Why Two Traits Can Share Genetic Variation
Two traits may have some of the same genetic influences, but shared influences do not automatically mean the traits move in the same direction. A genetic variant may increase both traits, increase one while decreasing the other, or have effects that differ across variants. Additive genetic covariance summarizes the net direction and magnitude of the association between additive genetic values for two traits in a specified population. Additive genetic correlation expresses that covariance relative to the genetic variability of each trait, making it dimensionless.
These are properties of a defined genetic component in a defined population, not properties of the traits in every population. The allele frequencies, linkage disequilibrium, and genetic effects represented in the target population matter. In particular, genetic covariance has units tied to both traits, while genetic correlation does not. The two quantities answer related but distinct questions.
Definitions and Genetic-Effect Representation
A useful representation makes the population assumptions explicit. Let \(X\) be a vector of genotype dosages at the variants included in the genetic model, centered at their means in the target population. Let \(\beta_1\) and \(\beta_2\) be the corresponding additive effects, measured as change in trait value per additional allele. Write \(G_1=X^\mathsf{T}\beta_1\) and \(G_2=X^\mathsf{T}\beta_2\). With \(\Sigma_X=\operatorname{Cov}(X)\), it follows that
This representation describes the additive component included in the model. If \(X\) contains only a subset of variants, these quantities describe the additive genetic values captured by that subset, not necessarily all additive genetic variation. The covariance matrix \(\Sigma_X\) allows variants to be correlated through linkage disequilibrium. Only when the genotypes are mutually uncorrelated is it valid to replace this matrix by a diagonal matrix and sum variant contributions independently. The model also assumes that the genetic value is represented as a linear additive function of genotype dosage; dominance and interactions are not included in this definition.
Worked Example: Covariance From Two Independent Variants
Suppose a hypothetical target population has two independent variants. For variant \(j\), the dosage variance is \(2p_j(1-p_j)\), where \(p_j\) is its allele frequency. Let the frequencies be \(p_1=0.20\) and \(p_2=0.50\). Suppose the effects per allele are \(\beta_1=(0.40,0.20)^\mathsf{T}\) trait units for trait 1 and \(\beta_2=(0.30,-0.10)^\mathsf{T}\) trait units for trait 2.
Because the variants are independent, \(\Sigma_X\) is diagonal. Their dosage variances are \(2(0.20)(0.80)=0.32\) and \(2(0.50)(0.50)=0.50\). The genetic covariance is therefore
The covariance is \(0.0284\) trait-unit-squared, where the product of units is one trait-1 unit times one trait-2 unit. The first variant contributes positively; the second contributes negatively. The additive genetic variances are
Thus the additive genetic correlation is \(0.0284/\sqrt{(0.0712)(0.0338)}=0.579\) to three significant figures. In this hypothetical model the net genetic association is positive and moderately strong on the correlation scale, although one of the two variant contributions opposes the other. This calculation relies on the stated independence assumption; with linkage disequilibrium, the off-diagonal terms in \(\Sigma_X\) must also be included.
Population and Sample Quantities
An estimand must say which population it describes. A population genetic covariance refers to the covariance of the additive genetic values among individuals in a specified target population. The corresponding population genetic correlation uses that population’s covariance and genetic variances. Changing the target population can change these quantities, for example if allele frequencies or linkage disequilibrium differ. The genetic effects and the trait scales also need to be understood as applying to the population and measurement definition under discussion.
A finite cohort can instead be the target of a descriptive summary. If its \(n\) individuals have genetic values \(G_{i1}\) and \(G_{i2}\), define their cohort means as \(\bar G_1=n^{-1}\sum_i G_{i1}\) and \(\bar G_2=n^{-1}\sum_i G_{i2}\). The finite-cohort covariance, using the cohort itself as the entire target, is
This is a descriptive estimand for those particular individuals. By contrast, the conventional sample covariance
is a sample statistic. Under independent random sampling from a target population with finite second moments, it estimates that population’s covariance and is unbiased for it. The sample correlation is \(r_g=S_{12}/\sqrt{S_1^2S_2^2}\), where \(S_1^2\) and \(S_2^2\) are the sample variances calculated with denominator \(n-1\). It estimates the population genetic correlation, but it is not generally an unbiased estimator of that correlation. In real genetic analyses, genetic values are generally not observed directly; a method estimates the target quantity from genotype and phenotype data. The definitions do not, by themselves, specify an estimation method.
Worked Example: A Cohort Summary and a Sample Estimate
Suppose five hypothetical individuals have known additive genetic values \(G_1=(-2,-1,0,1,2)\) for trait 1 and \(G_2=(-1,0,1,0,2)\) for trait 2. Treat these as the full cohort for a descriptive calculation, or as a random sample when estimating a larger target population. The means are \(\bar G_1=0\) and \(\bar G_2=0.4\).
