Why Fit Local-Ancestry and SNP Models?
In an admixed cohort, the ancestry inherited at a particular genomic region can vary from one person to another. A phenotype association with that local ancestry is not the same question as an association with a particular SNP in the region. Local ancestry summarizes which ancestral backgrounds are present at the locus; a SNP genotype records the number of copies of a specified allele. Their associations can differ because allele frequencies and linkage disequilibrium patterns vary across ancestral backgrounds, and because local ancestry can capture genetic or environmental differences not represented by one SNP.
This distinction matters when interpreting an estimate. A local-ancestry coefficient describes a conditional difference in phenotype per additional ancestry copy, under the model. A SNP coefficient describes a conditional difference per additional copy of the coded allele. Neither coefficient, by itself, establishes a causal effect. The earlier material on identifiability of ancestry-specific effects is relevant: estimating ancestry-specific effects requires sufficient variation and overlap in ancestry and genotype, not just a fitted regression.
Posterior Dosage and Model Specification
For haplotype \(h\) of person \(i\), let \(q_{ih}\) be the posterior probability that the haplotype has the ancestry of interest, given the available genotype and reference information. The posterior local-ancestry dosage is \(L_i=q_{i1}+q_{i2}\), and can be fractional. It is not a probability that the person has a particular total ancestry count; it is the expected count of ancestry copies. Using this expectation retains more information than converting each haplotype to its most probable ancestry and then counting hard calls.
For a quantitative phenotype \(Y_i\), a basic local-ancestry association model is
Here \(\mathbf{C}_i\) contains prespecified adjustment variables, such as global ancestry proportions, age, sex, or technical covariates. The coefficient \(\beta_L\) is the conditional mean phenotype difference for one additional copy of the specified local ancestry, holding the included covariates fixed. A SNP association model that adjusts for local ancestry can be written
In this model, \(\beta_G\) is the conditional mean phenotype difference per additional effect-allele copy, at fixed local ancestry and covariates. The local-ancestry coefficient is now conditional on the SNP genotype as well. If the scientific question concerns different SNP effects by ancestry, an interaction term such as \(G_iL_i\) must be considered; the additive model above assumes no such interaction. A main-effect local-ancestry model does not test ancestry-specific SNP effects.
Worked Example: Calculating Posterior Ancestry Dosage
Suppose a hypothetical individual has posterior probability 0.80 that the first haplotype carries ancestry A and probability 0.35 that the second haplotype carries ancestry A. The expected number of ancestry-A copies is
The dosage is 1.15 copies, not a hard count. A most-probable-ancestry call would label the first haplotype A and the second haplotype not A, giving a hard count of 1. That loses uncertainty information. If the two haplotype ancestry states are conditionally independent, the posterior variance of the copy count is \(0.80(1-0.80)+0.35(1-0.35)=0.3875\) copies squared. Independence is needed for this variance calculation, but not for the expected dosage, which follows by adding expectations.
This example illustrates dosage construction, not a claim that ancestry assignment is error-free. In practice, posterior probabilities depend on the ancestry reference panels and inference method; poorly matched references or weakly informative regions can make the dosage uncertain or biased.
Worked Local-Ancestry Association
Consider the following deliberately small, hypothetical cohort. \(L\) is the dosage of ancestry A, \(G\) is the dosage of the effect allele, and \(Z\) is a centered genome-wide ancestry covariate in arbitrary standardized units. The phenotype is measured in units. This constructed example makes the arithmetic transparent: the centered predictors are orthogonal, a simplifying feature that is unusual in real admixed data.
