The Baseline-LD Model for Joint Annotation Analysis

Why Jointly Model Functional and LD-Related Features?

A functional annotation can mark variants in coding regions, conserved sequence, or other genomic categories. But annotated variants also differ in properties such as allele frequency and local LD structure. Those properties matter because an association statistic at one SNP can reflect effects at other correlated SNPs. If a model includes functional annotations but omits relevant LD-related features, a functional coefficient may partly reflect those omitted genomic patterns rather than the functional annotation alone.

The Baseline-LD model addresses this by fitting functional annotations and LD-related genomic features together in a single annotation-based model. “Together” is essential: a coefficient is estimated while accounting for the other included annotation predictors. The model does not make correlated annotations independent, nor does it turn an observational annotation association into evidence of a causal biological mechanism.

Definition: In an additive annotation variance model, each SNP \(k\) has an annotation value \(a_c(k)\) for each annotation \(c\), and its effect-size variance is modeled as \(\operatorname{Var}(\beta_k)=\sum_{c=1}^{C}\tau_c a_c(k)\). The Baseline-LD model uses a joint set of functional and LD-related annotations in this sum. The coefficient \(\tau_c\) describes the modeled change in per-SNP effect variance per unit of annotation \(c\), conditional on the other included annotations.

An annotation value need not be binary. A functional annotation may be an indicator, while an LD-related feature may be a continuous score. The annotations need not be mutually exclusive: one SNP can have nonzero values for several annotations. Their contributions are represented by a sum, not by assigning each SNP to exactly one category.

From SNP Annotations to Annotation LD Scores

A GWAS statistic at a tested SNP can reflect effects at nearby causal SNPs because genotypes are correlated through LD. For tested SNP \(j\), the annotation-specific LD score for annotation \(c\) is the sum of the squared LD correlations between \(j\) and SNPs \(k\), weighted by their annotation values.

\ell(j,c)=\sum_{k} r_{jk}^{2}a_c(k)

Here \(r_{jk}\) is the correlation between the genotypes at SNPs \(j\) and \(k\), and \(a_c(k)\) is the annotation value of SNP \(k\). Thus \(\ell(j,c)\) describes how much LD SNP \(j\) has with SNPs carrying annotation \(c\). It is a feature calculated for each tested SNP, not the annotation value of that tested SNP alone.

In Baseline-LD, LD-related genomic features are themselves included as annotations in the per-SNP variance model. Their annotation LD scores are then constructed by the same rule as for functional annotations. This distinction matters: a feature describing a candidate causal SNP is used to model its effect variance, while the LD score aggregates that feature across SNPs correlated with the tested SNP.

Worked Example: Calculating Two Annotation LD Scores

Consider a hypothetical tested SNP \(j\) and three SNPs \(k=1,2,3\). Suppose their squared LD correlations with \(j\) are \(0.80\), \(0.10\), and \(0.00\). A functional annotation has values \(1,0,1\) across the three SNPs, and an LD-related feature has values \(1,2,1\).

For the functional annotation,

\ell(j,\text{functional})=0.80(1)+0.10(0)+0.00(1)=0.80.

For the LD-related feature,

\ell(j,\text{LD feature})=0.80(1)+0.10(2)+0.00(1)=1.00.

The second score is larger because the SNP with squared correlation \(0.10\) has feature value \(2\). The calculation uses the annotation values of all three SNPs, not just the annotation values at \(j\).

The Joint Regression and What Its Coefficients Mean

Under the additive variance model, assume SNP effects have mean zero and are uncorrelated across SNPs, with variances given by the annotation sum. For a tested SNP \(j\), its marginal genetic effect is a sum of correlated-SNP effects weighted by their genotype correlations. Its variance is therefore the sum of \(r_{jk}^{2}\operatorname{Var}(\beta_k)\) across SNPs \(k\). Substituting the annotation variance model gives a sum of annotation-specific LD scores.

\begin{aligned} \operatorname{Var}\left(\sum_k r_{jk}\beta_k\right) &=\sum_k r_{jk}^{2}\operatorname{Var}(\beta_k)\\ &=\sum_k r_{jk}^{2}\sum_{c=1}^{C}\tau_c a_c(k)\\ &=\sum_{c=1}^{C}\tau_c\ell(j,c). \end{aligned}

In a standard LD score regression approximation, this motivates the joint regression

\operatorname{E}[\chi_j^2]\approx 1+\eta+N\sum_{c=1}^{C}\tau_c\ell(j,c),

where \(\chi_j^2\) is the association statistic at SNP \(j\), \(N\) is the GWAS sample size, and \(\eta\) is an intercept excess that can capture average inflation not explained by the modeled polygenic signal. The exact regression implementation uses observed association statistics and fits an intercept along with the annotation predictors. The intercept is not an annotation contribution to heritability.

This equation explains the central adjustment in Baseline-LD. Each \(\tau_c\) is estimated from the joint pattern across all included annotation LD scores, rather than from a separate regression using only annotation \(c\). When two annotation predictors are correlated, the model estimates their coefficients from the information that distinguishes them, given the other predictors. Consequently, a coefficient is a conditional association in this specified model. It is not generally equal to the association obtained by fitting that annotation alone.

Worked Example: How Joint Fitting Changes a Coefficient

For a small hypothetical regression illustration, let \(F\) and \(L\) denote two annotation LD-score predictors, for a functional annotation and an LD-related feature. Let \(y\) be a rescaled regression response. The four rows are deliberately chosen for transparent arithmetic; they are not GWAS results.

