Direct And Indirect Genetic Effects

Why Separate Direct and Indirect Genetic Effects?

A person's phenotype can be associated with their own genotype and with the genotypes of people around them. For example, a genotype could affect a person's own learning-related traits, while classmates' genotypes could affect the shared learning environment. These are conceptually distinct routes to phenotypic differences. A model that includes only an individual's genotype may combine them rather than estimate either route separately.

The distinction is about whose genotype is the source of the effect on the focal person's phenotype. It does not by itself identify a molecular mechanism, establish causation, or specify a particular social relationship. A direct genetic effect may operate through many biological or behavioral pathways within the focal individual. An indirect genetic effect is attributable to the genotype of another person and reaches the focal phenotype through that person's influence or through an environment associated with that genotype.

Definition: A direct genetic effect is the effect on an individual's phenotype of changing that individual's genotype while holding the relevant genotypes of other people fixed. An indirect genetic effect is the effect on that individual's phenotype of changing one or more other people's genotypes while holding the individual's own genotype fixed. The effects are defined relative to a specified phenotype, set of social partners, and genotype contrast.

These definitions describe estimands: precisely specified quantities a study aims to estimate. They are not interchangeable with the coefficients from any regression that happens to include genotype variables. The model, genotype coding, partner set, and assumptions determine what a coefficient represents.

Potential Outcomes and the Two Contrasts

Let \(Y_i(g_i,\mathbf{g}_{N_i})\) denote the phenotype that individual \(i\) would have under own genotype \(g_i\) and social-partner genotypes \(\mathbf{g}_{N_i}\). Here \(N_i\) is the defined set of partners whose genotypes could affect \(i\)'s phenotype. A genotype may be coded as an allele count, such as 0, 1, or 2 copies of a specified allele. For a quantitative phenotype, effects are measured in that phenotype's units.

For two own-genotype values \(g\) and \(g'\), the direct effect at a fixed partner-genotype configuration \(\mathbf{g}_{N_i}\) is the contrast \(Y_i(g,\mathbf{g}_{N_i})-Y_i(g',\mathbf{g}_{N_i})\). For two partner-genotype configurations \(\mathbf{h}\) and \(\mathbf{h}'\), the indirect effect at fixed own genotype \(g_i\) is \(Y_i(g_i,\mathbf{h})-Y_i(g_i,\mathbf{h}')\). These contrasts make explicit what is held constant.

A population-average direct effect averages the direct contrast over a stated distribution of partner genotypes and other relevant characteristics. An average indirect effect similarly averages the partner-genotype contrast over a stated distribution of own genotypes and other characteristics. If effects vary across those backgrounds, changing the averaging distribution can change the reported average. Thus an estimand should state both the contrast and the population or distribution over which it is averaged.

Interpretation: A direct-effect estimand compares own genotypes at fixed partner genotypes; an indirect-effect estimand compares partner genotypes at fixed own genotype. If the effects depend on the fixed background, report that background or the distribution used to average over it.

A Linear Model for Distinguishing the Effects

A simple model can represent the two sources of genetic effect. Let \(G_i\) be the focal individual's allele count, and let \(S_i\) be a defined summary of partner genotypes. For example, \(S_i\) could be the sum of allele counts across specified partners, or a weighted average if partners contribute with different prespecified weights. Let \(\mathbf{C}_i\) contain measured covariates.

Y_i=\mu+\beta_D G_i+\beta_I S_i+\mathbf{C}_i^\mathsf{T}\boldsymbol{\gamma}+\varepsilon_i

In this additive model, \(\beta_D\) is the expected phenotype difference for a one-allele increase in own genotype, comparing individuals with the same \(S_i\) and covariates. Its units are phenotype units per allele. The coefficient \(\beta_I\) is the expected phenotype difference for a one-unit increase in the specified partner-genotype summary, comparing individuals with the same own genotype and covariates. Its units depend on how \(S_i\) is defined. If \(S_i\) is a sum of partner allele counts, a one-unit change is one additional allele across that sum; if it is an average, it is one allele in the average.

Interpreting these coefficients as causal effects requires more than fitting the model. The partner set and genotype summary must represent the relevant exposure; the functional form must be adequate; and factors associated with both genotype and phenotype, such as ancestry or social selection, must be addressed appropriately. The error term must not retain a source of confounding that biases the contrasts. In particular, genotypes are not automatically independent of environmental or social conditions in a sampled population. Without suitable design and assumptions, coefficients describe adjusted associations rather than causal effects.

