Interpreting Coefficients with Overlapping Annotations

Why Joint Coefficients Need Conditional Interpretation

In annotation-based heritability analysis, a variant may belong to more than one category. A variant can, for example, be both near a gene and within a conserved region. When annotations overlap, the fitted coefficient for one annotation is not automatically the heritability of that category, nor is it a comparison of variants in that category with all other variants. It describes the annotation’s fitted contribution conditional on the other annotations included in the model.

The distinction matters because the joint model assigns a contribution to every annotation recorded for a variant. A variant in two annotations receives both fitted components. To obtain an annotation’s heritability contribution, sum that annotation’s fitted component over the variants carrying it. The resulting category totals can overlap; they are not necessarily mutually exclusive shares.

Definition: In a joint additive annotation variance model, let $a_c(j)$ indicate whether variant $j$ belongs to annotation $c$, and let $\tau_c$ be its fitted coefficient. The fitted per-variant genetic variance is $$ \operatorname{Var}(\beta_j)=\sum_c \tau_c a_c(j). $$ For a binary annotation, $a_c(j)=1$ when variant $j$ is in category $c$, and $a_c(j)=0$ otherwise. The coefficient $\tau_c$ is the fitted change in per-variant genetic variance associated with membership in annotation $c$, holding the other modeled annotation values fixed. The annotation-specific fitted heritability component is $h_c^2=\tau_c\sum_j a_c(j)$.

Here $\beta_j$ denotes the genetic effect of variant $j$, with genotypes standardized and the phenotype on the scale used for the analysis. Thus $\operatorname{Var}(\beta_j)$ is a contribution to genetic variance on that phenotype scale. For a binary annotation, $\tau_c$ has the same units: phenotype variance per annotated variant under this parameterization. It is not a per-person change in phenotype, and it is not an effect of changing a person’s annotation status.

“Conditional” describes the model comparison implied by the coefficient. If two variants differ in membership of annotation $c$, while having the same values for every other annotation in the model, their fitted per-variant variances differ by $\tau_c$. This is a model-based contrast, not a causal effect. In observational genetic data, annotations can be correlated, and the fitted coefficient does not by itself establish a biological mechanism.

How the Coefficients Enter Stratified LD Score Regression

Stratified LD Score Regression uses the relationship between association statistics and linkage disequilibrium (LD) to estimate the annotation coefficients. For variant $j$, the annotation-specific LD score is $l(j,c)=\sum_k r_{jk}^2 a_c(k)$, where $r_{jk}$ is the correlation between variants $j$ and $k$. Under the additive variance model, the expected association statistic is approximated by:

E[\chi_j^2]\approx 1+N\sum_c \tau_c l(j,c)+I,

where $N$ is the GWAS sample size and $I$ represents the contribution of the fitted intercept beyond the baseline value 1. In practice, the regression uses the specified intercept convention and weights from the fitted procedure. The key interpretive point is that each annotation LD score has its own slope $\tau_c$ in the joint regression: the slope describes its contribution after accounting for the other included annotation LD scores.

Justification: Under the standard approximation, the genetic part of the expected association statistic at $j$ is proportional to $N\sum_k r_{jk}^2\operatorname{Var}(\beta_k)$. Substituting the additive annotation model gives $$ N\sum_k r_{jk}^2\sum_c \tau_c a_c(k) =N\sum_c\tau_c\sum_k r_{jk}^2a_c(k) =N\sum_c\tau_c l(j,c). $$ The second equality reorders finite sums. It shows why the joint regression has annotation-specific LD scores as predictors and why each coefficient corresponds to a conditional component of per-variant variance.

This derivation also clarifies why the coefficient is not simply a category total. The regression coefficient describes the fitted contribution for one unit of annotation membership at a variant. The total fitted component for a category depends both on that coefficient and on how many variants carry the annotation.

Worked Example: Interpreting a Coefficient for an Overlapping Variant

Suppose a hypothetical joint model contains two binary annotations, A and B. Its fitted coefficients are $\hat{\tau}_A=0.000030$ and $\hat{\tau}_B=0.000020$, in standardized phenotype-variance units per variant. Variant $q$ belongs to both annotations. Calculate its fitted genetic variance contribution and interpret each coefficient.

For variant $q$, $a_A(q)=1$ and $a_B(q)=1$. Substitute these values in the fitted variance model:

\widehat{\operatorname{Var}}(\beta_q) =\hat{\tau}_A a_A(q)+\hat{\tau}_B a_B(q) =(0.000030)(1)+(0.000020)(1) =0.000050.

The fitted contribution for this variant is $0.000050$ standardized phenotype-variance units. The annotation A component is $0.000030$, conditional on B being included; the B component is $0.000020$, conditional on A being included. The model adds the two components for this doubly annotated variant. Neither coefficient alone is the total contribution for variant $q$, and neither coefficient alone is the total heritability of its annotation.

For comparison, a variant in A but not B would receive $0.000030$ under this model, while a variant in B but not A would receive $0.000020$. This comparison follows from the fitted equation; it does not imply that the annotations caused those differences.

Summing Fitted Contributions Within a Category

For binary annotation $c$, each variant in the category receives the fitted component $\hat{\tau}_c$. Summing that component over all variants gives:

\hat{h}_c^2 =\sum_j \hat{\tau}_c a_c(j) =\hat{\tau}_c\sum_j a_c(j).