Row	\(F\)	\(L\)	\(y\)
1	1	3	3
2	1	1	1
3	3	3	7
4	3	5	9

Fit \(y=b_0+b_FF+b_LL\) by ordinary least squares. The means are \(\bar F=2\), \(\bar L=3\), and \(\bar y=5\). The centered predictors are \(f=(-1,-1,1,1)\) and \(l=(0,-2,0,2)\), and the centered response is \(y-\bar y=(-2,-4,2,4)\). Their required cross-products are

f^\mathsf{T}f=4,\quad l^\mathsf{T}l=8,\quad f^\mathsf{T}l=4,\quad f^\mathsf{T}(y-\bar y)=12,\quad l^\mathsf{T}(y-\bar y)=16.

The joint normal equations are therefore

\begin{pmatrix}4&4\\4&8\end{pmatrix} \begin{pmatrix}b_F\\b_L\end{pmatrix} = \begin{pmatrix}12\\16\end{pmatrix}.

The determinant is \(4(8)-4(4)=16\), so the solution is \(b_F=2\) and \(b_L=1\). The intercept is \(b_0=\bar y-b_F\bar F-b_L\bar L=5-2(2)-1(3)=-2\). If \(F\) were fitted alone with an intercept, its slope would instead be \(f^\mathsf{T}(y-\bar y)/(f^\mathsf{T}f)=12/4=3\). That single-predictor slope combines the pattern associated with \(F\) and the correlated pattern associated with \(L\). The joint slope \(b_F=2\) is adjusted for \(L\). This illustrates conditional estimation; the toy coefficients are not Baseline-LD estimates or biological effect sizes.

Reading the Model as a Variance Partition

The fitted coefficients describe the modeled per-SNP effect variance, not a direct fraction of heritability by themselves. To obtain a SNP-specific modeled variance, multiply each coefficient by that SNP’s annotation value and sum across annotations. Summing these variances over SNPs gives the model-implied total for the SNP set, subject to the model and scaling conventions.

Worked Example: Combining Fitted Annotation Contributions

Suppose a hypothetical fitted model has a functional coefficient \(\tau_F=0.004\) and an LD-feature coefficient \(\tau_L=0.002\). For three SNPs, let the functional annotation values be \(1,0,0\), and the LD-feature values be \(2,1,3\). The modeled per-SNP effect variances are:

\begin{aligned} \operatorname{Var}(\beta_1)&=0.004(1)+0.002(2)=0.008,\\ \operatorname{Var}(\beta_2)&=0.004(0)+0.002(1)=0.002,\\ \operatorname{Var}(\beta_3)&=0.004(0)+0.002(3)=0.006. \end{aligned}

The sum across these three SNPs is \(0.008+0.002+0.006=0.016\), in the model’s effect-variance scale. This calculation shows how the joint model combines contributions when annotations overlap. It does not mean that \(\tau_F\) alone is the total heritability of the functional category, or that the LD-feature coefficient is a separate, non-overlapping share.

Assumptions and Interpretation Boundaries

The derivation and interpretation rely on several assumptions. The effect-variance model is additive in the included annotation values; the effects are modeled as mean zero and uncorrelated across SNPs; and the LD values used to construct annotation LD scores adequately represent LD in the GWAS population. The regression approximation also relies on appropriate alignment of the GWAS statistics, SNP set, annotations, and LD reference. Model coefficients describe the included predictors and do not automatically account for omitted or poorly measured genomic features.

Functional annotations and LD-related features can be correlated. Joint fitting adjusts coefficients for the included predictors, but it cannot manufacture information that the data do not contain. If two predictors are very similar, their separate coefficients can be imprecisely determined or sensitive to model specification. A coefficient should therefore be read as conditional on the particular annotation set and coding used—not as an intrinsic property of a biological label.

The LD-related annotations provide a way to account for systematic differences in per-SNP effect variance associated with genomic features that also shape association-statistic patterns. They do not imply that LD itself is a biological cause of heritability. Nor does a positive functional coefficient establish that the annotated process causes a trait: the regression is an association model, and its interpretation depends on its assumptions and included annotations.

Key takeaway: Baseline-LD places functional and LD-related genomic features in one additive effect-variance model, converts each feature into annotation LD scores, and fits their coefficients jointly. Each coefficient is an adjusted, conditional model parameter—not an isolated enrichment measure or causal effect.

Common Mistakes

Reading an annotation LD score as the tested SNP’s annotation value. It is a sum over SNPs weighted by squared LD correlations.
Interpreting a joint coefficient as a single-annotation result. The coefficient is estimated conditional on the other annotations included in the model.
Treating overlapping annotations as exclusive categories. A SNP can contribute to several terms in the additive variance model.
Calling a fitted annotation coefficient a causal biological effect. The model estimates conditional associations under its assumptions; it does not establish causation.

Check Your Understanding

Answer these questions using the model definitions and interpretations above.

Why can an annotation LD score differ from the annotation value of the tested SNP?
In the joint regression, what does it mean to interpret a functional coefficient conditional on an LD-related feature?
If two annotation predictors are correlated, why can a single-predictor coefficient differ from the joint coefficient?
For a SNP with two nonzero annotation values, how is its modeled effect variance calculated?
Why does including LD-related annotations not establish that LD causes trait heritability?