Handling Missing Genotypes in Score Calculation

Why Missing Genotypes Matter for a Score

A polygenic risk score (PRS) combines effect-allele dosages with corresponding weights. If a target genotype is missing, its contribution cannot be calculated directly. Treating that missing value as zero, dropping the affected variant, or dropping the person are different choices: each defines a different calculation and can change the score’s scale, variation, or target population.

Let \(J\) be the set of variants used in a score, let \(G_{ij}\) be person \(i\)’s dosage at variant \(j\), and let \(\beta_j\) be the effect weight for that variant. Dosages are expected effect-allele counts on the 0-to-2 scale. For a person with complete genotypes, the score is

A missing genotype is not a dosage of zero: zero means that no effect alleles were observed. Write \(R_{ij}=1\) when the dosage is observed and \(R_{ij}=0\) when it is missing. The missingness pattern \(R_{ij}\) indicates which terms in the sum are unavailable.

For a biallelic variant, a commonly used population mean dosage is \(\mu_j=2p_j\), where \(p_j\) is the effect-allele frequency in a suitable reference population. An empirical mean among observed target dosages is another possible estimate. Neither is automatically appropriate for every target cohort: the mean should represent the population to which the imputation is being applied, and its source should be documented.

Mean-Dosage Imputation: What It Does

With mean-dosage imputation, the score is calculated over the original variant set, substituting \(\mu_j\) only where a dosage is missing:

This preserves the score’s list of weighted variants. It does not recover the unknown genotype of a particular person. Rather, it assigns that person the mean dosage for the missing term, so that term contributes the same amount to everyone missing the same variant.

A Justification Under Missing Completely at Random

The properties of mean imputation depend on how genotypes become missing and on how well \(\mu_j\) represents their mean. For a simple result, suppose that at variant \(j\), the dosage \(G_{ij}\) has mean \(\mu_j\) and variance \(\sigma_j^2\), and the observation indicator \(R_{ij}\) is independent of \(G_{ij}\), with \(P(R_{ij}=1)=q_j\). This is a missing-completely-at-random model for that dosage: the chance of observing it does not depend on the genotype. Also suppose the imputation value is exactly \(\mu_j\).

The imputed dosage has the same expectation as the original dosage:

Its variance is smaller. Since \(G^*_{ij}-\mu_j=R_{ij}(G_{ij}-\mu_j)\), and \(R_{ij}^2=R_{ij}\),

Thus, under these assumptions, the imputed dosage has the correct mean but only the fraction \(q_j\) of the original dosage variance. For a score, the expected contribution from this variant remains \(\beta_j\mu_j\), while its variance contribution is reduced. If genotypes at distinct variants are independent and their missingness indicators satisfy the corresponding independence assumptions, the score variance is \(\sum_{j\in J}\beta_j^2q_j\sigma_j^2\). With linkage disequilibrium, cross-variant covariances also contribute, so this sum is not generally the full score variance.

This calculation explains an important limitation: mean imputation can preserve an average under the stated model while compressing individual differences. If missingness depends on genotype, or the imputation mean does not match the missing individuals’ genotype mean, even the expectation need not be preserved.

Variant Exclusion and Sample Exclusion

Variant exclusion removes the same term from every person’s score. If \(J'\subset J\) is the retained set, then the score becomes \(\sum_{j\in J'}\beta_jG_{ij}\). This avoids assigning any dosage to the excluded variant, but changes the score definition. People may still have different amounts of usable information if additional missing genotypes remain at the retained variants.

Sample exclusion instead keeps the score definition fixed but removes people who do not meet a completeness rule. For example, a complete-case rule may retain only people with observed dosages at every variant in \(J\). It produces scores for the retained participants, not for the full target cohort. If missingness is associated with ancestry, genotyping quality, or another participant characteristic, the retained group may differ systematically from those excluded.

Choosing and Auditing a Strategy

These approaches answer different practical needs. Mean imputation retains participants and the planned variants; excluding a problematic variant may be reasonable when its missingness is substantial or its genotypes are unreliable; excluding samples can be useful when a predefined quality rule identifies insufficient genotype data. No strategy repairs poor genotype quality or guarantees comparable scores across datasets.

Before calculating scores, inspect missingness by variant and by participant, and report the fraction of missing dosages. Check whether missingness varies across relevant participant characteristics, such as ancestry groups or genotyping batches. Compare score distributions and the number of contributing variants under the chosen procedure. For comparisons between cohorts, use the same effect-allele coding, weights, variant set, and imputation rule wherever possible. If cohort-specific dosage means are used, the imputed contribution can differ between cohorts even when the weights match.

A common mistake is to replace every missing dosage by zero because zero is a valid genotype count. That reasoning confuses “not observed” with “observed to carry zero effect alleles.” The correction is to make the missing-data rule explicit: impute a stated mean, exclude a stated variant set, or exclude participants under a stated completeness criterion. Another mistake is to compare raw scores after one cohort has had variants removed. A common set of weights alone does not make scores comparable if the contributing variants differ.