Why the Target Population Is Part of the Estimand
A genetic association coefficient is not fully described by naming a variant and an outcome. It also depends on which people the coefficient describes and on whether it summarizes an association before or after conditioning on covariates. This matters when allele frequencies, covariate distributions, or outcome distributions differ across populations. A coefficient estimated in one cohort is not automatically a coefficient for another population, even if both analyses use the same variant and outcome.
Here, \(G\) denotes the number of copies of a specified allele, or an allele dosage on the corresponding 0-to-2 scale, and \(Y\) denotes a quantitative outcome. Let \(P\) represent a clearly specified target population, and let \(C\) denote a set of measured covariates. In a non-admixed cohort, \(P\) might be the population represented by a particular sampling frame. In an admixed cohort, \(P\) might be the enrolled cohort itself, with its particular distribution of ancestry composition and other characteristics. These labels describe target groups; they do not imply that individuals can be assigned to discrete biological categories.
“Marginal” does not mean that the analysis has established an unconfounded or causal effect. It means that the association is defined without conditioning on the specified covariates. The slope is a descriptive population quantity: it summarizes the linear projection of \(Y\) on \(G\) under the joint distribution of \(G\) and \(Y\) in \(P\). If that relationship is not linear, the slope remains a best linear summary, not a claim that every additional allele copy has the same effect at every genotype value.
Conditional Effects and the Role of Covariates
A conditional association asks a different question: how does the expected outcome vary with genotype among individuals held comparable on specified covariates? Under a linear conditional-mean model with a common genotype slope,
\(\beta_{\mathrm{C}}(P)\) is the conditional association slope for population \(P\). It is measured in outcome units per allele copy, conditional on the covariates in \(C\). The model assumes that the conditional mean is linear in \(G\) and in the included covariates, and that the genotype slope is constant over the covariate values represented in \(P\). If the genotype association varies with \(C\), a single conditional slope may not describe the relationship adequately; the model must represent that variation or state which covariate distribution is used to summarize it.
When the conditional-mean model is correct and includes an intercept, the coefficient can also be expressed through residuals. Regress \(G\) on the intercept and \(C\) in population \(P\), and call the residual \(G_{\mathrm{res}}\). Regress \(Y\) on the same terms, and call that residual \(Y_{\mathrm{res}}\). Then
This expression makes a key requirement visible: after accounting for \(C\), there must still be genotype variation. If genotype is completely determined by the covariates, the conditional slope cannot be identified from that population's data.
Worked Example: Different Marginal Slopes in Two Populations
Worked Example: Different Marginal Slopes in Two Populations
Suppose two hypothetical non-admixed cohorts are analyzed for the same allele dosage \(G\) and the same quantitative outcome \(Y\), measured in standardized outcome units. Assume the following are the exact population moments: in population A, \(\operatorname{Var}(G)=0.80\) and \(\operatorname{Cov}(G,Y)=1.20\); in population B, \(\operatorname{Var}(G)=1.00\) and \(\operatorname{Cov}(G,Y)=0.40\). Define each population's marginal effect by the linear-projection slope.
For population A,
For population B,
The estimated population quantities are therefore 1.50 and 0.400 standardized outcome units per additional allele copy, respectively. Each is an association in its own target population. The difference does not, by itself, show that a biological effect differs between populations: the marginal slopes can also differ because the populations have different distributions of other factors related to genotype and outcome. This calculation assumes positive genotype variance in each population, as the supplied variances confirm.
Why a Pooled Slope Is Not Necessarily a Population Average
A pooled analysis uses the joint distribution in the combined sample. That distribution includes both within-population variation and differences between population means. Write \(A\) for a population label with groups \(a\), and let \(\pi_a=P(A=a)\) be each group's proportion in the target mixture. The law of total covariance gives
The corresponding variance decomposition is
Consequently, the pooled marginal slope is the ratio of these two quantities. It includes between-group covariance and between-group genotype variation, so it is not generally a weighted average of the within-group slopes. This result follows by writing each variable as its within-group deviation plus its group mean. The within-group deviations have mean zero conditional on \(A\), so their cross-products with group means have expectation zero. The remaining covariance separates into the average within-group covariance and the covariance of group means; the same argument gives the variance decomposition.
Worked Example: A Pooled Slope Includes Between-Population Differences
Consider a hypothetical mixture containing equal proportions of populations A and B. Suppose their within-population moments are \(\operatorname{Var}_{A}(G)=0.80\), \(\operatorname{Cov}_{A}(G,Y)=0.40\), \(\operatorname{Var}_{B}(G)=1.20\), and \(\operatorname{Cov}_{B}(G,Y)=0.40\). Their genotype means are 0.50 and 1.50, and their outcome means are 2.00 and 6.00 standardized units. The within-population slopes are \(0.40/0.80=0.500\) in A and \(0.40/1.20=0.333\) in B.
The average within-population covariance is \(0.5(0.40)+0.5(0.40)=0.40\). The between-population covariance of the means is
Thus the pooled covariance is \(0.40+1.00=1.40\). The average within-population genotype variance is \(0.5(0.80)+0.5(1.20)=1.00\), and the between-population variance of genotype means is \(0.5(0.50-1.00)^2+0.5(1.50-1.00)^2=0.25\). The pooled variance is therefore 1.25, giving
The pooled slope, 1.12 standardized outcome units per allele copy, is neither within-population slope and is not their equal-weight average, \(0.417\). Much of its value comes from differences in the population means. The calculations assume the stated groups and mixture proportions define the target mixture and that the supplied moments describe those groups.
