Deriving Marginal Effect Changes Across Populations

Why a Tag-Variant Slope Can Change

A variant used to tag a genetic signal need not be causal. Its association with a trait depends partly on how well it tracks causal variants in the population being studied. Because allele frequencies and linkage disequilibrium (LD) can differ across populations, the same underlying causal effects can produce different marginal regression slopes at the tag variant. The distinction is important: a changed marginal slope does not, by itself, show that the causal effect changed.

The earlier material on population-specific association estimands distinguishes a population-specific regression association from a causal effect. Here we derive that association for a tag variant under an additive model and show exactly which population features enter the expression.

Model and Assumptions

Index populations by \(k\), individuals within a population by \(i\), and causal variants by \(j=1,\ldots,m\). Let \(T_{ik}\) be the allele dosage at a tag variant, and let \(G_{ijk}\) be the dosage at causal variant \(j\). Each dosage counts copies of a specified effect allele, from 0 to 2. Consider the additive trait model

Y_{ik}=\mu_k+\sum_{j=1}^{m}\beta_{jk}G_{ijk}+\varepsilon_{ik}.

Here \(\beta_{jk}\) is the per-allele causal effect at variant \(j\) in population \(k\), in units of the trait per additional effect allele. The error has mean zero and is uncorrelated with every genotype in that population: \(\operatorname{Cov}(G_{ijk},\varepsilon_{ik})=0\). The model assumes additive effects on the stated trait scale, with no unmodelled dominance or genotype-by-environment terms that alter the conditional mean. Population-specific intercepts allow different mean trait levels.

The marginal slope is the ordinary least-squares slope from regressing \(Y\) on \(T\) within population \(k\), without including the causal genotypes as predictors. The derivation is for a population-specific slope; covariate adjustment can be handled by applying it to suitably residualized variables when the adjustment model is appropriate. For the allele-frequency expressions below, assume diploid genotypes arise from random pairing of haplotypes (Hardy–Weinberg proportions). These assumptions are not claims that every real data set satisfies the model: they specify when the formula has its stated interpretation.

Definition: The population-\(k\) marginal tag slope is \(b_{Tk}=\operatorname{Cov}(T_k,Y_k)/\operatorname{Var}(T_k)\), provided \(\operatorname{Var}(T_k)>0\). It describes the expected change in the population-specific linear regression prediction of the trait for one additional tag-allele copy; it is not necessarily a causal effect.

Deriving the Marginal Slope

For a regression with an intercept, the population slope is covariance divided by predictor variance. Substitute the additive trait model into the covariance. Constants have zero covariance, and the error term contributes zero under the stated genotype–error assumption:

\begin{aligned} b_{Tk} &=\frac{\operatorname{Cov}(T_k,Y_k)}{\operatorname{Var}(T_k)}\\ &=\frac{\operatorname{Cov}\left(T_k,\sum_{j=1}^{m}\beta_{jk}G_{jk}+\varepsilon_k\right)} {\operatorname{Var}(T_k)}\\ &=\frac{\sum_{j=1}^{m}\beta_{jk}\operatorname{Cov}(T_k,G_{jk})} {\operatorname{Var}(T_k)}. \end{aligned}

The last equality follows from covariance linearity and the fact that each \(\beta_{jk}\) is a fixed coefficient within population \(k\). This is the general covariance form. It shows that a causal effect contributes to the tag slope in proportion to the tag’s covariance with that causal genotype, with all contributions scaled by tag-genotype variance. Contributions can reinforce or oppose one another.

To express the result using allele frequencies and LD, let \(p_{Tk}\) and \(p_{jk}\) be the frequencies of the specified alleles at the tag and causal variant \(j\). Define haplotype LD as \(D_{Tj,k}=P(T=1,G_j=1)-p_{Tk}p_{jk}\), where the probabilities refer to one haplotype. Under random pairing of haplotypes, a diploid genotype is the sum of two haplotype allele indicators. Thus its variance is \(2p(1-p)\), and the covariance of two diploid dosages is \(2D\). Substitution gives

b_{Tk} =\sum_{j=1}^{m}\beta_{jk} \frac{D_{Tj,k}}{p_{Tk}(1-p_{Tk})}.

This allele-frequency formula requires a polymorphic tag, so that \(0<p_{Tk}<1\). If the tag is monomorphic, its variance is zero and a marginal slope is undefined. The LD terms must also be calculated with consistent effect-allele coding: reversing an allele changes the sign of the corresponding dosage covariance.

