Maternal And Paternal Genetic Effects

Why Parent and Offspring Genotypes Must Be Considered Together

An offspring’s genotype is partly inherited from each parent. A parent’s genotype can also be associated with the offspring’s outcome through pathways that do not require the relevant allele to be present in the offspring. For example, a maternal genotype could be related to the prenatal environment or to later caregiving; a paternal genotype could be related to the environment the family provides. These are distinct from an effect of the offspring’s own genotype, even though all three genotypes are correlated.

The statistical challenge is therefore not simply to ask whether a parent’s genotype predicts an offspring outcome. It is to distinguish that association from the offspring’s inherited genotype and from shared family circumstances. A coefficient for a parental genotype in a joint model is a conditional association; interpreting it as a causal maternal or paternal effect requires additional assumptions about confounding and the family process.

These effects are also different from parent-of-origin effects. Imprinting concerns whether an allele’s effect depends on whether the offspring inherited it from the mother or father. A maternal genetic effect concerns an association with the mother’s genotype, potentially through a pathway outside the offspring’s inherited copy. Both can occur, but they are not interchangeable.

Definition: A direct offspring genetic effect is an effect of the offspring’s own genotype on the offspring’s outcome. A maternal genetic effect is an effect of the mother’s genotype on the offspring’s outcome through a pathway not reducible to the offspring’s own genotype; a paternal genetic effect is defined analogously for the father. A parental effect may operate through prenatal conditions, caregiving, or other family pathways. The term describes a pathway of influence, not a claim that the parent’s genotype acts directly on the offspring’s cells.

A Joint Parent–Offspring Model

For one biallelic variant, let \(G_D\), \(G_M\), and \(G_F\) denote the offspring’s, mother’s, and father’s allele dosages, respectively. Each dosage counts copies of the same designated allele and takes values 0, 1, or 2. Centering the dosages can make the intercept easier to interpret; it does not change the meaning of the slopes. Let \(Y\) be the offspring outcome, and let \(C\) represent measured covariates such as age, sex, or ancestry variables.

Y_i = \beta_0 + \beta_D G_{D,i} + \beta_M G_{M,i} + \beta_F G_{F,i} + C_i^\top\gamma + \varepsilon_i.

In this additive linear model, \(\beta_D\) is the expected outcome difference per additional offspring allele, comparing offspring with different dosages at the same parental dosages and covariates. The coefficients \(\beta_M\) and \(\beta_F\) are the expected outcome differences per additional maternal or paternal allele dosage, conditional on the offspring genotype and the other parent’s genotype. If the outcome is measured in centimeters, for example, the coefficients are in centimeters per allele copy.

A standard interpretation of these slopes requires a correctly specified conditional mean: the included terms must capture the relevant genotype–outcome relationships, including nonlinearity or interactions if they are present. For ordinary least-squares uncertainty estimates, observations must also satisfy the error assumptions used by the chosen variance estimator. With one offspring per independent family, independence across families may be plausible; with siblings, outcomes within a family are not generally independent, so the analysis must account for clustering or model the family covariance.

The joint model helps separate effects because it conditions on all three genotypes. It does not by itself remove every source of bias. In particular, parental genotypes can be correlated with ancestry, assortative mating, family circumstances, and one another. If a measured or unmeasured family factor is associated with both parental genotype and the offspring outcome, a parental coefficient can reflect that factor as well as a genetic effect.

Worked Example: Bias From Omitting a Correlated Maternal Genotype

Suppose a hypothetical study uses centered, standardized offspring and maternal genotype predictors, each with variance 1 and correlation 0.50. The true conditional model for a standardized offspring outcome is \(Y = 0.20G_D + 0.40G_M + \varepsilon\), where the error has mean zero conditional on both genotypes. What is the expected slope if the analyst regresses \(Y\) on \(G_D\) alone?

In a regression with an intercept, the population slope from regressing \(Y\) on \(G_D\) alone is \(\operatorname{Cov}(G_D,Y)/\operatorname{Var}(G_D)\). Under the stated model and error assumption,

\operatorname{Cov}(G_D,Y) = 0.20\operatorname{Var}(G_D) + 0.40\operatorname{Cov}(G_D,G_M) = 0.20(1) + 0.40(0.50) = 0.40.

