How Ancestry-Specific LD Changes Association Patterns

Why the Same Causal Variant Can Produce Different Association Patterns

A variant can affect a phenotype in two ancestry groups while the surrounding association results look different. The reason may be that nearby markers are correlated with the causal variant to different degrees in the two groups. A marker that is a strong proxy in one group can be a weak proxy in another; consequently, marginal effect estimates and test statistics can differ even when the causal variant and its effect are shared.

Here, a marginal association means an association estimated for one marker at a time, without conditioning on the causal variant or the other nearby markers. The focus is the relationship between that single-marker result and ancestry-specific linkage disequilibrium (LD), not a claim that the marker itself causes the phenotype.

Definition: For a causal variant \(c\) and a tested marker \(j\), let \(G_c\) and \(G_j\) denote centered, standardized genotype scores, each with mean 0 and variance 1 in the ancestry being considered. Their LD correlation is \(r_{jc}=\operatorname{Corr}(G_j,G_c)\). In the one-causal-variant model \(Y=\gamma G_c+\varepsilon\), where \(\gamma\) is the causal effect and \(E(\varepsilon\mid G_c,G_j)=0\), the population marginal slope for marker \(j\) is \(\beta_j^{\mathrm{marg}}=\gamma r_{jc}\).

The standardized genotype convention makes the arithmetic transparent: a slope is the expected phenotype change per one-standard-deviation increase in that genotype score. In actual analyses, genotype dosage may be coded as allele count, and the coefficient’s units then depend on that coding. Correlations and the proxy-versus-causal distinction remain central, but effect sizes must be interpreted using the actual coding and allele alignment.

Why the Marginal Slope Depends on LD

The result follows from the definition of the population least-squares slope. Regressing \(Y\) on \(G_j\) alone gives slope \(\operatorname{Cov}(G_j,Y)/\operatorname{Var}(G_j)\). Substituting the one-causal-variant model and using the stated conditional mean-zero assumption yields:

\begin{aligned} \beta_j^{\mathrm{marg}} &=\frac{\operatorname{Cov}(G_j,Y)}{\operatorname{Var}(G_j)}\\ &=\frac{\operatorname{Cov}(G_j,\gamma G_c+\varepsilon)}{1}\\ &=\gamma\operatorname{Cov}(G_j,G_c)+\operatorname{Cov}(G_j,\varepsilon)\\ &=\gamma r_{jc}. \end{aligned}

Because both genotype scores have variance 1, their covariance is their correlation. Also, \(E(\varepsilon\mid G_c,G_j)=0\) implies \(\operatorname{Cov}(G_j,\varepsilon)=0\). Thus LD scales the marginal slope: if \(r_{jc}=0\), this model gives a zero marginal slope at marker \(j\); if \(|r_{jc}|\) is large, the proxy’s slope can be close in magnitude to the causal effect. Its sign follows the signed correlation, assuming consistent effect-allele orientation.

Worked Example: Computing a Proxy Slope From a Genotype Pattern

Consider a hypothetical ancestry group with 16 people and standardized two-level genotype scores coded \(+1\) and \(-1\). At causal variant \(c\), people 1–8 have score \(+1\) and people 9–16 have score \(-1\). At proxy \(P_1\), two people whose causal score is \(+1\) and two whose causal score is \(-1\) have their scores flipped; all other scores match \(G_c\). Both genotype scores remain balanced, with mean 0 and variance 1.

The scores differ at four of 16 people. For balanced \(+1/-1\) scores, each mismatch contributes \(-1\) to the product \(G_jG_c\), while each match contributes \(+1\). Therefore:

r_{1c}=\frac{12(1)+4(-1)}{16}=\frac{8}{16}=0.5.

Suppose the hypothetical causal effect is \(\gamma=0.20\) phenotype units per genotype-score standard deviation, and the one-causal-variant model assumptions hold. The marginal slope at \(P_1\) is:

\beta_1^{\mathrm{marg}}=\gamma r_{1c}=0.20(0.5)=0.10 \quad\text{phenotype units per genotype-score standard deviation.}

The proxy’s expected marginal slope is half the causal slope because its correlation with the causal genotype is 0.5. This does not mean the biological effect at the proxy is half the causal effect: the proxy’s marginal association reflects correlation with \(c\).