The displayed model also assumes no interaction between own and partner genotypes: it assigns the same direct effect at every value of \(S_i\), and the same indirect effect at every value of \(G_i\). This restriction is useful for illustration but may be unrealistic. When effects vary with context, the estimand must identify the context or the averaging distribution.

Worked Example: Reading Direct and Indirect Coefficients

Suppose a hypothetical study models a quantitative phenotype measured in score points. Its specified partner-genotype summary is the sum of allele counts for two social partners. The fitted conditional mean is \(\widehat{Y}_i=50+0.8G_i+0.3S_i\), with other covariates held fixed. Interpret the two genotype contrasts for a focal individual with \(G_i=1\) and \(S_i=2\).

The predicted score is \(50+0.8(1)+0.3(2)=51.4\) points. Increasing the focal person's allele count by one while keeping the partner summary at 2 changes the prediction by \(0.8(2-1)=0.8\) points. Thus the estimated direct contrast is 0.8 score points per allele, conditional on the specified partner summary and covariates.

Increasing the partner summary by one while keeping the focal genotype at 1 changes the prediction by \(0.3(3-2)=0.3\) points. The estimated indirect contrast is 0.3 score points per one-unit increase in the partners' summed allele count. The unit is not “per partner” unless the summary is defined on a per-partner scale.

These are model-based conditional contrasts. To treat them as causal effects, the study would also need the exposure, model, and confounding assumptions described above; the arithmetic alone does not establish those assumptions.

Why an Own-Genotype Association May Combine Effects

If an individual's genotype is correlated with the genotypes of their partners, a regression of phenotype on own genotype alone can reflect both direct and indirect contributions. This is a question of what a marginal association estimates, not a redefinition of the direct effect. A short derivation makes the distinction clear.

Assumptions for the derivation: Suppose the phenotype follows \(Y_i=\mu+\beta_DG_i+\beta_IS_i+\varepsilon_i\), the model has no interaction, and \(\operatorname{Cov}(G_i,\varepsilon_i)=0\). Consider the population least-squares slope from regressing \(Y_i\) on \(G_i\) alone, with an intercept. This slope is an association; the assumptions here do not by themselves make it causal.

\begin{aligned} b_{\mathrm{marginal}} &=\frac{\operatorname{Cov}(G_i,Y_i)}{\operatorname{Var}(G_i)}\\ &=\frac{\beta_D\operatorname{Var}(G_i)+\beta_I\operatorname{Cov}(G_i,S_i)+\operatorname{Cov}(G_i,\varepsilon_i)} {\operatorname{Var}(G_i)}\\ &=\beta_D+\beta_I\frac{\operatorname{Cov}(G_i,S_i)}{\operatorname{Var}(G_i)}. \end{aligned}

The second line follows by substituting the phenotype model into the covariance and using the fact that the covariance of \(G_i\) with a constant is zero. The final line uses the stated assumption that \(\operatorname{Cov}(G_i,\varepsilon_i)=0\). If own genotype and partner summary are uncorrelated, this particular source of difference between the marginal and direct coefficients vanishes. If they are correlated, the marginal slope includes a term proportional to the indirect coefficient. Other violations, such as confounding or misspecification, can also affect an estimated association.

Worked Example: A Marginal Slope Is Not the Direct Effect

In hypothetical population summaries, let \(\operatorname{Var}(G_i)=0.50\) allele-count units squared and \(\operatorname{Cov}(G_i,S_i)=0.25\) allele-count units squared. Suppose the conditional model has \(\beta_D=0.80\) score points per own allele and \(\beta_I=0.30\) score points per unit of partner summary, and assume \(\operatorname{Cov}(G_i,\varepsilon_i)=0\).

The marginal slope is \(b_{\mathrm{marginal}}=0.80+0.30(0.25/0.50)=0.80+0.15=0.95\) score points per own allele. The conditional direct coefficient remains 0.80 score points per allele; the additional 0.15 in the marginal slope is the indirect-effect term induced by the stated genotype covariance. This calculation illustrates a potential mixture in the marginal association, not evidence that the assumed model or covariance describes any real population.

