Why Weight Variants by Frequency?
A burden combines dosage values across a prespecified set of variants. An unweighted burden counts each minor-allele copy equally, but a weighted burden can give different influence to variants with different allele frequencies. Frequency-based weights are motivated by the idea that rarer alleles may be more informative for a trait in some settings. Rarity alone, however, does not establish that a variant is harmful or associated with the trait. The weight is a chosen scoring rule, not evidence of a biological effect.
This tutorial uses minor-allele frequency \(p_j\) for variant \(j\), with \(0 < p_j \leq 0.5\), and genotype dosage \(G_{ij}\) for participant \(i\). Dosage is the number of minor-allele copies, usually 0, 1, or 2 for a diploid genotype. The variant set, or mask, is fixed before calculating the burden; frequency weights do not decide which variants belong in that set.
The examples assume that included genotypes have passed quality checks, the participants have comparable genotype data, and allele frequencies come from a suitable reference or analysis sample. Frequencies should be estimated consistently for the ancestry and population being analyzed. A frequency estimate based on very few allele copies may be unstable, so its corresponding weight can also be unstable. The burden itself is a constructed predictor; calculating it does not test an association or establish causation.
Two Frequency-Based Weighting Rules
A direct inverse-frequency rule assigns larger weights to lower-frequency variants. One simple choice is the raw inverse-MAF weight \(w_j=1/p_j\). This grows rapidly as \(p_j\) approaches zero. Other procedures use a softened inverse-frequency rule, such as \(1/\sqrt{p_j}\), or a genotype-variance scaling such as \(1/\sqrt{p_j(1-p_j)}\). These are distinct choices, not interchangeable names for one formula. Here, raw inverse MAF is used when the tutorial says “inverse-frequency weight,” so the calculation and its strong rare-allele emphasis are explicit.
A beta-distribution weight evaluates a beta density at the variant’s MAF. A common choice for rare-variant weighting is the Beta(1,25) density:
where \(B(1,25)\) is the beta function, and \(B(1,25)=1/25\). This weight is large for very small positive MAFs and decreases as MAF increases. Unlike raw inverse MAF, it remains finite as \(p_j\) approaches zero: its limiting value is 25. A beta density evaluated at a frequency is a weighting function; it is not the probability that a variant is causal, nor is it a posterior probability.
| Rule | Weight | Behavior as MAF gets very small |
|---|---|---|
| Unweighted burden | \(w_j=1\) | No additional frequency emphasis |
| Raw inverse MAF | \(w_j=1/p_j\) | Grows without bound |
| Beta(1,25) | \(w_j=25(1-p_j)^{24}\) | Approaches 25 and stays finite |
Worked Calculations
Worked Example: Calculating Weights at Two Frequencies
Suppose two hypothetical variants are included in a fixed mask. Variant 1 has MAF \(p_1=0.01\), and variant 2 has MAF \(p_2=0.20\). Calculate both raw inverse-frequency and Beta(1,25) weights.
Raw inverse-frequency weights. Apply \(w_j=1/p_j\):
Thus, a minor-allele copy at variant 1 contributes 100 burden units, compared with 5 units for a copy at variant 2. The contribution ratio is \(100/5=20\).
Beta(1,25) weights. Apply \(w_j=25(1-p_j)^{24}\):
The beta weights also favor the rarer variant, but the contribution ratio is about \(19.64195/0.11806\approx166.4\), which is much larger than the inverse-frequency ratio for these two frequencies. A comparison of formulas alone does not imply that one choice is universally more aggressive: the relative emphasis depends on the frequencies being compared and on the weight rule.
Worked Example: How Weights Change One Participant’s Burden
For the same hypothetical variants, suppose participant A has dosages \(G_{A1}=1\) and \(G_{A2}=2\). Assume these are valid diploid genotype calls and the stated frequencies are the frequencies used to assign weights. Calculate the unweighted, raw inverse-frequency, and beta-weighted burdens.
Unweighted burden. Each copy has weight 1:
Raw inverse-frequency burden. Use weights 100 and 5:
Beta-weighted burden. Use the weights from the preceding calculation:
The unweighted burden records three minor-allele copies. The weighted values are on different numerical scales because they use different weights; 110 and 19.87807 are not counts of allele copies. Under each weighted rule, the single copy at the rarer first variant makes a much larger contribution than either copy at the second variant.