From LD to Association-Test Strength

A slope alone does not determine the strength of statistical evidence: its standard error also matters. To give a precise comparison, consider the following additional assumptions. Genotype scores are independent draws from an ancestry-specific distribution with \(G_j,G_c\in\{-1,+1\}\), both balanced, and correlation \(r_{jc}\). The phenotype follows \(Y=\gamma G_c+\varepsilon\), with independent errors \(\varepsilon\sim N(0,\sigma_\varepsilon^2)\). The sample is large, and the ordinary homoskedastic regression standard error is used.

Under these conditions, \(E(G_c\mid G_j)=r_{jc}G_j\). Since \(G_c\) is a two-level score, \(\operatorname{Var}(G_c\mid G_j)=1-r_{jc}^2\) for either value of \(G_j\). The residual from the marginal regression is \(\gamma(G_c-r_{jc}G_j)+\varepsilon\), whose conditional variance given \(G_j\) is constant. Hence the marginal regression is homoskedastic in this specific setup, with residual variance \(\sigma_\varepsilon^2+\gamma^2(1-r_{jc}^2)\). For standardized scores, the large-sample standard error of the marginal slope is approximately the square root of this variance divided by \(n\), giving the approximate noncentrality parameter below.

\lambda_j\approx \frac{\sqrt{n}\,\gamma r_{jc}} {\sqrt{\sigma_\varepsilon^2+\gamma^2(1-r_{jc}^2)}}.

This expression is not a general formula for arbitrary genotype distributions and error structures. In particular, merely assuming that \(\varepsilon\) is uncorrelated with genotype does not establish the usual standard error or this noncentrality parameter. The formula here relies on the stated two-level genotype model, independent homoskedastic errors, and ordinary regression standard errors. With other genotype distributions or heteroskedastic marginal residuals, a robust variance calculation may be needed, and LD alone is not enough to determine test strength.

Worked Example: A Different Lead Proxy in Two Ancestry Groups

Suppose the same causal effect, \(\gamma=0.20\) phenotype units per genotype-score standard deviation, is present in two hypothetical ancestry groups. Both have \(\sigma_\varepsilon=0.50\), and each has \(n=10{,}000\) independent observations. Assume the two-level genotype and error conditions just described. In Group A, the correlations with the causal variant are \(r_{1c}=0.75\) for \(P_1\) and \(r_{2c}=0.50\) for \(P_2\). In Group B, these correlations are reversed: \(r_{1c}=0.50\) and \(r_{2c}=0.75\). Such balanced two-level genotype patterns are possible; for example, among 16 people, flipping four causal scores gives correlation 0.5, while flipping two gives correlation 0.75.

For a proxy with \(r=0.75\), the marginal slope and approximate noncentrality are:

\begin{aligned} \beta^{\mathrm{marg}}&=0.20(0.75)=0.150,\\ \lambda&\approx \frac{\sqrt{10{,}000}(0.20)(0.75)} {\sqrt{0.50^2+0.20^2(1-0.75^2)}}\\ &=\frac{15}{\sqrt{0.25+0.04(0.4375)}}\\ &=\frac{15}{\sqrt{0.2675}}\approx 29.0. \end{aligned}

For a proxy with \(r=0.50\), the corresponding calculations are:

\begin{aligned} \beta^{\mathrm{marg}}&=0.20(0.50)=0.100,\\ \lambda&\approx \frac{10}{\sqrt{0.25+0.04(0.75)}}\\ &=\frac{10}{\sqrt{0.28}}\approx 18.9. \end{aligned}

In Group A, \(P_1\) has the larger expected association statistic under these assumptions; in Group B, \(P_2\) does. The causal effect has not changed in the setup: the LD relationships changed. The calculated \(\lambda\) values describe approximate noncentralities, not observed test statistics or p-values. An observed statistic varies from sample to sample, and a p-value cannot be recovered from LD and these model parameters alone without specifying the testing procedure and observing the data.

