Functional Weights From Variant Annotations

Why Translate Annotations Into Weights?

A functional annotation describes a variant in biological terms: for example, a variant may be synonymous, missense, or protein-truncating. A burden calculation needs numbers, however, so each included variant must be assigned a numeric weight. That translation is a modeling choice: it makes the annotation usable in a score, but it does not turn the annotation into a measured effect on the trait.

The key is to make the mapping explicit and apply it consistently. If annotations are uncertain, a single fixed score can hide that uncertainty. Representing uncertain weights with a probability distribution makes it possible to report the expected burden and the amount of score variation attributable to annotation uncertainty. The resulting uncertainty is about the annotation-based score, not a confidence interval for a genetic effect.

Definition: For variant $j$, a functional weight $w_j$ is a prespecified numeric value assigned from its functional annotation. If person $i$ has alternate-allele count $G_{ij}$ at that variant, their weighted burden is $B_i=\sum_j w_jG_{ij}$, where the sum is over the variants included in the analysis.

From Ordered Labels to a Scoring Rule

Functional labels can often be ordered by anticipated consequence, but an ordering alone does not specify numeric distances. Saying that one annotation is more severe than another does not establish that its weight should be twice as large, or even that the difference between two adjacent categories is meaningful. A scoring rule therefore needs to state both its categories and the numeric value assigned to each.

For example, an analysis might prespecify weights of 0 for synonymous variants, 0.6 for damaging missense variants, and 1 for protein-truncating variants. These values encode a chosen ranking and scale; they are not universal biological constants. Another defensible analysis might use a different mapping, but it should specify the mapping before interpreting the resulting score. Functional consequence labels and transcript choice can affect the category assigned, while uncertainty in prediction can make that category uncertain.

The weight is separate from the genotype count. Under an allele-count definition, $G_{ij}$ is 0, 1, or 2 copies of the alternate allele. A carrier indicator instead records whether a person carries at least one alternate allele; that is a different burden definition and should be stated explicitly. In either case, a weight of 0 contributes nothing, and larger weights contribute more to the score per counted allele.

Worked Example: Applying a Prespecified Annotation Scale

Suppose a hypothetical analysis includes three variants and uses the following rule: synonymous variants receive weight 0, damaging missense variants receive weight 0.6, and protein-truncating variants receive weight 1. The table gives the annotations and alternate-allele counts for one hypothetical person.

Variant	Annotation	Weight $w_j$	Allele count $G_{ij}$
1	Protein-truncating	1	1
2	Damaging missense	0.6	2
3	Synonymous	0	1

The weighted burden is the sum of the product of each weight and its allele count:

B_i=\sum_j w_jG_{ij} =(1)(1)+(0.6)(2)+(0)(1) =1+1.2+0 =2.2.

The score is 2.2 on this prespecified scale. It is not 2.2 alternate alleles: the allele counts have been transformed by the functional weights. Nor does a score of 2.2 mean the person has a trait effect of a particular size. The score summarizes weighted variant counts under the chosen rule.

Representing Annotation Uncertainty

If the annotation for a variant is uncertain, its weight can be treated as a random variable $W_j$ over the possible scoring values. This is a mathematical representation of uncertainty in the annotation or prediction. Its probabilities should come from an explicitly chosen and appropriate source, such as a calibrated prediction model or a stated expert assessment; an uncalibrated prediction score does not automatically provide probabilities.

Write $m_j=E(W_j)$ for the mean weight and $v_j=\operatorname{Var}(W_j)$ for its variance. For a person whose genotype counts are treated as fixed, the weighted burden is now a random score $B_i=\sum_j G_{ij}W_j$, because its weights are uncertain. Its expected score is the sum of the genotype counts times their mean weights. Its variance depends on the genotype counts squared, as well as on the variances and covariances of the weights.

Result: Conditional on fixed genotype counts, $$ E(B_i\mid G_i)=\sum_j G_{ij}m_j, $$ and $$ \operatorname{Var}(B_i\mid G_i) =\sum_j G_{ij}^{\,2}v_j+ 2\sum_{j<k}G_{ij}G_{ik}\operatorname{Cov}(W_j,W_k). $$ If weight uncertainties are independent across variants, every covariance term is 0, so only the sum of squared-count variance contributions remains.

This variance describes how much the score would vary as uncertain annotations were resolved or resampled under the stated weight distributions, with the genotype counts held fixed. It does not include variation in genotypes across people, uncertainty from sampling participants, or uncertainty in an association estimate.

Why the Genotype Count Is Squared

The variance formula follows from the variance of a weighted sum. For fixed genotype counts, each $G_{ij}$ is a constant multiplying $W_j$. Multiplying a random variable by a constant multiplies its variance by the square of that constant. Expanding the variance of the sum also produces covariance terms between pairs of weights.

\begin{aligned} \operatorname{Var}(B_i\mid G_i) &=\operatorname{Var}\left(\sum_j G_{ij}W_j\right)\\ &=\sum_j \operatorname{Var}(G_{ij}W_j) +2\sum_{j<k}\operatorname{Cov}(G_{ij}W_j,G_{ik}W_k)\\ &=\sum_j G_{ij}^{\,2}\operatorname{Var}(W_j) +2\sum_{j<k}G_{ij}G_{ik}\operatorname{Cov}(W_j,W_k). \end{aligned}

In particular, if one variant’s weight has variance $v_j$, a person with one copy contributes $1^2v_j=v_j$ to the score variance from that variant. A person with two copies contributes $2^2v_j=4v_j$. Thus two copies give four times the variance contribution of one copy, and twice the standard-deviation contribution, when all else is equal. The mean contribution doubles, but the variance contribution does not merely double.