The sum $\sum_j a_c(j)$ is the number of variants assigned to the category. This formula is the fitted annotation component under the model, not necessarily a fraction of heritability. To report a fraction, a denominator and a consistent scale must also be specified. In a joint model, variants with overlapping memberships contribute to more than one category total.

Worked Example: Calculating Two Category Components

Consider four hypothetical variants in a modeled set. Variant 1 belongs to A only, variants 2 and 3 belong to both A and B, and variant 4 belongs to B only. Suppose the fitted coefficients are $\hat{\tau}_A=0.0040$ and $\hat{\tau}_B=0.0015$. Calculate the fitted heritability component for each category and the total fitted genetic variance across the four variants.

There are three variants in A and three in B. Therefore,

\hat{h}_A^2=(0.0040)(3)=0.0120, \qquad \hat{h}_B^2=(0.0015)(3)=0.0045.

The per-variant fitted contributions are $0.0040$ for variant 1; $0.0040+0.0015=0.0055$ for each of variants 2 and 3; and $0.0015$ for variant 4. Summing across variants gives:

$$ 0.0040+0.0055+0.0055+0.0015=0.0165. $$

The sum of the category components agrees: $0.0120+0.0045=0.0165$. This equality holds because all four variants’ fitted variance is represented by these two annotations in this example. The two category totals are not disjoint shares: each of the two overlapping variants contributes to both $h_A^2$ and $h_B^2$, and its total per-variant contribution is the sum of both components.

Reading a Joint Regression Prediction

The joint coefficients also determine the genetic part of the expected association statistic through annotation-specific LD scores. The following calculation illustrates how the coefficients work together in a regression prediction. It does not turn a coefficient into a category total: the LD scores describe how annotation membership across variants is correlated with the tested variant through LD.

Worked Example: Combining Coefficients and Annotation LD Scores

In a hypothetical quantitative-trait GWAS with $N=10{,}000$, suppose a fitted joint model has $\hat{\tau}_A=0.000030$, $\hat{\tau}_B=0.000020$, an intercept contribution $I=0.02$, and a tested variant with $l(j,A)=2.0$ and $l(j,B)=1.5$. Use the stated approximation to calculate its expected chi-square statistic.

Substitute the values:

E[\chi_j^2]\approx 1+0.02+10{,}000\big[(0.000030)(2.0)+(0.000020)(1.5)\big].

The annotation terms are $0.000030\times2.0=0.000060$ and $0.000020\times1.5=0.000030$. Their sum is $0.000090$; multiplying by $10{,}000$ gives $0.90$. Thus,

E[\chi_j^2]\approx 1+0.02+0.90=1.92.

Under this approximation, the expected chi-square statistic is 1.92 for a variant with those annotation LD scores. Both fitted coefficients contribute, each weighted by its corresponding LD score. The value 1.92 is a prediction for the association statistic, not an estimate of category heritability and not an individual-level effect.

When the Estimates Need Careful Wording

A fitted coefficient is conditional on the annotations in the model, their coding, and the phenotype scale. Changing which annotations are included can change the conditional comparison that a coefficient represents. A positive coefficient indicates a positive fitted variance component per unit of annotation membership, after accounting for the other included predictors. The coefficient does not state that every variant in the category has the same observed effect; it specifies a component in the model for genetic effect variance.

The underlying variance model requires nonnegative per-variant variances. However, regression estimates are not always constrained to make every estimated coefficient nonnegative. A negative fitted coefficient can occur from sampling variability or model limitations; it should be described as a negative estimate of the conditional component, not as literal negative genetic variance for a variant. If a combination of fitted coefficients implies negative per-variant variance for some annotation pattern, that is a warning about the fitted model’s implications, not a biological interpretation.

A frequent error is to say, “The coefficient for A is the heritability explained by A, so the A and B percentages are separate pieces that must add to the total.” This confuses a per-variant coefficient with a category sum and ignores overlapping membership. The correction is to calculate $\hat{h}_c^2=\hat{\tau}_c\sum_j a_c(j)$ for each category, state that the result is conditional on the other model terms, and explicitly note that overlapping category totals may count shared variants in more than one component. Neither the fitted partition nor an association between annotation membership and variance establishes causation.

Key takeaway: In a joint annotation model, each coefficient is a conditional per-variant variance contribution. Multiply it by the number of variants in its binary annotation to obtain that category’s fitted component. When annotations overlap, a variant receives each applicable component, so category totals are not automatically exclusive shares.

Check Your Understanding

Use the joint-model interpretation and summation rule to answer these questions.

A variant belongs to A and C in a model with fitted coefficients $\hat{\tau}_A=0.002$ and $\hat{\tau}_C=0.001$. What is its fitted variance contribution, and what does each coefficient mean conditionally?
An annotation has 250 variants and a fitted coefficient of $0.00004$. Calculate its fitted heritability component.
Why can two overlapping annotation components sum to more than the total contribution of the variants in their union?
In the expected chi-square equation, what role does the annotation-specific LD score play alongside its coefficient?
Why is it incorrect to describe a fitted annotation coefficient as a causal effect on phenotype?