Context-Specific Effects and Averaging

When the direct effect changes with partner-genotype exposure, a single direct coefficient may not summarize the relationship. For example, a model with a product term is

Y_i=\mu+\beta_DG_i+\beta_IS_i+\beta_{DI}G_iS_i+\varepsilon_i.

Holding \(S_i\) fixed, a one-unit increase in \(G_i\) changes the model's expected phenotype by \(\beta_D+\beta_{DI}S_i\). Holding \(G_i\) fixed, a one-unit increase in \(S_i\) changes it by \(\beta_I+\beta_{DI}G_i\). In this model, \(\beta_D\) alone is the direct slope only at \(S_i=0\), and \(\beta_I\) alone is the indirect slope only at \(G_i=0\). A population-average effect would average the relevant conditional contrast over a stated distribution of the other genotype variable.

Worked Example: Reporting a Direct Effect at a Specified Context

Consider a hypothetical fitted model for a phenotype in score points, with \(G_i\) measured as allele count and \(S_i\) as the mean partner allele count: \(\widehat{Y}_i=40+0.6G_i+0.2S_i+0.1G_iS_i\). Calculate the direct contrast for one additional own allele when \(S_i=1\), and compare it with the contrast when \(S_i=2\).

At \(S_i=1\), the predicted change for a one-allele increase is \(0.6+0.1(1)=0.7\) score points per allele. At \(S_i=2\), it is \(0.6+0.1(2)=0.8\) score points per allele. The model therefore estimates different direct effects in the two partner-genotype contexts. Reporting only \(0.6\) as “the direct effect” would omit the interaction and would describe the direct slope only at \(S_i=0\).

Effects Are Not Variance Components

A direct or indirect effect is a contrast in expected phenotype under specified genotype changes. A variance component instead describes variation across individuals or genotypes in a population under a particular model and set of assumptions. The two concepts are related but not identical: a nonzero effect does not guarantee substantial population variance if the relevant genotype variation is small.

For example, in the additive part of the model, the genetic contribution represented by \(\beta_DG_i+\beta_IS_i\) has variance

\operatorname{Var}(\beta_DG_i+\beta_IS_i) =\beta_D^2\operatorname{Var}(G_i)+\beta_I^2\operatorname{Var}(S_i) +2\beta_D\beta_I\operatorname{Cov}(G_i,S_i).

The covariance term shows why direct and indirect contributions cannot generally be converted to separate variance components by simply squaring their coefficients. Their population contributions depend on how own and partner genotypes vary and covary in the population being studied. This quantity is not, by itself, a heritability estimate: a heritability definition also requires a specified phenotype variance and a clearly defined genetic component.

Common Mistakes and Practical Interpretation

Calling every genotype coefficient a direct effect. A model containing only own genotype cannot separate it from an indirect contribution correlated with own genotype. The correction is to state what variables and contrasts the coefficient actually represents.
Using “direct” to mean a direct molecular mechanism. Here, “direct” distinguishes own genotype from other people's genotypes. It does not exclude mediation through the focal individual's own traits or environment.
Leaving the partner exposure undefined. “A one-unit increase in social genotype” is ambiguous without specifying which partners are included and whether their genotypes are summed, averaged, or weighted.
Treating an adjusted association as automatically causal. Conditioning on partner genotypes does not alone eliminate population structure, social selection, or other confounding. Causal interpretation depends on design and assumptions.
Reporting an interaction coefficient as the effect in every context. With effect modification, report the conditional contrast at a stated partner-genotype value or average it over a stated distribution.

Key takeaway: Direct and indirect genetic effects answer different contrast questions. Specify whose genotype changes, whose genotype is held fixed, how partners are defined, and the population or context for averaging. Then distinguish the resulting estimand from a marginal association or a variance component.

Check Your Understanding

Use the definitions and models above to answer these questions.

In a potential-outcomes definition, what is held fixed when defining a direct genetic effect?
A partner-genotype summary is the sum of allele counts for three partners. What does a one-unit change in that summary mean?
Under the additive model, what term causes the marginal own-genotype slope to differ from the direct coefficient when the residual is uncorrelated with own genotype?
In a model with an own-genotype-by-partner-summary interaction, what information is needed to report a direct effect?
Why can the variance of the combined direct and indirect genetic contribution not generally be calculated by adding only the two coefficient-squared terms?