The centered values for trait 1 are \((-2,-1,0,1,2)\); those for trait 2 are \((-1.4,-0.4,0.6,-0.4,1.6)\). The products of paired centered values are \(2.8, 0.4, 0, -0.4,\) and \(3.2\), which sum to \(6.0\). Therefore the finite-cohort covariance is \(6.0/5=1.20\) trait-unit-squared. The conventional sample covariance is \(6.0/(5-1)=1.50\) trait-unit-squared.
For the sample correlation, the centered squares sum to \(10.0\) for trait 1 and \(5.20\) for trait 2. Thus \(S_1^2=10.0/4=2.50\) and \(S_2^2=5.20/4=1.30\), giving
The finite-cohort variances use denominator \(n\), giving \(2.00\) and \(1.04\); using them with the finite-cohort covariance also gives correlation \(1.20/\sqrt{(2.00)(1.04)}=0.832\). The shared denominator cancels in the correlation. The cohort covariance and sample covariance differ because they answer different questions and use different denominators.
What Covariance and Correlation Tell Us
Covariance retains both direction and scale. A positive value means that individuals with above-average additive genetic values for one trait tend, on average, to have above-average additive genetic values for the other. A negative value means that above-average values for one tend to accompany below-average values for the other. The magnitude depends on the measurement units and the amount of genetic variation in each trait. If trait 1 is converted from meters to centimeters, for example, its genetic values and the covariance change by a factor of 100; the correlation does not.
Correlation standardizes covariance by the genetic standard deviations. It has no physical units and lies between \(-1\) and \(1\), provided both genetic variances are positive. Its sign gives the direction of the net linear association between additive genetic values, while its magnitude gives the strength of that standardized association. It is not the proportion of causal variants shared, and it does not imply that one trait causes the other.
The bound on the correlation follows directly from the Cauchy–Schwarz inequality. Center the genetic values by writing \(U=G_1-E(G_1)\) and \(V=G_2-E(G_2)\). For finite, positive variances, the inequality gives
Dividing both sides by the positive square root of the variance product proves \(|\rho_g|\leq 1\). Equality holds only when the centered genetic values are linearly proportional almost surely in the target population. If either genetic variance is zero, the correlation is undefined because its denominator is zero, even though the covariance is zero.
Worked Example: Zero Correlation Despite Effects at Shared Variants
Consider two independent variants, each with dosage variance \(0.50\). Let the effects be \(\beta_1=(1,1)^\mathsf{T}\) for trait 1 and \(\beta_2=(1,-1)^\mathsf{T}\) for trait 2, in their respective trait units per allele. Since \(\Sigma_X=0.50I\), where \(I\) is the two-by-two identity matrix,
Each trait has genetic variance \(0.50(1^2)+0.50(1^2)=1.00\), so \(\rho_g=0/\sqrt{(1.00)(1.00)}=0\). Both variants affect both traits, but their cross-trait contributions cancel exactly: one acts in the same direction for both traits, while the other acts in opposite directions. A zero genetic correlation therefore does not prove that the traits have no shared genetic influences.
Common Mistakes and Practical Interpretation
- Equating covariance with correlation. Covariance is unit-dependent and its magnitude depends on the scale of both traits. Correlation is standardized and dimensionless; it is not a covariance expressed in different units.
- Reading zero correlation as no shared effects. Contributions with opposing signs can cancel, as in the worked example. A zero net association is not evidence that every cross-trait genetic effect is zero.
- Ignoring the population definition. Population allele frequencies and linkage disequilibrium affect the covariance of genetic values. A result from one target population need not be identical in another.
- Calling a sample statistic a population truth. A cohort’s sample correlation describes its observed genetic values and estimates a target-population parameter only under an appropriate sampling and modeling framework.
- Interpreting genetic correlation causally. The correlation describes co-variation in additive genetic values. It does not establish that genetic influences on one trait cause changes in the other trait.
When reporting either estimand, name the traits, state the population and genetic component being described, and retain covariance units. For correlation, report the value on its dimensionless scale and explain its direction. These choices make clear whether the result describes net shared additive genetic variation and what population it applies to.
Check Your Understanding
Use the definitions and calculations above to check your understanding.
- In the genotype-effect representation, how does linkage disequilibrium enter the formula for additive genetic covariance?
- Why does converting one trait from meters to centimeters change genetic covariance but not genetic correlation?
- For a finite cohort, what is the difference between the covariance with denominator \(n\) and the conventional sample covariance with denominator \(n-1\)?
- Can genetic correlation be calculated when one trait has zero additive genetic variance? Explain.
- Give one reason that zero additive genetic correlation does not establish that the two traits have no shared genetic effects.