Nontransmitted Alleles And Genetic Nurture

Why Use Alleles That the Child Did Not Inherit?

A child’s genotype can be associated with an outcome for more than one reason. The child’s alleles may have a direct biological effect, while parental genotypes may also influence the environment the child experiences. For example, parental traits related to education or health may shape a child’s opportunities. A parental allele that the child did not inherit cannot act through that child’s genotype, but it may still be associated with the environment provided by the parent. This association is one way to study genetic nurture, an indirect genetic effect of relatives on an individual’s outcome.

A nontransmitted allele is not automatically a causal environmental influence. It is a marker observed in a family, and its association with the child’s outcome may reflect ancestry, mating patterns, or measurement error. The central task is therefore to distinguish the contrast that Mendelian transmission supplies from the additional assumptions needed to interpret an estimate as genetic nurture.

Scores and a Regression Model

For a set of loci, a polygenic score is a weighted sum of allele dosages. Use the same allele coding and weights for the child and parents. Let \(P_T\) be the sum of the parental transmitted-allele scores, and let \(P_N\) be the sum of their nontransmitted-allele scores. With weights \(w_j\) across loci \(j\), these can be written as \(P_T=\sum_j w_j(T_{mj}+T_{fj})\) and \(P_N=\sum_j w_j(N_{mj}+N_{fj})\). Under this construction, \(P_T\) is the child’s score at those loci. The total parental score is \(P_T+P_N\).

A basic trio regression for an outcome \(Y\) is

Here \(i\) indexes children, \(X_i\) is a set of measured covariates, and \(\varepsilon_i\) is the residual. The coefficient \(\beta_D\) is the conditional association of the child’s score with the outcome, holding the parental nontransmitted score fixed. The coefficient \(\gamma_N\) is the conditional association of the nontransmitted score, holding the child’s score fixed. If the model and identification assumptions are credible, \(\gamma_N\) can be interpreted as evidence of an indirect parental genetic effect. It is not, by itself, proof that a particular parental behavior mediates the association.

For a causal interpretation, the score must represent relevant genetic variation, the model must describe the associations adequately, and important sources of confounding must be controlled or removed by design. In particular, the nontransmitted score can remain associated with family background even though those alleles are not present in the child. Covariate adjustment is not a guarantee that all such structure has been removed.

What a Transmission Contrast Identifies

Conditional on a parent’s genotype, which allele is transmitted is determined by Mendelian segregation. For a parent with dosage \(G\), the transmitted dosage \(T\) has conditional expectation \(G/2\): if \(G=0\), \(T=0\); if \(G=1\), either allele is transmitted with probability one-half; and if \(G=2\), \(T=1\). Define the transmission deviation \(Z=T-G/2\). Then \(E(Z\mid G)=0\), and the nontransmitted dosage satisfies \(N=G-T=G/2-Z\).

This identity shows both the usefulness and the limitation of the contrast. Suppose, for one parent and one locus, the outcome model contains a direct transmitted-allele effect \(\beta_D T\) and an indirect nontransmitted-allele effect \(\gamma_N N\). Holding that parent’s genotype fixed, a change in transmission from \(T\) to \(T+\Delta\) changes \(N\) to \(N-\Delta\). The corresponding change in the modeled outcome is

Thus, the within-parent transmission contrast identifies a difference between the transmitted and nontransmitted coefficients in this model; it does not, on its own, separate the two coefficients. Separating them requires information across parental genotypes and families, together with assumptions about population structure, parental environments, mating patterns, and the score model. The standard trio regression uses both transmitted and nontransmitted score variation; interpreting its coefficients depends on those assumptions.

Assumptions and Sources of Confounding

The family structure helps define useful contrasts, but several problems can make a nontransmitted-score association misleading.

• Ancestry and population structure. Allele frequencies and social or environmental conditions can vary across ancestry groups. If ancestry is associated with both parental scores and the child’s outcome, a nontransmitted-score association may partly reflect that shared structure. Adjustment for ancestry-related covariates can help, but a small set of covariates may not capture all relevant differences. Within-family transmission comparisons remove some between-family structure because the parent’s genotype is held fixed, but they do not automatically solve every modeling or measurement problem.
• Assortative mating. When partners choose each other partly on correlated traits, their genotypes and scores may be correlated. Then maternal and paternal transmitted and nontransmitted scores need not behave like independent draws. A score from one parent may also track the other parent’s genotype and the shared family environment. Analyses should represent both parents and account for the intended mating and covariance structure; ignoring it can alter estimates and their interpretation.
• Parental genotype measurement. The nontransmitted dosage is calculated by subtracting the transmitted allele from the parental dosage. Errors in either quantity can therefore misclassify the nontransmitted allele. Incomplete parent data, incorrect parent-child links, or uncertain genotype calls can add further error. Ordinary nondifferential measurement error often weakens a score association, but errors in a reconstructed trio score need not act as simple independent noise, so the direction and size of bias are not guaranteed.
• Model and score limitations. A score captures only the variants and weights used to construct it. If the weights are poorly matched to the study population, or if relevant effects are omitted or modeled incorrectly, the fitted coefficients may not correspond to the genetic effects of interest. A nontransmitted score can also tag unmeasured parental characteristics rather than a specific environmental pathway.

Practical Interpretation and a Common Pitfall

A careful report distinguishes three claims: the observed association, the genetic-nurture interpretation, and a specific causal pathway. The first is a statistical result from the fitted model. The second requires the nontransmitted score to represent parental genetic influences on the child’s environment rather than residual ancestry, mating, or measurement structure. The third requires additional evidence about the environmental process itself. Reporting these levels separately prevents an association from being overstated.

A useful analysis checks that parental genotypes and transmissions are measured consistently, includes both parents when available, and states how ancestry and mating structure are handled. It should also report the score construction, the outcome and score units, the regression covariates, and uncertainty for the coefficients. The conditional coefficient on the nontransmitted score should be interpreted alongside the transmitted-score coefficient, not in isolation.

• Common mistake: “The child did not inherit these alleles, so their association with the child’s outcome must be a parental environmental effect.”
• Correction: Noninheritance rules out a direct effect through those alleles in that child, but it does not rule out confounding or measurement error. The nontransmitted-score association is consistent with genetic nurture only when the design and assumptions support that interpretation.