Genetic Correlation Versus Phenotypic Correlation

Why Two Correlations Can Tell Different Stories

Two traits can be similar in their additive genetic influences without having a strong observed correlation. Conversely, traits with only modest genetic correlation can be strongly correlated in measured individuals. The reason is that observed similarity reflects more than genetic covariance: environmental influences, other residual factors, and sometimes the way traits are measured also contribute.

The genetic correlation describes the correlation between additive genetic values for two traits in a specified population. The phenotypic correlation describes the correlation between the traits as observed in individuals. Neither correlation is a substitute for the other. Their relationship depends on the genetic and residual variances and covariances on the scales being analyzed.

Definition: Let \(Y_1\) and \(Y_2\) be two traits, with additive genetic values \(G_1,G_2\) and residual components \(E_1,E_2\). Under the decomposition \(Y_i=G_i+E_i\), the genetic correlation is \(r_g=\operatorname{Cov}(G_1,G_2)/\sqrt{\operatorname{Var}(G_1)\operatorname{Var}(G_2)}\). The phenotypic correlation is \(r_P=\operatorname{Cov}(Y_1,Y_2)/\sqrt{\operatorname{Var}(Y_1)\operatorname{Var}(Y_2)}\).

Here, “residual” means the part not represented by the additive genetic value in the specified model. It can include environmental effects, non-additive influences, and unexplained variation. A shared environment can make \(E_1\) and \(E_2\) covary. The word residual does not imply that these components are unrelated across traits.

The familiar decomposition below requires that genetic values are uncorrelated with residuals, including across traits: \(\operatorname{Cov}(G_1,E_2)=\operatorname{Cov}(G_2,E_1)=0\). Under that assumption, the total covariance is the sum of genetic and residual covariance. If these cross-covariances are nonzero, additional terms are needed, and the simple decomposition does not apply as written.

How Component Correlations Combine

Write \(\sigma_{g_i}^2=\operatorname{Var}(G_i)\), \(\sigma_{e_i}^2=\operatorname{Var}(E_i)\), and let \(h_i^2=\sigma_{g_i}^2/\operatorname{Var}(Y_i)\) denote the additive genetic variance proportion for trait \(i\) on the scale in use. Define \(r_e=\operatorname{Cov}(E_1,E_2)/\sqrt{\sigma_{e_1}^2\sigma_{e_2}^2}\) when both residual variances are positive. If the component assumptions above hold, then:

\operatorname{Cov}(Y_1,Y_2) = \operatorname{Cov}(G_1,G_2)+\operatorname{Cov}(E_1,E_2)

Dividing by the product of the phenotypic standard deviations expresses the phenotypic correlation as a weighted combination of the two component correlations:

r_P = r_g\sqrt{h_1^2h_2^2} + r_e\sqrt{(1-h_1^2)(1-h_2^2)}

This formula assumes the variance is partitioned into the stated genetic and residual components, so \(h_i^2\) and \(1-h_i^2\) are the respective proportions of total variance. It also assumes the same trait scales and population define all the components. It is not generally correct to average \(r_g\) and \(r_e\) equally: each is weighted by the standard deviations contributed by its component.

Justification: By definition, \(\operatorname{Cov}(G_1,G_2)=r_g\sigma_{g_1}\sigma_{g_2}\) and \(\operatorname{Cov}(E_1,E_2)=r_e\sigma_{e_1}\sigma_{e_2}\). Since the component cross-covariances are zero, their sum is \(\operatorname{Cov}(Y_1,Y_2)\). Also, \(\sigma_{g_i}/\sigma_{y_i}=\sqrt{h_i^2}\) and \(\sigma_{e_i}/\sigma_{y_i}=\sqrt{1-h_i^2}\). Dividing the summed covariance by \(\sigma_{y_1}\sigma_{y_2}\) gives the displayed expression.

The two terms can reinforce one another or oppose one another. If residual covariance has the opposite sign to genetic covariance, the observed correlation can be much smaller than the genetic correlation, or even have the opposite sign. If residual covariance has the same sign, the phenotypic correlation can be stronger. These are statements about the components of covariance, not claims that one component causes the other.

Worked Example: Opposing Residual Covariance Cancels Genetic Similarity

Suppose two hypothetical traits are each standardized to phenotypic variance 1. Their additive genetic variance proportions are \(h_1^2=0.40\) and \(h_2^2=0.25\), their genetic correlation is \(r_g=0.60\), and their residual correlation is \(r_e=-0.30\). Assume genetic values are uncorrelated with both traits’ residuals, as required for the decomposition.

The genetic variances are 0.40 and 0.25, and the residual variances are \(1-0.40=0.60\) and \(1-0.25=0.75\). The genetic covariance is:

\operatorname{Cov}(G_1,G_2) =0.60\sqrt{0.40(0.25)} =0.60\sqrt{0.10} \approx 0.1897

The residual covariance is:

\operatorname{Cov}(E_1,E_2) =-0.30\sqrt{0.60(0.75)} =-0.30\sqrt{0.45} \approx -0.2012

The total covariance is approximately \(0.1897-0.2012=-0.0115\). Because both phenotypic variances equal 1, the phenotypic correlation is also approximately \(-0.0115\). Thus these hypothetical traits have a positive genetic correlation of 0.60 but an essentially zero, slightly negative phenotypic correlation. The opposing residual covariance nearly cancels the genetic covariance.

Shared Environments Can Strengthen Observed Similarity

A residual covariance can arise when individuals experience influences that affect both traits. For example, a common environment or a shared feature of the assessment process may contribute to both measurements. In the component model, such a contribution appears in \(\operatorname{Cov}(E_1,E_2)\), provided it is not represented as part of the genetic component.

