Why Mating Patterns Matter for Heritability
Many familiar interpretations of family resemblance use a random-mating reference: for example, under a simple additive model, parent–offspring and full-sibling phenotypic covariances are each one-half of the additive genetic variance. Positive assortative mating—when individuals with similar trait values mate more often than expected by chance—can change the genetic covariance structure that supports those reference relationships.
Two consequences need to be kept separate. Assortative mating can change the population’s genetic architecture, including the additive genetic variance present in the population. It can also cause an estimator that assumes random mating to fit the wrong covariance structure, so that its estimate no longer has the intended interpretation. The direction and size of that estimation error depend on the design, assumptions, and target quantity; there is no universal correction that can be applied using only the spouse correlation.
The target population matters. An estimate intended to describe the current population should reflect its actual mating history and genetic covariance structure. An estimate intended to describe a hypothetical random-mating population targets a different quantity. Calling a result “biased” without specifying which quantity is being estimated can therefore be ambiguous.
How Assortative Mating Alters Genetic Resemblance
A trait can be influenced by many genetic variants. If people with similar phenotypes tend to mate, their genotypes may also be correlated. When genetic contributions at different loci become correlated, the variance of their sum includes covariance terms. This is the population-level link between assortative mating and genetic covariance across loci.
For a simple additive genetic value \(A=\beta_1G_1+\beta_2G_2\), with centered genetic variables \(G_1\) and \(G_2\), the variance is
If the covariance term is positive, it raises the variance of this genetic sum relative to a calculation that sets that term to zero. The result depends on the signs of the effects and covariances; assortment does not guarantee that every covariance term is positive. The earlier material on assortative mating and gametic disequilibrium develops how nonrandom mating can generate such across-locus covariance.
At the pedigree level, assortment can also change the relationship between the genetic values of the two parents. An offspring receives genetic material from both. If the parents’ genetic values are correlated, the offspring’s genetic value can covary with each parent through both the direct transmission from that parent and the association with the mate’s genetic value. As a result, a pedigree covariance need not equal the random-mating reference coefficient multiplied by one common additive variance.
A simple reference model makes the contrast explicit. With additive genetic values, random mating, and no shared environmental covariance between relatives, the expected additive covariance for parent–offspring and full-sibling pairs is \(0.5V_A\). A model that uses these coefficients to interpret all observed family covariances assumes that the random-mating relationship structure is adequate. Assortment can violate that assumption even when the trait’s direct genetic effects are additive.
Worked Example: A Midparent Regression Under Assortment
Worked Example: A Midparent Regression Under Assortment
Consider a hypothetical population with phenotypic value \(P=A+E\), where \(A\) is additive genetic value and \(E\) is an environmental contribution. Suppose \(V_A=40\), \(V_P=100\), and therefore \(V_E=60\). Assume that the covariance between mates is entirely genetic, with \(C_A=20\), and that their environmental contributions are independent. There are no shared environmental effects between parents and offspring, and the additive transmission assumptions of the simple model below hold. What is the expected regression slope of offspring phenotype on the mean of the two parents’ phenotypes?
Let \(P_m\) and \(P_f\) be the maternal and paternal phenotypes, and let \(M=(P_m+P_f)/2\) be the midparent phenotype. Under the stated additive transmission model, the offspring’s covariance with either parent’s phenotype is \((V_A+C_A)/2\): one part comes from transmission by that parent, and the other reflects the covariance between the parents’ genetic values. Thus
Because the mates’ phenotypic covariance is \(C_A=20\) under these assumptions,
The regression slope is covariance divided by predictor variance:
The population narrow-sense heritability defined from the stated variances is \(V_A/V_P=40/100=0.400\). In this simplified example, treating the midparent regression slope as though it were automatically \(h^2\) would give 0.500 rather than 0.400. This calculation demonstrates a possible mismatch under assortative mating; it is not a general correction formula for real pedigree data.
Derivation: Why the Random-Mating Slope Can Fail
The previous example follows from the covariance and variance of the midparent value. Under the same simplified assumptions, write \(C_A=\operatorname{Cov}(A_m,A_f)\), where \(A_m\) and \(A_f\) are the parents’ additive genetic values. The offspring’s additive value has covariance \((V_A+C_A)/2\) with either parent’s phenotype: the transmitted contribution has covariance \(V_A/2\) with that parent, and the mate’s correlated genetic value contributes \(C_A/2\). With no environmental covariance between offspring and parents, averaging the two parental covariances gives
For parents with the same phenotypic variance \(V_P\), and with spouse phenotypic covariance equal to \(C_A\) because the environmental contributions are independent, the variance of their mean is
Therefore the expected regression slope is
When mating is random, \(C_A=0\), and this reduces to \(V_A/V_P\). If \(C_A>0\) and \(V_E=V_P-V_A>0\), then the slope exceeds \(V_A/V_P\), since
This derivation requires the specific assumptions just stated. In real data, spouse resemblance may have environmental as well as genetic components, family environments may be correlated, and mating may not follow this simple model. Consequently, the observed regression slope should not be converted into heritability by this expression unless its assumptions are justified.