Marginal and Conditional Effects in an Admixed Cohort
An admixed cohort is itself a valid target population for an association estimand. Its marginal slope averages over that cohort's actual joint distribution of genotype, outcome, and ancestry composition. It should not be described as an effect “in” one ancestral source population merely because the cohort has ancestry contributions from multiple sources. A conditional slope can instead describe the association after accounting for specified measured covariates. The choice of covariates and the population in which the slope is defined must be stated. Defining these estimands does not prescribe a particular ancestry-adjustment method.
Worked Example: Covariate-Conditional Association in a Hypothetical Admixed Cohort
Suppose four hypothetical individuals from one admixed cohort have a measured covariate \(C\), coded 0 or 1, and the following values. \(Y\) is a quantitative outcome in standardized units.
| Individual | \(G\) (allele copies) | \(C\) | \(Y\) (standardized units) |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | 1 | 0 | 2 |
| 3 | 2 | 1 | 7 |
| 4 | 3 | 1 | 9 |
Assume the conditional-mean model \(E(Y\mid G,C)=\alpha+\beta_{\mathrm{C}}G+\gamma C\) is appropriate for this illustrative data set. The values follow \(Y=2G+3C\), so the conditional coefficient is \(\beta_{\mathrm{C}}=2\) standardized units per allele copy, with \(\gamma=3\) standardized units for \(C=1\) versus \(C=0\). This can be checked by subtracting the within-\(C\) means. For \(C=0\), the means are \(\bar{G}=0.5\) and \(\bar{Y}=1\); for \(C=1\), they are \(\bar{G}=2.5\) and \(\bar{Y}=8\). Within either covariate group, a one-copy difference corresponds to a two-unit difference in \(Y\).
The marginal slope is different. Across all four individuals, \(\bar{G}=1.5\) and \(\bar{Y}=4.5\). The mean of the products \((G-\bar{G})(Y-\bar{Y})\) is \((6.75+1.25+1.25+6.75)/4=4.00\); the mean of \((G-\bar{G})^2\) is \((2.25+0.25+0.25+2.25)/4=1.25\). Therefore,
Here \(G\) and \(C\) vary together, and \(C\) is also associated with \(Y\), so the marginal slope differs from the conditional one. This small example is an illustration of the estimands, not evidence about any actual cohort. Its conditional slope is identifiable in this example because genotype varies within both covariate groups. In a real analysis, the covariates, sample size, model fit, and uncertainty would also need careful assessment.
Assumptions, Diagnostics, and Limitations
An estimand is defined by its population and mathematical target; estimating it from data adds further assumptions. For a marginal linear-projection slope, genotype must vary in the target population, and the sampled individuals must support inference to the stated target. A conditional slope additionally requires residual genotype variation after conditioning on \(C\). If a common conditional slope is reported, the model should be checked for departures from linearity and for variation of the genotype association across covariate values. These conditions concern whether the chosen summary is estimable and adequate; they do not make it causal.
Diagnostics should be interpreted in relation to the intended estimand. Check genotype distributions and sample sizes within the populations being reported; sparse or absent genotype categories can make a slope unstable or unidentified. For conditional estimates, inspect overlap in genotype across covariate values and whether residual genotype variation is substantial. Compare fitted and observed outcome patterns across genotype and covariate groups, and examine whether a small number of observations dominate the fitted slope. In an admixed cohort, be explicit about whether the target is the enrolled cohort as observed or a population the sample is intended to represent.
A common alternative is to report a pooled association adjusted for covariates rather than separate population-specific coefficients. That may be a useful summary for a specified cohort and model, but it is not automatically a population-specific effect; its meaning depends on the covariates, model, and population mixture. Separate estimates make the target groups visible but can be imprecise, particularly when allele variation is limited. A pooled estimate can improve precision under an appropriate common-effect model, but assuming a common slope is a substantive restriction, not a consequence of pooling. Neither approach should be selected solely because its result is more statistically significant.
- Pitfall: “The pooled coefficient is the average genetic effect across ancestries.” This is generally wrong because the pooled slope can include between-group covariance, as well as within-group associations.
- Correction: State the target population and whether the coefficient is marginal or conditional. If reporting a pooled estimate, explain the population mixture and model that define it; do not label it ancestry-specific without a matching estimand.
- Pitfall: “The conditional coefficient is the causal effect of changing genotype while holding ancestry fixed.” Conditioning alone does not establish causality, and the coefficient's interpretation depends on the model, covariates, and assumptions.
Check Your Understanding
Answer each question by naming the estimand or showing the calculation. Solutions follow the questions.
- A target population has \(\operatorname{Cov}(G,Y)=0.60\) and \(\operatorname{Var}(G)=0.75\). What is its marginal slope, and what are its units if \(Y\) is measured in milligrams?
- Why does a pooled slope generally fail to equal the weighted average of two population-specific slopes?
- In a conditional model, what condition on genotype variation is needed to estimate a conditional slope?
- A study of an admixed cohort reports a marginal slope. What population does that estimate describe?
Solutions. (1) \(\beta_{\mathrm{M}}=0.60/0.75=0.800\) milligrams per additional allele copy. (2) The pooled covariance and variance each include between-population differences in means as well as within-population variation, so their ratio is not generally a weighted average of within-population slopes. (3) Genotype must retain positive residual variance after conditioning on the covariates; otherwise the conditional slope is unidentified. (4) It describes the joint genotype-outcome distribution of the specified admixed target cohort, including that cohort's ancestry-composition distribution, not automatically an ancestral source population.