An equivalent representation uses the correlation \(r_{Tj,k}\) between the two diploid genotypes. Since \(\operatorname{Cov}(T,G_j)=r_{Tj,k}\sqrt{\operatorname{Var}(T)\operatorname{Var}(G_j)}\), the contribution from variant \(j\) is \(\beta_{jk}r_{Tj,k}\sqrt{\operatorname{Var}(G_j)/\operatorname{Var}(T)}\). This makes clear that even an unchanged LD correlation does not guarantee an unchanged tag slope: the ratio of genotype variances can change.

Worked Examples

Worked Example: One Causal Variant, Different LD

Suppose a hypothetical trait has one causal variant with a shared effect of \(2.00\) trait units per effect-allele copy. In population A, the causal-allele frequency is \(0.30\), the tag-allele frequency is \(0.20\), and haplotype \(D_{T,C}=0.080\). In population B, the corresponding frequencies are \(0.50\) and \(0.40\), with \(D_{T,C}=0.020\). Calculate both marginal tag slopes.

The one-variant formula is \(b_T=\beta D_{T,C}/[p_T(1-p_T)]\). In population A, the tag variance factor in the denominator is \(0.20(1-0.20)=0.160\), so

b_{T,A}=\frac{2.00(0.080)}{0.160}=1.00 \quad\text{trait units per tag-allele copy}.

In population B, \(p_T(1-p_T)=0.40(0.60)=0.240\), giving

b_{T,B}=\frac{2.00(0.020)}{0.240}=0.1667 \quad\text{trait units per tag-allele copy}.

The LD values are feasible for the stated frequencies: in each population, the specified positive \(D\) is no greater than the relevant maximum \( \min\{p_T(1-p_C),(1-p_T)p_C\}\). Under the assumed additive model and genotype–error independence, the causal effect is unchanged, but the tag slope is much smaller in population B because the tag-causal covariance relative to tag variance is smaller. The slope difference is not evidence, on its own, of causal-effect heterogeneity.

Worked Example: Opposing Causal Contributions

In a hypothetical population, suppose the tag and two causal genotypes each have variance \(0.50\). Let their covariances with the tag be \(0.20\) and \(0.10\), respectively, and let the two causal effects be \(1.00\) and \(-0.50\) trait units per allele. Calculate the tag slope.

Use the covariance formula, with the tag variance as the denominator:

\begin{aligned} b_T &=\frac{(1.00)(0.20)+(-0.50)(0.10)}{0.50}\\ &=\frac{0.20-0.05}{0.50}\\ &=0.300 \quad\text{trait units per tag-allele copy}. \end{aligned}

The second causal variant reduces the tag slope because its effect is negative while its covariance with the tag is positive. If its covariance were stronger, it could cancel the first contribution or make the total slope negative. Therefore, a tag slope is a sum of linked causal contributions, not a direct readout of any single causal effect. The given variances and covariances are hypothetical population summaries; using the formula presumes they arise from a coherent genotype distribution and the stated trait model.

Worked Example: Same LD Correlation, Different Allele Frequencies

Consider one causal variant with effect \(\beta=1.00\) trait unit per allele. In hypothetical population A, both causal and tag allele frequencies are \(0.20\), and their genotype correlation is \(r=0.40\). In population B, the causal frequency is \(0.50\), the tag frequency remains \(0.20\), and the genotype correlation is also \(0.40\). Find each tag slope using the correlation form.

Under random pairing, genotype variance is \(2p(1-p)\). In A, both genotype variances are \(2(0.20)(0.80)=0.320\). Thus

b_{T,A} =\beta r\sqrt{\frac{\operatorname{Var}(G_C)}{\operatorname{Var}(T)}} =(1.00)(0.40)\sqrt{\frac{0.320}{0.320}} =0.400.

In B, the causal-genotype variance is \(2(0.50)(0.50)=0.500\), while the tag variance remains \(0.320\). Therefore,

b_{T,B} =(1.00)(0.40)\sqrt{\frac{0.500}{0.320}} =0.500.

The assumed correlations are feasible with these frequencies. For example, they imply haplotype \(D=0.40\sqrt{p_C(1-p_C)p_T(1-p_T)}\), which is \(0.064\) in A and approximately \(0.0800\) in B; both satisfy the haplotype-frequency bounds. The slopes differ despite the same causal effect and the same \(r\), because the causal and tag genotype variances change differently across populations.