| Person | \(L\) | \(G\) | \(Z\) | \(Y\) |
|---|---|---|---|---|
| 1 | 0 | 0 | -1 | 6.9 |
| 2 | 0 | 2 | -1 | 9.7 |
| 3 | 0 | 0 | 1 | 9.1 |
| 4 | 0 | 2 | 1 | 9.5 |
| 5 | 2 | 0 | -1 | 10.5 |
| 6 | 2 | 2 | -1 | 10.9 |
| 7 | 2 | 0 | 1 | 10.3 |
| 8 | 2 | 2 | 1 | 13.1 |
Worked Example: Adjusting a Local-Ancestry Association
Fit \(Y_i=\beta_0+\beta_L L_i+\gamma_Z Z_i+\varepsilon_i\). This asks whether local ancestry is associated with the phenotype after adjusting for the genome-wide ancestry covariate. The sample means are \(\bar{Y}=10\), \(\bar{L}=1\), and \(\bar{Z}=0\). The centered \(L\) values are four \(-1\)s and four \(1\)s, so \(\sum_i(L_i-\bar L)^2=8\). The least-squares slope for \(L\), after adjustment for \(Z\), is
The centered \(L\) and \(Z\) columns have cross-product zero, so adjustment for \(Z\) does not change this slope in this particular dataset. Similarly, \(\widehat{\gamma}_Z=0.50\), and the intercept is \(\widehat{\beta}_0=\bar Y-\widehat{\beta}_L\bar L-\widehat{\gamma}_Z\bar Z=10-1.20(1)-0.50(0)=8.80\). The fitted model is therefore \(\widehat{Y}=8.80+1.20L+0.50Z\).
In context, the fitted conditional mean phenotype is 1.20 units higher per additional copy of ancestry A, comparing people with the same value of \(Z\). This is an association in the specified model, not evidence that changing a person's local ancestry would cause a 1.20-unit phenotype change. The model leaves substantial variation unexplained, including variation associated with \(G\); the local-ancestry association is not a SNP effect.
Worked SNP Association and a Short Derivation
To understand adjusted coefficients, it helps to express least squares as a comparison after removing the fitted contributions of other predictors. Let \(\mathbf{C}\) be the design matrix for the intercept and adjustment variables, and define \(M_C=I-\mathbf{C}(\mathbf{C}^{T}\mathbf{C})^{-1}\mathbf{C}^{T}\), assuming those columns are linearly independent. This matrix subtracts the linear projection onto the covariates. In a regression of \(Y\) on \(G\) and \(\mathbf{C}\), the genotype coefficient is
To justify the expression, the least-squares residual sum of squares for a proposed genotype coefficient \(b\) is minimized over the covariate coefficients by projecting \(Y-bG\) off the columns of \(\mathbf{C}\). The remaining sum of squares is \((Y-bG)^TM_C(Y-bG)\). Differentiating with respect to \(b\) gives \(-2G^TM_CY+2bG^TM_CG=0\), which yields the displayed ratio when \(G^TM_CG>0\). Thus the adjusted estimate compares the parts of phenotype and genotype not linearly explained by the adjustment variables.
Worked Example: Fitting the SNP Model Alongside Local Ancestry
Now fit \(Y_i=\alpha_0+\beta_GG_i+\alpha_LL_i+\delta_ZZ_i+\varepsilon_i\) to the same eight hypothetical observations. The centered columns for \(G\), \(L\), and \(Z\) are mutually orthogonal and each has sum of squares 8. The centered cross-product of \(G\) with \(Y\) is 6.4, so the genotype estimate, adjusted for \(L\) and \(Z\), is
The local-ancestry estimate remains 1.20 and the \(Z\) estimate remains 0.50 because of the constructed orthogonality. The intercept is \(10-0.800(1)-1.20(1)-0.50(0)=8.00\), giving \(\widehat{Y}=8.00+0.800G+1.20L+0.50Z\). The SNP estimate is a 0.800-unit higher fitted phenotype per additional effect-allele copy, conditional on local ancestry and \(Z\). It is not the unadjusted difference between allele groups, nor is it automatically causal.
The residuals are the constructed interaction pattern, with four values of \(0.60\) and four of \(-0.60\). Their sum of squares is \(8(0.60^2)=2.88\). Although this verifies the fitted coefficients, the tiny constructed dataset is not a sound basis for a conventional small-sample \(t\)-test: its residual pattern does not establish normally distributed errors. The example therefore reports estimates but does not attach a significance claim or confidence interval.
