Peer Genetic Effects In Social Groups

Why Peers’ Genotypes Can Matter

An individual’s phenotype can be related not only to their own genotype but also to the genotypes of people around them. Peers may influence behavior, resources, or other features of a shared environment. For example, a person’s genotype could be associated with how they respond to a group, while peers’ genotypes could be associated with the group’s behavior. A peer-genotype coefficient describes the latter association in a specified model; by itself, it does not establish a biological or social mechanism, or a causal effect.

The key modeling task is to define who counts as a peer and how their genotypes are summarized. The group definition and exposure measure must match the hypothesized pathway. An average across assigned teammates, for instance, answers a different question from an average across friends or household members.

Definition: Let $g$ index groups and $i$ individuals within group $g$. Let $x_{ig}$ be individual $i$’s genotype dosage, coded as the number of copies of a specified allele, from 0 to 2. For a group of $m_g$ members, the leave-one-out mean peer genotype is $$ z_{ig}=\frac{1}{m_g-1}\sum_{j\ne i}x_{jg}. $$ It summarizes the genotypes of the other group members, not the individual’s own genotype. A peer genetic effect is the modeled association between this exposure and the individual’s phenotype, conditional on the other terms in the model.

A more general social network can use nonnegative weights $w_{ijg}$ to describe how much individual $j$ contributes to individual $i$’s exposure:

z_{ig}=\sum_{j\ne i}w_{ijg}x_{jg},\qquad \sum_{j\ne i}w_{ijg}=1.

The leave-one-out mean is the special case in which every other group member has equal weight. The weights must exclude the individual’s own genotype and should be set using the intended definition of social exposure, rather than chosen to improve a result.

A Model for Direct and Peer Effects

A basic linear model separates an individual’s own genotype from their peers’ genotype exposure:

y_{ig}=\beta_0+\beta_D x_{ig}+\beta_P z_{ig}+\boldsymbol{\gamma}^{\mathsf T}\mathbf{c}_{ig}+u_g+\varepsilon_{ig}.

Here $y_{ig}$ is the phenotype, $\beta_D$ is the direct-genotype coefficient, and $\beta_P$ is the peer-genotype coefficient. The vector $\mathbf{c}_{ig}$ contains measured covariates, with coefficient vector $\boldsymbol{\gamma}$. The term $u_g$ represents a group-level influence shared by members, and $\varepsilon_{ig}$ is an individual residual. A group fixed-effect model replaces the group-level term with a separate intercept for each group.

If the model is correctly specified, $\beta_P$ is the expected phenotype difference per one-unit increase in the peer-genotype exposure, comparing individuals at the same own genotype and covariate values, and with the same modeled group terms. Its units are phenotype units per unit of peer dosage. A one-unit increase in $z_{ig}$ means a one-copy increase in the average peer dosage; it does not mean that every peer’s dosage rose by one.

The coefficients are interpretable as conditional associations under a linear model. A causal interpretation requires additional design and assumptions: in particular, the peer exposure must not stand in for unaccounted causes of the phenotype, and group formation must not create a spurious association between peer genotypes and outcomes. Adjusting for a group-level term does not automatically address all such concerns.

Model conditions: Before interpreting $\beta_P$, specify the phenotype, group membership, peer weights, and covariates. The conditional mean must be adequately represented by the model, including its assumed linear dosage effects. The genotype and peer-exposure columns must contain enough independent variation to distinguish their coefficients. For standard uncertainty estimates, the dependence of outcomes within groups must also be handled appropriately, for example by a method that accounts for group clustering. These conditions support estimation; they do not, by themselves, prove causation.

Worked Example: Calculating a Peer Exposure

Suppose a hypothetical group has four members with genotype dosages 0, 1, 1, and 2. Consider the member with dosage 1. Their three peers have dosages 0, 1, and 2, so the leave-one-out exposure is

z_{ig}=\frac{0+1+2}{3}=1.

Suppose a fitted model, in hypothetical task-score points, has $\beta_D=1.2$ points per own dosage and $\beta_P=0.8$ points per unit of peer mean dosage. Holding covariates and group terms fixed, the peer-genotype contribution for this individual is

\beta_P z_{ig}=(0.8)(1)=0.8\text{ points}.

If the peer mean were 1.5 rather than 1.0, the model’s predicted phenotype would differ by

0.8(1.5-1.0)=0.4\text{ points}.

This is a model-based comparison in the stated units, not evidence that changing the peers’ genotypes would cause a 0.4-point change.

What the Group Structure Can Identify

Identification depends on the joint pattern of own genotypes and peer exposures, not just the number of individuals. If the peer exposure is an exact linear combination of own genotype and group indicators, the model cannot separately estimate $\beta_D$ and $\beta_P$. More observations with the same uninformative structure do not resolve that problem.

For equal-sized groups of size $m$, let $S_g=\sum_j x_{jg}$ be the group’s total dosage. Then

z_{ig}=\frac{S_g-x_{ig}}{m-1} =\frac{S_g}{m-1}-\frac{x_{ig}}{m-1}.

With group fixed effects, the first term is constant within each group and is absorbed by its group intercept. The remaining within-group variation in $z_{ig}$ is exactly $-1/(m-1)$ times the variation in $x_{ig}$. Thus, in equal-sized groups, the two predictors cannot be separately estimated alongside group fixed effects. This is a structural limitation, not a small-sample inconvenience.