Worked Example: Covariance Between Genetic Contributions
Worked Example: Covariance Between Genetic Contributions
Suppose a hypothetical additive score is \(A=G_1+G_2\), where \(G_1\) and \(G_2\) are centered genetic contributions with variance 1 each. Under a random-mating reference calculation, suppose their covariance is set to zero. In the current population, suppose positive assortment is associated with \(\operatorname{Cov}(G_1,G_2)=0.15\). Compare the two variances.
Under the reference calculation,
In the population with the specified covariance,
The covariance term contributes 0.300 variance units, making the variance of this genetic score 2.30 rather than 2.00. This is a change in the variance represented by the specified population model, not by itself proof that any particular heritability estimate is biased. Bias depends on whether the estimator’s assumptions and target account for the population covariance.
Diagnosing Absorption in Pedigree and Population Data
A random-mating additive model often explains family resemblance using a single \(V_A\) and expected relationship coefficients. If assortment raises or otherwise alters observed covariances, a fitted model may increase its additive variance estimate to accommodate them. In that sense, the estimate can absorb covariance created by the mating pattern: it assigns the excess resemblance to the model’s additive component rather than representing the changed covariance structure explicitly.
A useful diagnosis compares multiple patterns rather than relying on one estimate. Under the simple random-mating reference, parent–offspring and full-sibling covariances are both \(0.5V_A\). If their implied values of \(V_A\) differ substantially, or if observed resemblance does not fit the expected pattern across relationship types, the common-variance model may be inadequate. Such disagreement is a warning, not a unique signature of assortative mating: shared environments, sampling error, and other departures from the model can also produce it.
Population evidence can complement pedigree patterns. Positive spouse phenotypic correlation is consistent with phenotypic assortment, while evidence of correlated genetic contributions across loci is consistent with a changed genetic covariance structure. Neither fact alone proves that an estimated heritability has a particular bias. Spouse resemblance could arise partly from shared environments or social sorting, and genetic covariance can have causes other than the mating pattern being considered.
The practical question is therefore not simply, “Is the estimated heritability high?” It is whether the assumptions used to connect the observed data to the target \(V_A/V_P\) are credible. A careful report states the population, the heritability target, the mating assumptions, and the covariance patterns used to assess model fit.
Worked Example: Comparing Pedigree Covariances
In a hypothetical sample, the phenotypic variance is estimated as 100 squared trait units. After accounting for the stated sampling uncertainty, the observed parent–offspring covariance is 26 squared trait units and the full-sibling covariance is 31 squared trait units. A simple random-mating additive model predicts each covariance to be \(0.5V_A\). What additive variances would each covariance imply under that model, and what does their difference tell us?
For parent–offspring pairs, solve \(26=0.5V_A\):
For full siblings, solve \(31=0.5V_A\):
The two covariance patterns imply different additive variances under the same model. This discrepancy is evidence that the single random-mating covariance structure may not describe both relationship types well. Assortative mating is one possible explanation, but these calculations do not establish it as the cause. Without additional evidence and a model that addresses alternative sources of covariance, averaging 0.520 and 0.620 would not resolve the model mismatch.
Common Mistake and Practical Interpretation
Wrong reasoning: “The spouses are correlated, so the heritability estimate must be inflated by that correlation.” This treats spouse phenotypic correlation as if it were entirely genetic and assumes the same simple relationship between mating and the estimator in every design.
Correction: Spouse correlation is a diagnostic clue, not a universal adjustment factor. Establish what the estimate targets, determine which genetic and environmental covariances the design can represent, and check whether the observed family and population patterns fit the model. A result may reflect a genuine change in current-population \(V_A\), misspecification of a random-mating estimator, or both.
Assortative mating matters because a heritability estimate is conditional on a population and a covariance model. Interpreting it well requires distinguishing the genetic variance present in that population from the assumptions used to estimate it. No single pedigree covariance, spouse correlation, or numerical estimate can make that distinction by itself.
Check Your Understanding
Use the assumptions in each question explicitly when answering.
- In the simple additive reference model, what covariance is expected for parent–offspring pairs when \(V_A=50\), and what random-mating assumption supports that calculation?
- Why can positive covariance between two additive genetic contributions increase the variance of their sum?
- In the midparent derivation, why does spouse environmental covariance matter for interpreting the regression slope?
- If parent–offspring and full-sibling covariances imply different \(V_A\) values under a common random-mating model, what can be concluded—and what cannot be concluded—from that discrepancy alone?
- Explain why a positive spouse phenotypic correlation is not, by itself, a correction factor for heritability.