Why Regress a Phenotype on Local Ancestry?
At a genomic locus in an admixed individual, the two inherited chromosome copies may have different ancestral origins. A local ancestry dosage records how many copies at that locus are assigned to a specified ancestry. Regression of a phenotype on this dosage asks whether the phenotype varies, on average, with the number of copies assigned to that ancestry. Covariates can be included to compare individuals with similar measured characteristics or ancestry structure.
This is an association analysis, not a causal experiment. A nonzero association may reflect genetic effects, environmental or social differences correlated with ancestry, residual population structure, or other factors. The coefficient describes a model-based conditional difference; it does not by itself identify a causal mechanism.
Linear Regression for a Quantitative Phenotype
For a quantitative phenotype \(Y_i\), a basic covariate-adjusted model is
Here \(\mathbf{C}_i\) is a vector of covariates, such as age, sex, study site, or ancestry principal components; \(\beta_0\) is the intercept; and \(\beta_L\) is the local ancestry coefficient. With \(L_i\) coded as 0, 1, or 2, \(\beta_L\) is the expected phenotype difference per additional A-assigned copy, conditional on the included covariates. The model predicts a difference of \(2\beta_L\) between dosages 2 and 0, holding covariates fixed.
Ordinary least squares (OLS) estimates the coefficients by minimizing the sum of squared residuals. For the usual standard errors and t tests to have their conventional interpretation, assume a correctly specified conditional mean, independent errors with mean zero and constant variance, and no exact linear dependence among predictors. Exact small-sample t inference additionally assumes normally distributed errors. In genomic cohorts, relatedness or other dependence can violate independence; an appropriate mixed model or other dependence-aware method may then be needed.
Covariate adjustment changes the question being answered: the model estimates a local dosage association conditional on the specified covariates. Covariates should be selected using subject-matter knowledge and the intended estimand, not simply because they change a p-value. In particular, adjustment cannot guarantee removal of unmeasured confounding, and conditioning on a variable affected by the process under study can alter the association being estimated.
Dosage Coding and the Meaning of the Test
A dosage model treats 0, 1, and 2 as equally spaced values and assumes the conditional mean at dosage 1 lies halfway between the means at dosages 0 and 2. This is a one-degree-of-freedom linear trend test. It is not the same as testing whether all three dosage categories have the same mean.
If the trend assumption is not appropriate, represent dosage as a three-level factor, with dosage 0 as the reference. The model then includes separate indicators for dosage 1 and dosage 2. A joint two-degree-of-freedom test asks whether both category coefficients are zero. This allows a non-linear pattern but typically uses more degrees of freedom and may have lower power when a linear trend is a reasonable description.
For more than two ancestral populations, let \(L_{ik}\) be the number of copies assigned to ancestry \(k\). At a diploid locus, the dosages sum to two. Including an intercept and every ancestry dosage therefore creates exact collinearity. One standard parameterization omits a reference ancestry: each included coefficient then compares replacing one copy of the reference ancestry with one copy of the named ancestry, conditional on other model terms. The coefficients are not independent ancestry-specific causal effects.
Worked Example: An Unadjusted Per-Copy Association
Consider a hypothetical sample of six unrelated individuals, with a quantitative phenotype measured in score units. Their local ancestry dosages and phenotype values are:
| Individual | \(L_i\) | \(Y_i\) (score units) |
|---|---|---|
| 1 | 0 | 10 |
| 2 | 0 | 12 |
| 3 | 1 | 12 |
| 4 | 1 | 14 |
| 5 | 2 | 14 |
| 6 | 2 | 16 |
Fit \(Y_i=\beta_0+\beta_L L_i+\varepsilon_i\). The means are \(\bar L=1\) and \(\bar Y=13\). The OLS slope is
The intercept is \(\hat{\beta}_0=\bar Y-\hat{\beta}_L\bar L=13-2(1)=11.000\). The fitted means at dosages 0, 1, and 2 are therefore 11, 13, and 15 score units. Each additional A-assigned copy is associated with a 2.000 score-unit higher fitted mean in this unadjusted model.
The residuals are \((-1,1,-1,1,-1,1)\), so \(SSE=6\), \(df=6-2=4\), and \(MSE=6/4=1.5\). Since \(S_{LL}=4\), the slope standard error is \(SE(\hat{\beta}_L)=\sqrt{1.5/4}=0.6124\). Thus \(t=2.000/0.6124=3.266\), with 4 degrees of freedom; the two-sided p-value is approximately 0.031. Under the regression assumptions, this is evidence against \(H_0:\beta_L=0\) at \(\alpha=0.05\). The sample is deliberately small for arithmetic; its p-value is illustrative, and the normal-error assumption would require particular scrutiny in such a small sample.
Adjustment and Inference
The adjusted coefficient is the dosage association after accounting for the modeled covariates. A useful justification follows from the partial-regression identity. Let \(\mathbf{C}\) contain the intercept and all covariate columns, and let \(M_C=I-P_C\), where \(P_C\) projects onto the columns of \(\mathbf{C}\). Residualize both the phenotype and dosage against those covariates. The OLS dosage coefficient is
To see why, write the least-squares objective as \(\|\mathbf{Y}-\mathbf{C}\boldsymbol{\gamma}-\beta_L\mathbf{L}\|^2\). For any fixed \(\beta_L\), the best-fitting covariate component is the projection onto \(\mathbf{C}\). Removing that projection leaves the residualized outcome \(M_C\mathbf{Y}\) and residualized dosage \(M_C\mathbf{L}\). Minimizing the remaining squared residuals with respect to \(\beta_L\) gives the displayed slope formula. Thus adjustment compares the parts of phenotype and dosage not linearly explained by the included covariates; it does not make the dosage randomized.
