Effect Alleles, Dosage Coding, and Score Scaling

Why Dosage Coding and Score Scale Matter

A polygenic score combines allele dosages with estimated per-allele effects. Its numerical value depends on what each dosage counts and how that dosage is represented. Two analysts can use the same genotypes and biological association information yet report scores with different signs or scales if one counts the opposite allele, centers the genotypes, or standardizes them without transforming the coefficients.

These changes need not alter the prediction represented by the model. A consistent change of coding can be accompanied by a corresponding change in coefficients and intercept. The key is to distinguish a change in the representation of the predictors from a change in the underlying fitted value. The introductory material on defining the polygenic risk score estimand motivates why the target quantity and its units should be stated before score values are interpreted.

A dosage may be an expected allele count rather than an observed integer, for example when genotype uncertainty is represented by a dosage of 0.73. The interpretation of \(\beta_j\) remains a change per additional allele copy under the fitted model. A positive coefficient means that larger dosage of the designated effect allele is associated with a higher modeled trait value; a negative coefficient means the modeled value decreases as that dosage increases. This interpretation concerns association in the model and does not, by itself, establish causation.

Effect-Allele Orientation and Score Direction

The effect allele is the allele whose dosage is multiplied by the reported coefficient. A coefficient is not interpretable apart from this coding: switching the counted allele switches the dosage direction. At a diploid biallelic variant, if \(G_{ij}\) counts copies of one allele, the dosage \(G'_{ij}\) for the other allele is \(2-G_{ij}\). A coefficient for the alternative coding must also change sign, with an intercept adjustment if the model includes an intercept.

For a single variant, the original contribution is \(\beta G\). Under the alternative coding, use \(\beta'=-\beta\). Then \(\beta'G'=-\beta(2-G)=\beta G-2\beta\). The contributions differ by a constant, not by a change in how individuals are ordered. Centering both dosages at their corresponding means removes that constant and gives identical contributions. Across variants, the total offset is the sum of the per-variant constants.

The arithmetic above assumes that the dosages truly refer to opposite alleles at the same biallelic variant and use the same ploidy. In practice, the variant and effect-allele identity must be aligned between the estimates and the target genotypes; allele harmonization addresses that prerequisite. A sign change made without changing the counted allele, or a dosage reversal without reversing the coefficient, changes the score rather than merely recoding it.

Raw and Centered Scores

A raw score uses the observed dosage directly. Its origin depends on the coding: an individual with zero copies at every included variant has raw score zero, regardless of the average genotype in the target population. A centered score subtracts each variant’s reference mean dosage before weighting. If \(p_j\) is the frequency of the counted effect allele in a specified reference population, the expected dosage mean is \(2p_j\) for diploid genotypes. The centered score is therefore \(S_{i,c}=\sum_j\beta_j(G_{ij}-2p_j)\).

Subtracting these means changes the score only by a constant shared by all individuals, provided the frequencies and coefficients are fixed. In particular, \(S_{i,c}=S_i-\sum_j2p_j\beta_j\). The centered score has mean zero in the reference population when the dosage means are \(2p_j\) there. It does not generally have mean zero in a different target population, nor does centering alone make the score have unit variance.

Standardizing Dosages and Rescaling Effects

Standardizing a genotype dosage means subtracting its mean and dividing by a positive scale, commonly its standard deviation. Under Hardy–Weinberg equilibrium for a diploid biallelic variant with effect-allele frequency \(p_j\), the dosage mean is \(2p_j\), and the dosage variance is \(2p_j(1-p_j)\). The corresponding population-standardized dosage is \(Z_{ij}=(G_{ij}-2p_j)/\sqrt{2p_j(1-p_j)}\), when \(0<p_j<1\). The denominator is the standard deviation, not the variance.

If \(\beta_j\) is an effect per allele copy, the coefficient for \(Z_{ij}\) must be \(\gamma_j=\beta_j\sqrt{2p_j(1-p_j)}\). Then the weighted standardized contribution is exactly the centered contribution: \(\gamma_jZ_{ij}=\beta_j(G_{ij}-2p_j)\). This identity is a change of units for the predictor and its coefficient, not a new association estimate. In contrast, multiplying standardized genotypes by the original per-allele coefficients generally gives a different quantity.

The Hardy–Weinberg variance formula is a modeling choice about the genotype distribution. It should not be silently treated as the observed variance in every target sample. Population structure, relatedness, sampling variation, or dosage uncertainty can make the target-sample variance differ from \(2p_j(1-p_j)\). When using a frequency-based standardization, record the allele-frequency source and standardization convention so that the score’s scale is reproducible.

Comparing Score Conventions and Checking Their Use

A further operation is to standardize the completed score across a sample, for example by subtracting its sample mean and dividing by its sample standard deviation. That is different from standardizing each genotype before combining them. A score standardized this way is expressed in score standard-deviation units, not in the original trait units. Report which operation was used and which population supplied the mean and scale.

Useful checks include verifying that effect-allele labels match the dosages, that the sign of each coefficient is interpreted against the allele actually counted, and that score differences attributed to standardization are not caused by using coefficients on the wrong scale. If a variant is nearly monomorphic, its frequency-based standard deviation is near zero; at frequency zero or one, the formula cannot standardize it at all. Such a variant provides no dosage variation in that population, and division by its standard deviation is undefined or unstable.

Raw scores are convenient when coefficients are reported per allele and scores must retain their direct additive interpretation. Centered scores make the reference origin explicit and remove a constant offset. Standardized genotypes can put variants on comparable predictor scales, but that convenience is useful only when coefficients are converted to the matching units. These conventions are not interchangeable unless the transformations are tracked. Score magnitudes from different cohorts or software implementations should not be compared as if they shared a scale until allele orientation, centering frequencies, standard deviations, and any final score standardization have been checked.

Common Mistakes and Limitations

• Reading a coefficient without its allele. A positive coefficient is an increase per copy of the specified effect allele, not an increase per copy of whichever allele a data file happens to count.
• Flipping dosage but retaining the coefficient sign. For a diploid biallelic recoding \(G'=2-G\), the matching coefficient changes from \(\beta\) to \(-\beta\); otherwise the model contribution changes direction.
• Calling a centered score a standardized score. Centering changes the origin but does not divide by a standard deviation.
• Using per-allele coefficients with standardized dosages. This mixes units. For standard deviation \(s_j\), use \(\beta_js_j\) to preserve the centered contribution.
• Assuming every reported score has trait units. A weighted sum with per-allele coefficients has trait units, whereas a score subsequently divided by its sample standard deviation is dimensionless.

These transformations preserve a specified linear predictor only when the genotype transformation, coefficient transformation, and any intercept adjustment are all applied consistently. They do not resolve uncertainty in effect estimates, make different populations have the same genotype distribution, or establish that the score has the same predictive performance in every population.