Since \(\operatorname{Var}(G_D)=1\), the one-predictor slope is \(0.40/1=0.40\), not the conditional direct-effect coefficient \(0.20\). The extra \(0.20\) arises because offspring and maternal genotypes are correlated and the maternal genotype is omitted. This calculation illustrates omitted-variable bias; it does not show that every maternal association is causal.

Transmission and Nontransmission

In a parent–offspring trio, each parent’s two alleles can be described in terms of the allele transmitted to the child and the allele not transmitted at that locus. Let \(T_M\) and \(N_M\) be the maternal transmitted and nontransmitted allele indicators, and define \(T_F\) and \(N_F\) similarly for the father. Each is 0 or 1 for a single allele copy at the locus. The offspring dosage is the sum of the two transmitted indicators; each parental dosage is the sum of that parent’s transmitted and nontransmitted indicators.

G_D = T_M + T_F,\qquad G_M = T_M + N_M,\qquad G_F = T_F + N_F.

A nontransmitted allele at the studied locus is not part of the offspring’s genotype at that locus. Its association with the outcome, conditional on the offspring genotype, can therefore be informative about a parental pathway outside direct inheritance. It is not automatically a causal estimate: population structure, assortative mating, correlated parental environments, selection, and other family-level factors can still create associations. Transmission also induces dependence among the predictors, so estimates may be imprecise even when the model is appropriate.

One can express the joint model using transmitted and nontransmitted alleles instead of total parental dosage. For example, the maternal contribution is \(\beta_M(T_M+N_M)\), so both the transmitted and nontransmitted maternal indicators occur in that term. This parameterization makes the bookkeeping explicit, but it does not create new information or remove confounding by itself. A useful interpretation of a nontransmitted-allele coefficient depends on which other predictors are included and on the assumptions of the design.

Worked Example: Decomposing Genotypes in Hypothetical Trios

Consider four hypothetical parent–offspring trios at one locus. The table records each parent’s dosage, the allele copy that parent transmitted, and the resulting offspring dosage. The nontransmitted copy is calculated by subtracting the transmitted indicator from the parent’s dosage.

Trio	Maternal dosage	Maternal transmitted	Maternal nontransmitted	Paternal dosage	Paternal transmitted	Paternal nontransmitted	Offspring dosage
1	2	1	1	0	0	0	1
2	1	0	1	2	1	1	1
3	2	1	1	1	1	0	2
4	0	0	0	1	0	1	0

For trio 2, the mother has dosage 1 and transmits 0, so her nontransmitted indicator is \(1-0=1\). The father has dosage 2 and transmits 1, so his nontransmitted indicator is \(2-1=1\). The offspring dosage is \(0+1=1\). Across the table, the maternal nontransmitted dosages are \(1,1,1,0\), and the paternal nontransmitted dosages are \(0,1,0,1\).

These records illustrate how to construct predictors for a trio analysis; four trios are not enough to establish a reliable effect. In a larger study, a regression could include offspring dosage and maternal and paternal nontransmitted dosage, together with prespecified covariates. A nonzero conditional coefficient would be an association requiring the design assumptions and confounding assessment described above before receiving a causal interpretation.

What the Coefficients Do—and Do Not—Establish

The key comparison in the joint model is conditional. A maternal coefficient is not the raw difference in offspring outcomes between families with different maternal genotypes. It compares families at the same modeled offspring and paternal genotypes and covariates. This can reduce confounding by the offspring’s measured genotype, but it cannot guarantee that the remaining maternal-genotype variation is unrelated to unmeasured family influences.

A causal interpretation would require, among other conditions, that relevant common causes of parental genotype and offspring outcome are adequately controlled or otherwise handled by the design; that the model represents the effects of interest; and that there is enough independent variation in the predictors to estimate their separate contributions. At a single locus, inheritance makes parent and offspring genotypes correlated by construction. Strong correlation can inflate standard errors, while exact collinearity would prevent separate coefficients from being estimated at all.

Maternal and paternal coefficients should also be compared cautiously. A larger maternal estimate does not, by itself, prove an intrauterine mechanism: maternal and paternal environments may differ, measurements may have different precision, and the estimates may be confounded differently. Evidence about the pathway generally requires more than a regression coefficient, such as carefully chosen family comparisons or additional measurements that address specific alternative explanations.