For a single dosage coefficient, the usual test is \(H_0:\beta_L=0\) against \(H_a:\beta_L\ne0\), using \(t=\hat{\beta}_L/SE(\hat{\beta}_L)\). A confidence interval is \(\hat{\beta}_L\pm t_{1-\alpha/2,df}SE(\hat{\beta}_L)\). It describes uncertainty in the model coefficient; it is not a probability statement about whether the fixed coefficient lies in this particular interval.
Worked Example: Interpreting a Covariate-Adjusted Fit
Suppose a hypothetical analysis of 500 unrelated individuals fits the linear model with local dosage, age, sex, and ancestry principal components. The fitted dosage coefficient is 1.20 score units per A-assigned copy, with \(SE=0.40\). Assume the model's residual degrees of freedom are 490 and the standard-error assumptions are judged reasonable.
The test statistic is \(t=1.20/0.40=3.00\). For a two-sided test, the p-value is approximately 0.0028 with 490 degrees of freedom. At \(\alpha=0.05\), the data are inconsistent with a zero conditional dosage coefficient under the model.
Using \(t_{0.975,490}\approx1.965\), the 95% confidence interval is \(1.20\pm1.965(0.40)=1.20\pm0.786\), or approximately \((0.414,1.986)\) score units per copy. The fitted conditional mean difference between dosage 2 and dosage 0 is \(2(1.20)=2.40\) score units, with the corresponding coefficient-scaled interval \((0.828,3.972)\). These statements concern association conditional on the listed covariates; they do not establish that ancestry itself causes the phenotype difference.
Alternatives, Diagnostics, and Limits
The appropriate model depends on the outcome scale and the scientific contrast. For a binary outcome, logistic regression can model the log odds as a function of dosage and covariates; its dosage coefficient is a conditional log-odds change per copy, and exponentiating it gives an odds ratio per copy. A categorical dosage model is useful when a constant per-copy trend is doubtful. For related participants or other dependence structures, ordinary independent-error standard errors may be unsuitable, so a dependence-aware model should be considered.
| Approach | What it estimates or tests | Main trade-off |
|---|---|---|
| Linear dosage model | One conditional mean change per additional ancestry copy | Concise and often efficient, but assumes a linear dosage trend |
| Categorical dosage model | Separate differences for dosage 1 and 2 versus dosage 0 | Allows non-linearity but uses an additional parameter |
| Logistic dosage model for a binary outcome | Conditional log-odds change per copy | Matches a binary outcome, but the coefficient is not a risk difference |
Inspect the dosage counts and phenotype distribution before interpreting a fit. Sparse or absent dosage categories make comparisons unstable or unidentified. Check residual plots for nonlinearity, unequal variance, and influential observations; assess whether the dosage is strongly predictable from covariates, since near-collinearity inflates uncertainty. Consider relatedness and other dependence, and examine whether conclusions are sensitive to reasonable covariate specifications. A dosage association can also reflect unmeasured structure or environmental differences, and local ancestry estimates themselves may be uncertain. This model treats the supplied dosage as the predictor; it does not propagate that uncertainty.
Common Mistakes
- Calling the coefficient an effect of ancestry. A regression association is not proof of causation. Describe it as a conditional phenotype difference per dosage copy.
- Interpreting the coefficient as the full 0-to-2 contrast. Under linear dosage coding, the fitted contrast is \(2\hat{\beta}_L\), not \(\hat{\beta}_L\).
- Assuming adjustment solves population structure. Adjustment controls only modeled covariates in the specified form; omitted or poorly measured structure can remain.
- Treating dosage coding as assumption-free. The 0, 1, 2 trend imposes equal increments in the conditional mean. Use category indicators when that restriction is not defensible.
- Using ordinary standard errors with dependent observations. Relatedness or other dependence can invalidate the independence-based standard errors even when the fitted mean model looks plausible.
Check Your Understanding
Answer each question, then compare with the solutions below.
- In a linear dosage model, what phenotype contrast does a coefficient of 0.7 represent between dosage 2 and dosage 0?
- Why might a categorical dosage model be preferable to a linear dosage model?
- For three ancestry dosages that sum to two, why can an intercept and all three dosage columns not be included together?
- A binary-outcome logistic model has a dosage coefficient of 0.4. What is the per-copy odds ratio, approximately?
Solutions. (1) The fitted contrast is \(2(0.7)=1.4\) phenotype units, conditional on covariates. (2) It does not impose equal mean increments from dosage 0 to 1 and from 1 to 2, though it uses more parameters. (3) The dosage columns sum exactly to twice the intercept column, causing exact collinearity; omit a reference ancestry or use another identifiable parameterization. (4) The per-copy odds ratio is \(\exp(0.4)\approx1.492\), a conditional odds multiplier, not a risk difference or causal effect.