Deriving the Additive Annotation Variance Model

From Functional Annotations to Effect Variance

A functional annotation describes a property of a genetic variant, such as whether it lies in a specified genomic region. A variance model uses those properties to describe how much variation in genetic effects is expected across variants. The central modeling choice here is additive: each annotation that applies to a variant contributes a specified amount to the variance of that variant’s effect.

This is a model for the distribution of effects, not a claim that an annotation deterministically fixes a variant’s effect. In particular, a variance contribution is not an effect estimate, and it does not imply that every annotated variant has a nonzero effect. The model says how broadly effects are expected to vary under specified assumptions.

The annotation values are treated as fixed when specifying this conditional variance. If two annotations both apply to a variant, both terms enter the sum. Thus, the model allows overlapping annotations; it does not require each variant to belong to exactly one class. When annotations are mutually exclusive and exhaustive, each variant instead has one active class and its variance is the corresponding class contribution.

The units of \(\tau_c\) are the squared units of \(\beta_j\). For example, if genotype is measured in standardized units and the trait is measured in trait units, \(\beta_j\) is in trait units per standardized genotype unit, and \(\tau_c\) is in squared trait units per squared standardized genotype unit. The numerical scale of the effect depends on how genotypes are coded.

Deriving the Additive Form

A direct way to obtain the model is to represent a variant’s effect as a sum of annotation-specific random contributions. Let \(\beta_{jc}\) denote the contribution from annotation \(c\) to the effect of variant \(j\). If annotation \(c\) is absent, its contribution is zero. If it is present, assume that contribution has mean zero and variance \(\tau_c\):

Suppose the total effect is the sum of these contributions, \(\beta_j=\sum_{c=1}^{C}\beta_{jc}\), and that contributions from distinct annotations for the same variant are independent. Independence is a sufficient assumption for zero covariance. The variance of a sum is the sum of the variances plus twice each pairwise covariance, so the derivation is:

In the third line, an absent annotation gives a zero contribution and hence zero variance; a present annotation contributes variance \(\tau_c\). Independence makes every cross-covariance zero. This also shows why the additive equation is an assumption about how annotation-specific effects combine, rather than a consequence of annotation labels alone.

The formula specifies a variance, but does not by itself specify a complete probability distribution for \(\beta_j\). A common additional assumption is a normal distribution with mean zero and the stated variance. That distributional assumption is not needed to derive the variance equation itself.

Baseline Annotations and the Meaning of the Terms

A baseline annotation is an annotation that applies to every variant in the modeled set. Including one allows the model to assign nonzero variance even to variants lacking the other functional annotations. It also makes clear that “annotation-specific contribution” refers to a term in the variance model, not necessarily to a disjoint group of variants.

Connecting Variant Effects to Genetic Variance

The per-variant model describes effect variation. Under additional genotype assumptions, it also gives a simple expression for expected genetic variance. Let \(g_j\) be a centered genotype for variant \(j\), and let the genetic component of a trait be \(G=\sum_j g_j\beta_j\). If genotypes are uncorrelated across variants and each has variance one, then, conditional on the effects, \(\operatorname{Var}(G\mid\boldsymbol{\beta})=\sum_j\beta_j^2\). If the effects have mean zero, taking expectation over their distribution gives \(\operatorname{E}[\operatorname{Var}(G\mid\boldsymbol{\beta})]=\sum_j\operatorname{Var}(\beta_j)\). The genotype conditions matter: this expression is not generally valid without accounting for genotype covariance.

Assumptions, Scope, and Common Mistakes

For the additive variance formula to follow from the random-contribution construction, the annotation-specific contributions must have the stated variances and their cross-covariances must be zero. Independence is one way to ensure this. If contributions are correlated, the variance includes extra terms:

The simple additive model omits those cross-terms. It is also important to distinguish binary annotations from other codings. The equation here uses \(a_{jc}\in\{0,1\}\), so an active annotation contributes \(\tau_c\). A different nonnegative weighting scheme would require an explicit definition of how a weight scales the variance; it should not be silently substituted into the binary model.

A common mistake is to interpret \(\tau_c\) as the average effect of variants in annotation \(c\), or as a fixed increment in every variant’s effect. That reasoning confuses effect size with effect variance. The correction is that \(\tau_c\) adds to the variance of a distribution of effects; individual effects remain random under the model, and their mean is specified separately. Another mistake is to add contributions for overlapping annotations while also treating those annotations as mutually exclusive categories. The correction is to use the actual annotation indicators: when both are one, both terms enter the sum.