Direction, Allele Alignment, and Interpretation

LD can be negative. If the effect allele at a proxy tends to occur with the allele coded positively at the causal variant, the correlation may be positive; if it tends to occur with the other causal allele, the correlation may be negative. With allele coding held fixed, a negative correlation reverses the sign of the expected marginal slope. That sign reversal does not by itself imply a different biological mechanism or an opposite causal effect.

Worked Example: A Negatively Correlated Proxy

In a third hypothetical ancestry group, suppose \(r_{jc}=-0.60\), while \(\gamma=0.15\) phenotype units per genotype-score standard deviation. Under the one-causal-variant model, the marginal slope is:

\beta_j^{\mathrm{marg}}=\gamma r_{jc} =0.15(-0.60)=-0.090.

The negative sign is the expected direction for the proxy’s stated coding. If the proxy’s alleles were recoded so that the genotype score changed sign, the correlation and marginal slope would also change sign. The underlying fitted association would be the same after the corresponding coding change. Before comparing signed estimates across ancestry-specific summary statistics, the effect alleles and strand conventions must therefore be aligned.

What Changes and What Does Not

An ancestry-specific LD matrix records pairwise correlations among variants in a particular population or reference sample. For a fixed causal variant and shared \(\gamma\), changing the correlations changes the expected marginal slopes of its proxies. It can also reorder their association statistics, so the marker with the smallest p-value—the lead marker—need not be the same across groups.

This pattern is informative but not conclusive on its own. A changed lead marker does not establish that different causal variants are operating, and a shared lead marker does not prove that the same variant is causal. Marginal signals summarize both the causal effect and the correlation structure. Sample size, allele frequency, phenotype variance, measurement, imputation, and model assumptions can also affect observed evidence. The examples isolate LD by holding other features fixed; real comparisons should not assume those features are identical without checking.

A useful interpretation keeps three quantities distinct: the causal effect \(\gamma\), the ancestry-specific LD correlation \(r_{jc}\), and the marginal association result at marker \(j\). Under the stated model, the first two determine the expected marginal slope through \(\gamma r_{jc}\). Statistical precision then affects the observed test statistic. These distinctions are essential when interpreting association patterns before using a joint model that accounts for LD.

Common Pitfall

Wrong reasoning: “The lead marker changed between ancestry groups, so the causal effect must have changed.”
Correction: A changed LD matrix can change proxy slopes and reorder marginal association strengths even with the same causal variant and effect. Check ancestry-specific correlations and allele coding before attributing the pattern to an effect difference.
Wrong reasoning: “The marginal slope is \(\gamma r\), so the p-value is determined by LD alone.”
Correction: The slope relation concerns the population expectation under the one-causal-variant model. Test strength also depends on sample size and residual variation, among other features; a noncentrality formula requires assumptions about the genotype distribution, residual variance, and standard error.

Key takeaway: With standardized genotypes and one causal variant, the expected marginal slope at a proxy is \(\gamma r_{jc}\). Ancestry-specific LD can therefore change the size, sign, and ranking of proxy associations without changing the causal effect. Comparisons of test strength require more than LD alone.

Check Your Understanding

Use the one-causal-variant model and the assumptions stated in the tutorial where relevant.

If \(\gamma=0.12\) phenotype units per genotype-score standard deviation and \(r_{jc}=0.25\), what is the expected marginal slope, with units?
Two proxies have correlations 0.80 and 0.40 with the same causal variant. Which has the larger expected absolute marginal slope if both are standardized and the causal effect is nonzero?
What is the expected marginal slope under this model when a tested marker has zero correlation with the causal variant?
A proxy’s estimated slope is negative in one ancestry group and positive in another. Name two issues to examine before concluding that the causal effect differs.
Why does \(\beta_j^{\mathrm{marg}}=\gamma r_{jc}\) not by itself determine the p-value at marker \(j\)?