Partitioning by Allele Frequency and Functional Class

Why Partition by Both Frequency and Function?

A functional annotation may contain variants across a wide range of minor-allele frequencies. If effect-size variance differs across those frequencies, a single functional-class coefficient can combine distinct patterns. Conversely, a frequency bin may contain several functional classes with different effects. A joint partition makes both dimensions explicit: it asks how genetic variance is distributed across frequency-by-function cells.

The key is to keep three quantities separate. An annotation column records whether a variant belongs to a category. A fitted coefficient describes the modeled per-variant effect variance associated with that column. A heritability contribution sums the modeled variance across variants, and therefore depends on both the coefficient and how many variants the category contains. Enrichment compares a heritability share with a variant share; it is not itself an effect-size coefficient.

Constructing Joint Annotation Columns

First define frequency-bin boundaries that are mutually exclusive and cover the variants being analyzed. For example, bins might be \(0\leq q_j<0.05\), \(0.05\leq q_j<0.20\), and \(0.20\leq q_j\leq0.50\). The exact cut points are analysis choices and should be stated; boundary rules prevent variants at a cut point from being assigned inconsistently. Assign each variant using frequencies from a clearly specified reference or analysis sample.

For each bin \(b\), create an indicator \(F_{jb}\) equal to 1 if variant \(j\) is in bin \(b\), and 0 otherwise. For a functional class \(c\), define \(C_{jc}\) similarly. The joint indicator is their product. When the functional classes are mutually exclusive and exhaustive, the resulting joint columns form a partition: each variant belongs to exactly one frequency-by-function cell.

If the functional classes overlap, a variant may have more than one joint indicator equal to 1. That is allowed in an additive annotation model, but the columns no longer define disjoint cells. In that case, adding annotation-specific contributions is a model-based allocation across terms, not a simple division of the variants into nonoverlapping groups. A variant can be counted in more than one annotation’s SNP share, so enrichment denominators must be defined accordingly.

From Joint Indicators to Heritability Contributions

For a clear variance model, consider standardized genotype \(Z_j\), with mean 0 and variance 1, and a standardized phenotype \(Y\). Write the additive genetic component as \(G=\sum_j Z_j\beta_j\). Assume effect means are zero, effects at distinct variants are uncorrelated, and their variances follow an additive annotation model:

Here \(\tau_{bc}\) is the modeled effect variance per variant in the joint cell \(b,c\), when the joint columns form a disjoint partition. For overlapping annotations, the sum is over the included annotation terms and a variant’s modeled variance is the sum of the coefficients for its active terms. The model assumes that this specified additive structure is an adequate description of per-variant effect variance; it does not establish that functional labels cause the differences.

Under the random-effects assumptions, the expected genetic variance is the sum of per-variant effect variances. To see why, expand \(\operatorname{Var}(G)\). Since the standardized genotypes have unit variance and the effects at distinct variants are uncorrelated in the model, the expected cross-products between distinct effects are zero. Thus:

where \(n_{bc}=\sum_j A_{j,bc}\) is the number of variants in a disjoint cell. The cell’s heritability contribution is therefore \(h^2_{bc}=n_{bc}\tau_{bc}\), and total modeled SNP heritability is the sum of those cell contributions. With overlapping annotations, the corresponding term-level sum uses each annotation’s active variants and coefficient; it should not be described as a unique nonoverlapping partition unless the columns truly are disjoint.

This derivation also shows why a coefficient is not a contribution. Two cells can have the same \(\tau_{bc}\), yet the cell containing more variants contributes more total variance. Conversely, a cell with a larger coefficient can contribute less if it contains sufficiently few variants.

The standardized-genotype convention matters for frequency comparisons. If the per-allele dosage is \(G_j\), with variance \(2p_j(1-p_j)\), and its per-allele effect is \(\alpha_j\), then the standardized-genotype effect is \(\beta_j=\sqrt{2p_j(1-p_j)}\alpha_j\). Consequently, the variance contribution on the raw dosage scale is \(2p_j(1-p_j)\operatorname{Var}(\alpha_j)\). A model for standardized effects and a model for per-allele effects are not interchangeable without this frequency factor.

Estimating and Summarizing the Joint Model

The annotation columns are inputs to a joint variance model, not independent one-column analyses. For example, the joint frequency-by-function columns can be fitted with the stratified LD score regression framework introduced in the earlier material on annotation-specific LD scores and weighted stratified LD score regression. Each joint column must have its corresponding LD-score predictor, computed using the same variant definitions and reference information as the annotation. The fitted coefficients are conditional on the other columns in that model.

After fitting, obtain each cell contribution by summing the modeled per-variant heritability across variants in the cell. In a simple disjoint model with a constant fitted \(\hat{\tau}_{bc}\) within each cell, this is \(n_{bc}\hat{\tau}_{bc}\). More generally, calculate and sum the fitted per-variant contributions using the model’s specified annotations and scale. Then calculate the cell’s share of total heritability by dividing its contribution by total modeled heritability.

For a functional class such as coding, aggregate its joint contributions over the frequency bins before calculating its overall heritability share. If the classes form a disjoint partition, its SNP share is the number of coding variants divided by the total number of variants in the same analysis set. Enrichment is the coding heritability share divided by the coding SNP share. The numerator and denominator must refer to compatible variant sets and annotation definitions.

A Practical Construction and Interpretation Sequence

Common Pitfall: Calling Enrichment an Effect-Size Ratio

A common mistake is to see a coding enrichment of 2 and conclude that a coding variant has twice the effect variance of a noncoding variant. That conclusion does not follow. Enrichment is a ratio of shares: the class’s share of heritability divided by its share of variants. It depends on the fitted contributions and on the number of variants in the denominator. A frequency-specific coefficient, a summed heritability contribution, and an enrichment ratio answer different questions.

Frequency bins also do not automatically remove frequency-related differences in effect scale. If the scientific target is per-allele effect variance, account for the \(2p_j(1-p_j)\) genotype-variance factor. If the target is variance explained by standardized genotypes, state that scale and interpret coefficients accordingly. Finally, when functional annotations overlap, do not present overlapping term contributions as a unique partition or use a SNP-share denominator that silently counts a variant more than once.