When Does the Marginal Effect Change?

Comparing populations \(a\) and \(b\), the general expression is

b_{Ta}-b_{Tb} = \sum_{j=1}^{m}\beta_{ja} \frac{\operatorname{Cov}(T_a,G_{ja})}{\operatorname{Var}(T_a)} - \sum_{j=1}^{m}\beta_{jb} \frac{\operatorname{Cov}(T_b,G_{jb})}{\operatorname{Var}(T_b)}.

This difference can arise from changes in causal effects, tag-causal LD, tag variance, or the set and effects of causal variants represented in the model. If causal effects are shared across populations, so that \(\beta_{ja}=\beta_{jb}\), the slope can still change whenever the LD-to-tag-variance ratios change. Conversely, different population features can sometimes offset one another and yield similar slopes. Similar marginal slopes therefore do not establish identical LD or identical underlying causal architecture.

A useful special case is a single causal variant. If it is the tag itself, its dosage covariance equals its variance, so the marginal slope equals its causal effect (under the model assumptions). If it is a distinct variant, the tag slope is the causal effect multiplied by \(\operatorname{Cov}(T,G_C)/\operatorname{Var}(T)\). This factor can be positive, negative, or zero. Thus a tag can have a null marginal slope despite a nonzero causal effect when its covariance with the causal genotype is zero.

Assumption Checks and Common Pitfalls

The derivation is useful only when the target and assumptions are clear. Check that the compared slopes use the same trait scale and allele coding, that the tag varies in each population, and that the model’s additive causal effects and error-independence assumptions are plausible for the analysis. For the frequency-based version, check the genotype-frequency assumptions or use the covariance form estimated from the relevant population data. With covariate adjustment, the simple raw-genotype expression need not apply unchanged: the regression slope is based on variation remaining after adjustment.

Calling a tag slope a causal effect. A marginal tag slope includes LD-mediated contributions from causal variants. It equals a causal effect only in special cases, such as a single causal variant that is itself the tag under the stated model.
Blaming every slope difference on causal heterogeneity. A slope can change when causal effects are shared, because LD and genotype variances differ. The formula separates these possible sources algebraically, but the slope comparison alone does not identify which source changed.
Ignoring allele orientation. If an effect allele is reversed in one population, the associated dosage and LD signs change. Harmonize allele definitions before comparing signed slopes.
Applying the frequency formula without its conditions. The conversion from haplotype \(D\) to diploid genotype covariance assumes random pairing of haplotypes. If that assumption is unsuitable, use the covariance expression appropriate to the observed genotypes.

An alternative to the marginal regression is a joint regression that includes candidate causal variants. Under a correctly specified additive model with adequate genotype variation and no confounding, its coefficients target conditional effects rather than the combined tag association. But joint estimates can be unstable when variants are highly correlated, and a causal variant may be unmeasured or unknown. The marginal slope is simpler and directly describes the tag association; its limitation is that it conflates causal effects with population-specific tagging. This tutorial derives that relationship, rather than providing a test for whether effects differ across ancestries.

Key takeaway: The marginal tag slope is the sum of causal effects weighted by each causal genotype’s covariance with the tag, divided by tag-genotype variance. With shared causal effects, changes in allele frequencies and LD can still change the marginal slope.

Check Your Understanding

Try each problem, then check the solution.

A single causal variant has effect \(1.5\), tag frequency \(0.25\), and haplotype \(D=0.06\). What is the marginal tag slope?
Can a tag have a zero marginal slope while the causal effect is nonzero? State a condition under the one-causal-variant model.
With shared causal effects, name two population-specific quantities that can change the tag slope.
Why does a constant LD correlation not necessarily imply a constant marginal tag slope?

Solutions. (1) \(p_T(1-p_T)=0.25(0.75)=0.1875\), so \(b_T=1.5(0.06)/0.1875=0.480\) trait units per tag-allele copy. (2) Yes. If the tag and causal genotype have zero covariance, then \(b_T=\beta\operatorname{Cov}(T,G_C)/\operatorname{Var}(T)=0\) when the tag variance is positive, even if \(\beta\ne0\). (3) The tag-causal covariance and tag-genotype variance can change; equivalently, allele frequencies and LD can change. (4) The slope contribution is \(\beta r\sqrt{\operatorname{Var}(G_C)/\operatorname{Var}(T)}\); the variance ratio can change even when \(r\) and \(\beta\) remain fixed.