation and to account for both component covariances and component variances. As the second example showed, \(\rho_P=0.4\) can coexist with quite different genetic covariances and correlations.

A related error is assuming residual covariance equals zero just because the traits are measured on different individuals or in different units. Neither fact establishes zero covariance. The zero cross-component assumptions concern genetic-residual terms; they do not require residual covariance across traits to vanish. If the cross-component assumptions fail, the general four-term expansion applies, and the two-term decomposition alone is incomplete.

Key takeaway: Under zero genetic-residual cross-covariances, phenotypic covariance is the sum of genetic covariance and residual covariance. Phenotypic correlation standardizes that sum using total variances, so it cannot by itself identify either genetic covariance or genetic correlation.

Check Your Understanding

Answer each question using the decomposition and assumptions stated above.

Write the full covariance expansion for \(Y_1=G_1+E_1\) and \(Y_2=G_2+E_2\). Which two terms must vanish for the two-component covariance decomposition?
If the genetic covariance is \(0.25\) and the residual covariance is \(-0.10\), what is the phenotypic covariance under the decomposition assumptions?
Why does a phenotypic correlation of zero not necessarily imply zero genetic covariance?
What additional assumption is needed to write \(\operatorname{Var}(Y_j)=V_{Gj}+V_{Ej}\)?
Can two pairs of traits with the same phenotypic variances and phenotypic covariance have different genetic correlations? Explain using the component decomposition.