Worked Example: Interpreting Joint-Model Estimates

Suppose a hypothetical trio study models offspring birth weight in standard-deviation units using offspring, maternal, and paternal allele dosage and measured covariates. The fitted coefficients are \(\hat{\beta}_D=0.08\) with \(SE=0.10\), \(\hat{\beta}_M=0.30\) with \(SE=0.12\), and \(\hat{\beta}_F=0.05\) with \(SE=0.11\). Treat the following calculations as large-sample, two-sided Wald inference.

For the maternal coefficient, the Wald statistic is

z = \frac{\hat{\beta}_M}{SE(\hat{\beta}_M)} = \frac{0.30}{0.12} = 2.50.

A standard normal two-sided test gives \(p\approx 0.0124\). The approximate 95% confidence interval is

0.30 \pm 1.96(0.12) = 0.30 \pm 0.2352 = (0.0648,\ 0.5352).

Under the fitted model, this interval corresponds to an estimated conditional difference of about 0.065 to 0.535 outcome standard deviations per additional maternal allele copy, holding the included offspring and paternal dosages and covariates fixed. It excludes zero under this Wald procedure. This supports evidence of a conditional maternal-genotype association, not proof that a maternal biological or caregiving pathway caused the difference.

For comparison, the offspring statistic is \(0.08/0.10=0.80\), with an approximate two-sided \(p\)-value of 0.424; the paternal statistic is \(0.05/0.11\approx0.455\), with \(p\approx0.649\). These results do not establish that the direct or paternal effect is exactly zero. Their uncertainty intervals include both zero and nonzero values, and differences between coefficients should be tested directly rather than inferred merely from one \(p\)-value being smaller than another.

A Short Justification for the Omitted-Variable Formula

The bias calculation follows directly from the population regression slope. Suppose the true model is \(Y=\beta_DG_D+\beta_MG_M+\varepsilon\), with centered variables and \(\operatorname{Cov}(G_D,\varepsilon)=0\). If only \(G_D\) is used as a predictor, its population slope is \(\operatorname{Cov}(G_D,Y)/\operatorname{Var}(G_D)\). Substituting the true model and using the covariance rules gives

\begin{aligned} \frac{\operatorname{Cov}(G_D,Y)}{\operatorname{Var}(G_D)} &= \frac{\operatorname{Cov}(G_D,\beta_DG_D+\beta_MG_M+\varepsilon)} {\operatorname{Var}(G_D)}\\ &= \beta_D + \beta_M\frac{\operatorname{Cov}(G_D,G_M)}{\operatorname{Var}(G_D)}. \end{aligned}

Thus the omitted-variable term is zero if the maternal effect is zero or if maternal and offspring genotypes are uncorrelated. Parent–offspring inheritance generally makes the latter condition implausible. The formula is a population result for the stated linear model and error condition; it does not account for other omitted predictors or model misspecification.

Common Mistakes and a Practical Checklist

Calling a parental genotype association a direct offspring effect. If parental genotype is omitted, inherited correlation can contribute to the offspring-genotype slope. Include the relevant family genotypes or use an appropriate family design.
Treating a nontransmitted allele as an automatic causal control. It is absent from the offspring’s genotype at that locus, but its association can still reflect population structure, family-level confounding, or selection.
Equating a maternal coefficient with a prenatal mechanism. A conditional maternal association can arise through multiple pathways and can be confounded; the regression alone does not identify the pathway.
Interpreting a nonsignificant coefficient as no effect. A large standard error or collinearity can make an estimate imprecise. Report the estimate and interval, not only the test result.
Ignoring related observations. Siblings share parents and environments. Standard errors that assume independent offspring may be too small unless within-family dependence is addressed.

Before interpreting a family-based estimate, check how genotypes were coded, which genotypes and covariates were included, whether the predictors are sufficiently variable, and how related observations and uncertainty were handled. Then state the coefficient as a conditional association in the outcome’s units, and identify the assumptions needed for any causal claim.

Check Your Understanding

Use the model and distinctions in this tutorial to answer the following questions.

In the joint model, what does the maternal genotype coefficient compare, and why is it not automatically a causal effect?
A parent has dosage 2 and transmits one copy of the allele. What is the nontransmitted dosage at the locus?
Why can a regression of offspring outcome on offspring genotype alone produce a slope different from the conditional direct-effect coefficient?
What can a nontransmitted-allele association suggest, and name two reasons it might not identify a causal parental pathway.
Why should an analyst account for family clustering when the data include multiple siblings?