Worked Example: A Singleton and the Risk of Extreme Weighting
Consider a hypothetical sample of 100 diploid participants. At one variant, exactly one minor-allele copy is observed, giving an estimated MAF of \(1/(2\times100)=0.005\). At a second variant, 10 minor-allele copies are observed, giving an estimated MAF of \(10/200=0.05\). Suppose a hypothetical participant carries one copy at each variant. Compare their contributions under raw inverse MAF and Beta(1,25) weights.
For raw inverse MAF, the weights are \(1/0.005=200\) and \(1/0.05=20\). Since the participant has dosage 1 at each site, their raw inverse-frequency burden is:
For the beta weights, the contributions are:
Therefore, the beta-weighted burden is approximately \(22.166+7.29973=29.4657\). The singleton still contributes more than the second variant, but the contribution ratio is about \(22.166/7.29973\approx3.04\), rather than 10 under raw inverse MAF. In a sample this small, the singleton frequency is determined by one observed allele copy; a different sample could yield a substantially different estimate and weight.
Why the Weighting Rules Behave Differently
The distinction can be seen directly from how each formula changes with MAF. For raw inverse MAF, \(w(p)=1/p\), so reducing frequency from \(p\) to \(p/2\) doubles the weight. Repeatedly halving a small frequency keeps multiplying its weight by two. For the beta rule, \(w(p)=25(1-p)^{24}\), and its derivative is:
The negative derivative shows that the beta weight decreases with frequency. Its boundary behavior is different from the inverse rule:
The beta rule therefore limits the absolute weight assigned to an allele approaching zero frequency, while the raw inverse-MAF rule has no such cap. This does not make the beta rule immune to all problems: a very rare allele can still receive a large contribution relative to more common variants, and the selected beta parameters determine the resulting profile.
Define the mask independently of the participant’s burden calculation. The weighting formula does not replace variant-inclusion criteria.
Use a clearly specified, appropriate frequency source and record how frequencies were estimated. Very small allele counts make estimates particularly sensitive to sampling.
Calculate each \(w_j\) from its MAF, then calculate \(B_i=\sum_jw_jG_{ij}\) using the same weights for every participant.
Check whether a small number of ultra-rare sites dominate the burden. The total alone can conceal that pattern.
Interpretation, Scaling, and Practical Limits
A weighted burden is a numerical summary, not a universal biological quantity. Its scale depends on the chosen weight rule and, for a beta density, its parameters. Multiplying every weight by the same positive constant multiplies every participant’s burden by that constant; it does not change participant rankings. It does change the numerical scale of the burden, so any coefficient from a model using it would need to be interpreted on that scale. Choosing variant-specific weights changes relative contributions and can change participant rankings.
Frequency estimates deserve particular care for ultra-rare alleles. When a variant is seen once or a few times, a small change in the observed allele count can cause a large proportional change in \(1/p_j\). If frequencies are calculated separately in groups defined by outcome, weights may differ by group for reasons tied to the outcome itself; a common, prespecified frequency source avoids that particular complication. Population structure also matters: a frequency estimated in a population unlike the analyzed participants may not represent their allele frequencies.
Finally, frequency weighting is not a substitute for genotype quality control or for checking missingness and callability. A miscalled or selectively missing rare variant can receive a large inverse-frequency weight and distort a burden. The quality and missingness checks from earlier material remain necessary before constructing a score. Nor does weighting repair a mismatch between the variants in the mask and the scientific question.
Common Mistake: Treating Rarity as Proof of Importance
- Wrong reasoning: “This allele is rarer, so it must be more damaging and should dominate the participant’s score.”
- Correction: Frequency weighting encodes a chosen analysis preference. Rarity by itself does not show that an allele is functional, harmful, or associated with the trait.
- Wrong reasoning: “A finite beta weight means ultra-rare variants cannot dominate.”
- Correction: A beta weight has a finite limit as MAF approaches zero, but it can still be large relative to other included weights. Inspect per-variant contributions and assess whether conclusions depend on extremely rare sites.
- Wrong reasoning: “The weighted burden is the number of minor alleles carried.”
- Correction: Only the unweighted sum has that interpretation. A weighted burden is a weighted score whose units and scale come from its specified weights.
Check Your Understanding
Answer each question using the definitions and calculations above.
- A variant has MAF \(0.025\). What is its raw inverse-frequency weight, and what is its Beta(1,25) weight as a formula?
- A participant has dosage 2 at a variant with weight 4. What is that variant’s contribution to the participant’s burden?
- Why can raw inverse-MAF weights make a singleton variant influential in a small sample?
- Does a large beta-distribution weight mean that the variant is likely causal? Explain.
- How does multiplying every weight by the same positive constant affect burden rankings and the burden scale?