Worked Example: Expected Burden With Two Uncertain Weights

For a hypothetical person, variant A has allele count 2 and an uncertain weight equal to 1 with probability 0.8 or 0 with probability 0.2. Variant B has allele count 1 and an uncertain weight equal to 0.6 or 0.2, each with probability 0.5. Assume the two weights are independent.

For variant A, the mean and variance of its weight are:

m_A=(0.8)(1)+(0.2)(0)=0.8,\qquad v_A=(0.8)(1-0.8)^2+(0.2)(0-0.8)^2=0.16.

For variant B, they are:

m_B=(0.5)(0.6)+(0.5)(0.2)=0.4,\qquad v_B=(0.5)(0.6-0.4)^2+(0.5)(0.2-0.4)^2=0.04.

The expected burden is:

E(B_i\mid G_i)=(2)(0.8)+(1)(0.4)=1.6+0.4=2.0.

Independence makes the covariance term zero. Applying the squared-count variance formula gives:

\operatorname{Var}(B_i\mid G_i) =(2^2)(0.16)+(1^2)(0.04) =0.64+0.04 =0.68.

The standard deviation due to annotation uncertainty is $\sqrt{0.68}\approx0.825$ score units. Variant A alone contributes 0.64 to the variance because its two copies multiply the weight variance by $2^2=4$. This is an uncertainty summary under the stated discrete distributions and independence assumption, not an interval estimate for a trait effect.

When Uncertain Weights Are Correlated

The independence assumption is convenient, but it is not automatic. Two variants may have related annotation uncertainty, for example when their predictions depend on the same uncertain transcript choice or shared prediction process. Positive covariance increases the variance of the combined burden when both genotype counts are positive; negative covariance reduces it. Marginal means and variances alone are not enough to determine the variance of the total score when such dependence is present.

Worked Example: Adding a Covariance Term

Consider a hypothetical person with allele counts $G_{i1}=1$ and $G_{i2}=2$. Suppose the two uncertain weights have means $m_1=0.7$ and $m_2=0.5$, variances $v_1=v_2=0.09$, and covariance $\operatorname{Cov}(W_1,W_2)=0.03$. These values describe a possible positively associated uncertainty pattern; the covariance is specified as part of the example.

The expected burden is:

E(B_i\mid G_i)=(1)(0.7)+(2)(0.5)=0.7+1.0=1.7.

For the variance, include both individual variance contributions and the covariance contribution:

\begin{aligned} \operatorname{Var}(B_i\mid G_i) &=(1^2)(0.09)+(2^2)(0.09) +2(1)(2)(0.03)\\ &=0.09+0.36+0.12\\ &=0.57. \end{aligned}

If the covariance were incorrectly set to zero, the calculation would give $0.09+0.36=0.45$, understating the variance under the stated positive covariance. The standard deviation with the covariance included is $\sqrt{0.57}\approx0.755$ score units. In a real analysis, the dependence structure must be justified or acknowledged; it should not be inferred from the marginal weight variances alone.

A Practical Construction and Interpretation

A defensible functional-weighted burden starts with a transparent sequence. The included variants are defined by the analysis mask and other prespecified filters. Each included variant receives an annotation using the chosen transcript and consequence rules. The scoring map then assigns a fixed weight, or a distribution over possible weights when uncertainty is explicitly modeled. Finally, the weights are multiplied by each person’s chosen genotype measure and summed.

State the scoring map.
List the annotation categories and their numeric weights. Explain whether the values are ordinal scores, calibrated probabilities, or another scale.

Fix the genotype convention.
Specify whether $G_{ij}$ is an alternate-allele count or a carrier indicator, and use that definition consistently.

Calculate the score.
For each person, sum the product of each included variant’s weight and genotype measure.

Summarize annotation uncertainty if needed.
State the weight distributions and dependence assumptions, then calculate the expected score and variance conditional on genotype.

A functional weight should not be described as an estimated regression coefficient, a causal effect, or a probability of disease unless it was specifically constructed and validated as such. In particular, a numerical ranking of functional severity does not establish how strongly a variant changes a trait. The score is a constructed summary of genotypes under the selected annotation rule; testing or interpreting its relationship with a phenotype requires additional statistical modeling.

Common Mistakes

Treating the annotation order as a complete numeric scale. An ordering does not determine the spacing between weights. State the actual mapping and its rationale.
Calling an arbitrary score a probability. A weight of 0.6 is not a 60% probability of functional impact unless the score is explicitly defined and calibrated that way.
Using genotype counts linearly in the variance. The variance contribution is $G_{ij}^{\,2}v_j$. Two copies therefore contribute four times, not twice, the variance from one copy at the same uncertain weight.
Ignoring dependence without stating it. When weights are correlated, the covariance terms belong in the burden variance.
Interpreting score uncertainty as effect uncertainty. The variance here concerns the constructed annotation-weighted score conditional on genotype, not uncertainty in an association coefficient or causal effect.

Key takeaway: Functional annotations become burden weights only through an explicit scoring rule. Multiply each weight by the chosen genotype count and sum; when weights are uncertain, calculate the expected burden and include squared genotype counts and any relevant covariance terms in its variance.

Check Your Understanding

Use the definitions and calculations above to answer these questions.

A variant has weight 0.4, and a person has two alternate-allele copies. What is its contribution to the weighted burden?
Why does a two-copy genotype contribute four times the annotation-weight variance of a one-copy genotype at the same variant?
If an uncertain weight is 1 with probability 0.25 and 0 with probability 0.75, calculate its mean and variance.
What additional quantity is needed to calculate the variance of a burden when two uncertain variant weights may be dependent?
Explain why a functional weight of 0.6 should not automatically be interpreted as a 60% probability of trait effect.