Assumptions, Diagnostics, and Failure Modes
For conventional linear-model standard errors and tests, the conditional mean must be correctly represented by the model, errors must have mean zero given the predictors, observations must be independent (or their dependence must be handled), and the conditional error variance must be constant. Exact small-sample \(t\)-based inference additionally requires normally distributed errors. Heteroskedasticity-robust standard errors can allow nonconstant conditional variance, but do not fix a misspecified conditional mean, dependence, or ancestry-related confounding. In genomic cohorts, relatedness and population structure can violate independence or the mean specification and may require appropriate modeling.
Inspect the design and the fitted model before interpreting an association. Useful checks include:
- Overlap and collinearity: examine the distributions of \(L\) and \(G\) within ancestry and covariate strata. If certain allele dosages occur only on one local-ancestry background, the separate coefficients may be weakly identified or not estimable.
- Residual patterns: plot residuals against fitted values and key predictors to look for nonlinearity, changing spread, or influential observations. A linear coefficient can conceal a nonlinear relationship or an outlier-driven result.
- Dependence and structure: consider relatedness and remaining population structure. Adjusting for one global ancestry summary does not guarantee that all confounding is removed.
- Local-ancestry uncertainty: compare posterior dosage quality across regions and inspect whether inference is sensitive to reference panels or low-information segments. Treating uncertain ancestry as perfectly observed can understate uncertainty.
- Allele and ancestry coding: verify the effect allele, reference ancestry, locus coordinates, and dosage scale. Reversing a binary ancestry definition reverses the corresponding coefficient's direction.
A common mistake is to interpret a local-ancestry coefficient as the effect of a causal allele, or to claim that adding local ancestry proves population stratification has been eliminated. The correction is to state the coefficient's actual conditional contrast, identify the covariates in the model, and distinguish an association signal from a causal claim. Another mistake is to substitute the most probable ancestry call for a posterior dosage without noting the loss of uncertainty information. Posterior dosages improve representation of uncertain calls, but a regression using expected dosages does not automatically propagate all uncertainty in those dosages into its standard errors.
How the Models Relate—and Where They Stop
The local-ancestry-only model asks whether phenotype varies with ancestry inherited at the region, conditional on included covariates. Such an association can arise from differences in allele frequencies, effects of multiple variants, environmental or social structure correlated with ancestry, or other unmodeled factors. The SNP model estimates an allele association conditional on local ancestry and other covariates. Comparing adjusted and unadjusted SNP estimates can be informative, but a change in estimate does not by itself identify the mechanism.
A model with both \(G\) and \(L\) assumes additive contributions unless interactions or nonlinear terms are added. To ask whether the SNP association differs by local ancestry, one can fit an interaction model containing \(G_iL_i\), with enough data and genotype variation to estimate its terms. That is a distinct target from the two main-effect associations worked here. In real admixed cohorts, ancestry and genotype can be correlated, so the orthogonality that simplified the arithmetic above should not be expected. Large standard errors, unstable coefficients, or sensitivity to covariate choices may indicate weak overlap rather than absence of association.
The main alternatives are not interchangeable: hard local-ancestry calls are simpler but discard posterior uncertainty; posterior dosage uses expected copy count but may not fully propagate uncertainty; and a model that omits local ancestry may answer a different SNP-association question. Model choice should follow the scientific estimand and data quality, not a desire to report whichever coefficient is most significant.
Check Your Understanding
Answer each question, then compare with the solutions below.
- Two haplotypes have posterior probabilities 0.60 and 0.25 of carrying ancestry B. What is the posterior dosage of ancestry B?
- In a model containing \(G\), \(L\), and covariates, what does the coefficient of \(G\) represent?
- Why can a local-ancestry association remain after a SNP is included in the model?
- Name two conditions needed for conventional linear-model standard errors and tests, beyond a correctly specified conditional mean.
- What additional term is needed to model a SNP effect that varies with local ancestry?
Solutions. (1) \(0.60+0.25=0.85\) ancestry-B copies in posterior expectation. (2) The conditional mean phenotype difference per additional effect-allele copy, at fixed local ancestry and included covariates. (3) Local ancestry can capture variation beyond that single SNP, including other variants or residual structure; the association is not by itself causal. (4) Examples include independent observations, mean-zero errors conditional on predictors, and constant conditional error variance; exact small-sample \(t\)-inference also needs normal errors. (5) Include the product \(G_iL_i\), then assess whether the data support estimating that interaction.