A positive residual correlation does not establish which environmental factor is responsible, nor does it establish a causal pathway between the traits. It says that the residual components tend to vary together in the population and model under consideration.

Worked Example: Positive Residual Covariance Raises the Phenotypic Correlation

Consider two hypothetical standardized traits with phenotypic variance 1 each. Let \(h_1^2=0.50\), \(h_2^2=0.40\), \(r_g=0.20\), and \(r_e=0.70\). Assume again that genetic values are uncorrelated with the residuals, including across traits.

The genetic contribution to phenotypic covariance is:

r_g\sqrt{h_1^2h_2^2} =0.20\sqrt{0.50(0.40)} =0.20\sqrt{0.20} \approx 0.08944

The residual contribution is:

r_e\sqrt{(1-h_1^2)(1-h_2^2)} =0.70\sqrt{0.50(0.60)} =0.70\sqrt{0.30} \approx 0.3834

Adding the contributions gives \(r_P\approx0.08944+0.3834=0.4728\), or about 0.473. The phenotypic correlation is therefore substantially larger than the genetic correlation of 0.20. Under the stated model, this difference is accounted for mainly by the positive residual covariance, not by strong genetic correlation.

Measurement Processes Can Also Matter

The correlation between measured traits can reflect covariance introduced by how they are recorded. A common rater, shared instrument conditions, or an overlapping measurement procedure can create correlated measurement deviations. If those deviations are included in the residual component, they can alter the residual covariance and therefore the observed phenotypic correlation.

The next example keeps biological residuals and measurement errors separate so their roles are visible. It assumes measurement errors have mean zero and are uncorrelated with genetic values and biological residuals. The errors for the two traits are allowed to covary with each other. These assumptions are important: without them, further covariance terms would have to be included.

Worked Example: Correlated Measurement Errors Change the Observed Correlation

Suppose two hypothetical traits have genetic variances 4 and 9, with genetic covariance 3. Their biological residual variances are 6 and 16, with biological residual covariance 0. The measurement-error variances are 4 and 9, with measurement-error covariance 3. Assume all genetic, biological-residual, and measurement-error components are mutually uncorrelated, except for the stated covariance between the two measurement errors.

The genetic correlation is:

r_g=\frac{3}{\sqrt{4(9)}}=\frac{3}{6}=0.50

Without measurement error, the trait variances would be \(4+6=10\) and \(9+16=25\). Their covariance would be \(3+0=3\), so their biological phenotypic correlation would be:

r_{P,\mathrm{biological}} =\frac{3}{\sqrt{10(25)}} =\frac{3}{\sqrt{250}} \approx 0.1897

For the measured traits, the variances are \(4+6+4=14\) and \(9+16+9=34\). Their covariance is \(3+0+3=6\), including the covariance between measurement errors. Therefore:

r_{P,\mathrm{measured}} =\frac{6}{\sqrt{14(34)}} =\frac{6}{\sqrt{476}} \approx 0.2750

In this example, the measured correlation is about 0.275, while the genetic correlation remains 0.50. The correlated measurement errors raise the observed covariance relative to the biological covariance, while the added error variances also change the denominator. The example illustrates why the measurement process matters; it does not imply that the same adjustment or direction of change applies in every study.

Interpreting the Difference Carefully

A comparison of \(r_g\) and \(r_P\) is meaningful only when each quantity is clearly defined for the same traits, population, and relevant scale. A genetic correlation describes the relative pattern of additive genetic effects, not the fraction of individuals who share particular alleles and not the correlation between measured trait values. A phenotypic correlation describes observed co-variation, which may combine genetic, environmental, and measurement-related sources.

The formula also shows why a small phenotypic correlation does not imply a small genetic correlation. In the first example, substantial positive genetic covariance was nearly offset by negative residual covariance. Conversely, a large phenotypic correlation does not by itself establish a large genetic correlation: the second example has a phenotypic correlation of about 0.473 despite a genetic correlation of only 0.20.

These distinctions matter when results are used to describe shared genetic architecture. A phenotypic association alone cannot determine whether two traits share additive genetic influences, and a genetic correlation alone does not describe how strongly the measured traits co-vary in individuals. Interpretation should report which correlation is being discussed and identify the assumptions or components that connect them.

Common Mistake

Wrong reasoning: “The phenotypic correlation is close to zero, so the traits cannot have shared genetic influences.”
Correction: The phenotypic covariance is the sum of genetic and residual covariance only under the stated decomposition assumptions. Opposing components can cancel, as in the first example. A near-zero phenotypic correlation therefore does not, by itself, establish a near-zero genetic correlation.
Another error: Treating \(r_P\) as the unweighted average of \(r_g\) and \(r_e\). The component correlations are weighted by the component standard deviations relative to the total trait standard deviations; use the decomposition formula when its assumptions hold.

Key takeaway: Genetic and phenotypic correlations answer different questions. Under a genetic-plus-residual decomposition with zero genetic–residual cross-covariances, phenotypic correlation combines genetic and residual covariance, weighted by their contributions to the traits’ variances. Shared environments and measurement processes can therefore strengthen, weaken, or reverse the observed relationship.

Check Your Understanding

Use the component definitions and assumptions in this tutorial to answer the following.

If two traits have positive genetic covariance but negative residual covariance, what determines whether their phenotypic covariance is positive, negative, or zero?
For standardized traits with \(h_1^2=0.25\), \(h_2^2=0.64\), \(r_g=0.50\), and \(r_e=0\), calculate \(r_P\) under the stated decomposition.
Why can a large phenotypic correlation occur even when the genetic correlation is modest?
What additional covariance terms may be needed if a genetic value for one trait is correlated with the residual of the other trait?
In the measurement example, which assumptions allow the observed covariance to be written as the sum of genetic, biological-residual, and measurement-error covariances?