Worked Example: Why Pairs Cannot Separate the Effects With Group Fixed Effects

Consider a hypothetical pair with genotype dosages 0 and 2. For the member with dosage 0, the peer dosage is 2; for the member with dosage 2, the peer dosage is 0. Let the two observed task scores be 5 and 7, respectively. A pair-specific intercept is included.

One parameterization reproduces these scores with $\beta_D=1$, $\beta_P=0$, and group intercept 5: the predictions are $5+1(0)+0(2)=5$ and $5+1(2)+0(0)=7$.

A different parameterization gives the same predictions: $\beta_D=2$, $\beta_P=1$, and group intercept 3. The predictions are $3+2(0)+1(2)=5$ and $3+2(2)+1(0)=7$. The data therefore do not distinguish these coefficient pairs.

In general, for a pair, $z_{ig}=S_g-x_{ig}$. Substitution into the model gives

\alpha_g+\beta_Dx_{ig}+\beta_Pz_{ig} =(\alpha_g+\beta_PS_g)+(\beta_D-\beta_P)x_{ig}.

Only the direct-minus-peer contrast is identified within pairs when each pair has its own intercept; the separate coefficients are not. This is why a study needs informative exposure patterns, not merely many pairs.

Variation in group size can change this dependence. With group size $m_g$, peer mean dosage has a within-group slope of $-1/(m_g-1)$ against own dosage. If groups of different sizes contain sufficient genotype variation, these slopes differ, which can make the two coefficients distinguishable in a fixed-effect model. Other network structures can also produce informative variation. Whether they do so is determined by the actual design matrix, not by group size alone.

Worked Example: Contrasts Across Groups of Different Sizes

Consider two hypothetical groups and a model with group-specific intercepts. In a pair, dosages are 0 and 2, so the peer dosages are 2 and 0. The observed scores are 12 and 14, giving the within-pair contrast $14-12=2$. Under the model, the higher-dosage member has an own-genotype increase of 2 and a peer-exposure decrease of 2. Therefore,

2=2\beta_D-2\beta_P.

In a group of three, dosages are 0, 1, and 2. Their peer means are 1.5, 1, and 0.5, respectively. Suppose the scores are 21.5, 23, and 24.5. The contrast between the dosage-2 and dosage-0 members is $24.5-21.5=3$. Own dosage rises by 2 and peer mean falls by 1, so

3=2\beta_D-\beta_P.

The first contrast implies $\beta_D-\beta_P=1$. Subtracting this equation from $2\beta_D-\beta_P=3$ gives $\beta_D=2$; substituting back gives $\beta_P=1$. The two group sizes provide different within-group relationships between own genotype and peer mean, allowing these coefficients to be separated in this constructed example. This arithmetic illustrates identification under the specified model; it does not verify the model’s assumptions in an actual study.

Assumptions and Interpretation in Practice

A peer-genotype analysis requires a defensible account of group formation. People may enter groups through assignment, shared location, friendship, or other processes. If factors related to phenotype also influence group membership, peer genotype can be associated with those factors even when the peer genotype itself has no effect on the phenotype. Group covariates, group fixed effects, or a design with exogenous assignment may address some concerns, but each strategy has limits. For example, group fixed effects control shared group-level differences while also removing between-group information and, in uniform groups, can produce the non-identification just derived.

The model also imposes a particular exposure summary. A mean treats the same change in average peer dosage as equivalent regardless of which peer carries it. A weighted network exposure instead makes the assumed social connections explicit. Either summary may be inadequate if only certain peers matter, if effects are nonlinear, or if one peer’s genotype changes the influence of another. Such alternatives should be motivated and specified rather than inferred from the coefficient’s label.

When group members’ outcomes are dependent, treating all individual observations as independent can understate uncertainty. Inference should reflect the sampling and group structure, such as by using group-level clustering when its assumptions are appropriate. The number and diversity of groups matter: a large number of people concentrated in very few groups may provide limited information about between-group exposure patterns.

Common Mistakes

Including the individual in the peer average. This makes exposure partly a transformation of own genotype. Define peers with $j\ne i$, or use weights with $w_{ii}=0$.
Calling $\beta_P$ a causal effect automatically. The coefficient is conditional on the fitted model. Group sorting, omitted causes, or an unsuitable exposure definition can prevent a causal interpretation.
Assuming large sample size fixes collinearity. If group structure makes own genotype and peer exposure linearly dependent after adjustment, more observations with that same structure cannot separate their coefficients.
Interpreting a group association as an individual mechanism. A peer coefficient does not identify whether the pathway involves behavior, a shared environment, or another process.

Key takeaway: A peer genetic effect is defined relative to a particular group and peer-exposure measure. Estimating it separately from a direct genetic effect requires independent variation in own genotype and peer exposure after accounting for the model’s other terms. Interpret the coefficient conditionally, in phenotype units, and with attention to group formation and dependence.

Check Your Understanding

Use these questions to check your understanding of peer-genotype models.

In a group of five, the other four members’ dosages are 0, 1, 1, and 2. What is the leave-one-out peer mean for the focal individual?
What does a peer-genotype coefficient of 0.6 phenotype units per dosage unit mean in a linear model, holding its other terms fixed?
Why can’t a model with pair-specific fixed effects separate direct and peer coefficients when peer exposure is the other member’s dosage?
How can variation in group sizes help distinguish own genotype from peer mean when group fixed effects are included?
Name one reason a peer-genotype association might not have a causal interpretation, even if the model includes group terms.