Genotype Scaling and the Units of Score Weights

Why Genotype Scale Matters

A polygenic score is a weighted sum of genotype values. Its numerical value therefore depends on how those genotypes are coded, even when the underlying genetic contrast is unchanged. A weight of 0.10 is not interpretable by itself: it might describe the change in a phenotype per additional effect allele, per standard deviation of genotype, or per standard deviation of phenotype.

The key principle is simple: when genotype values are rescaled, weights must be transformed inversely to preserve the same genetic contribution. Centering has a related but distinct effect: it subtracts a constant contribution from every person. These distinctions clarify both the arithmetic of score construction and the units in which a score should be reported.

Definition: Let \(G_j\) be the dosage of the effect allele at variant \(j\), usually between 0 and 2. A weighted score is \(S=\sum_j w_jG_j\). If \(G_j\) is an allele count and \(w_j\) is a regression effect per additional effect allele, the contribution \(w_jG_j\) is in the phenotype units associated with that regression model. A centered dosage is \(G_j-\mu_j\), where \(\mu_j\) is its mean; a standardized dosage is \((G_j-\mu_j)/\sigma_j\), where \(\sigma_j\) is its standard deviation.

Dosage, Centering, and Standardization

For a biallelic variant, the usual effect-allele dosage \(G_j\) counts copies of the chosen effect allele: 0, 1, or 2 for an observed genotype. Imputed dosages can take intermediate values because they represent expected allele counts. In either case, one dosage unit represents one additional copy of that allele.

The mean dosage is often written \(\mu_j\). If the effect-allele frequency is \(p_j\), then \(\mu_j=2p_j\) under the usual allele-count definition. The genotype standard deviation \(\sigma_j\) may be calculated empirically in a sample. Under Hardy–Weinberg equilibrium, the theoretical variance of allele count is \(2p_j(1-p_j)\), so its standard deviation is \(\sqrt{2p_j(1-p_j)}\). Using an empirical standard deviation does not require that equilibrium assumption; it simply describes the observed dosage variation in that sample.

Centering and standardization change the values supplied to the score, not the biological allele contrast itself. A centered dosage has mean zero in the reference sample. A standardized dosage has mean zero and standard deviation one there. As a result, standardized-genotype weights describe effects per one standard deviation of that variant's dosage, rather than per additional allele.

Key result: If a phenotype regression uses \(Y=a+\beta G+\varepsilon\), then its equivalent regression using \(Z=(G-\mu)/\sigma\) is \(Y=(a+\beta\mu)+(\beta\sigma)Z+\varepsilon\). The standardized-genotype coefficient is \(\gamma=\beta\sigma\). Thus \(wG\) and \((w\sigma)Z\) differ only by the constant \(w\mu\); the centered contribution \(w(G-\mu)\) equals \((w\sigma)Z\).

Worked Example: Converting an Allele-Count Weight to a Standardized Weight

Suppose a hypothetical GWAS estimates an effect of \(0.12\) phenotype units per effect allele at one variant. In its reference sample, the effect-allele frequency is \(p=0.30\), and assume Hardy–Weinberg equilibrium for the theoretical genotype variance. Compare the raw-dosage, centered-dosage, and standardized-dosage contributions for a person with dosage \(G=1.6\).

The mean dosage is \(\mu=2p=0.60\). The genotype variance and standard deviation are

\operatorname{Var}(G)=2(0.30)(0.70)=0.42,\qquad \sigma=\sqrt{0.42}=0.6481.

With the raw dosage, the person's score contribution is \(0.12(1.6)=0.192\) phenotype units. With the centered dosage, it is

0.12(1.6-0.60)=0.12(1.00)=0.120\text{ phenotype units}.

The standardized dosage is \(Z=(1.6-0.60)/0.6481=1.543\). Its coefficient must be \(0.12(0.6481)=0.07777\) phenotype units per genotype standard deviation. The standardized contribution is therefore \(0.07777(1.543)=0.120\) phenotype units, up to rounding. Raw dosage gives a contribution larger by the constant \(0.12(0.60)=0.072\). Centering removes that baseline contribution; it does not change the contrast between people.

Why Inverse Weight Rescaling Works

The result follows directly from rewriting a regression equation. Suppose the genotype coding changes from \(G\) to \(G^*=a+bG\), where \(b\ne0\). Solving for the original coding gives \(G=(G^*-a)/b\). Substituting into the model \(Y=\alpha+\beta G+\varepsilon\) yields

Y=\left(\alpha-\frac{\beta a}{b}\right)+\frac{\beta}{b}G^*+\varepsilon.

The new slope is \(\beta^*=\beta/b\), and the intercept changes to account for the shift \(a\). The fitted value is unchanged because

\left(\alpha-\frac{\beta a}{b}\right)+\frac{\beta}{b}(a+bG)=\alpha+\beta G.

For standardization, \(a=-\mu/\sigma\) and \(b=1/\sigma\), so \(\beta^*=\beta\sigma\). For centering alone, \(a=-\mu\) and \(b=1\), so the slope stays \(\beta\) and the intercept becomes \(\alpha+\beta\mu\). These identities require a nonzero scale factor and consistent allele orientation. They are algebraic equivalences; they do not depend on a particular distribution of genotypes.

For a multi-variant score, the same argument applies variant by variant. Centering every dosage changes the score by a fixed amount, \(\sum_j w_j\mu_j\), provided the same means and weights are used for every individual. Therefore, centered and uncentered scores have the same differences between individuals and the same ranking, although their numerical zero points differ. If a model includes an intercept, this constant can be absorbed into it. If a score is reported or used without a recalibrated intercept, however, its absolute values are different.

Worked Example: Preserving a Score After Rescaling Dosage

Consider a hypothetical single-variant regression with an intercept of \(2.0\) trait units and a dosage effect of \(0.30\) trait units per allele. A person has dosage \(G=1\), and the dosage is recoded as \(G^*=2G\). Find the equivalent coefficient and intercept, then verify the fitted value.

Here \(a=0\) and \(b=2\), so the transformed coefficient is \(\beta^*=\beta/b=0.30/2=0.15\) trait units per recoded dosage unit. The intercept remains \(2.0\). The original fitted value is \(2.0+0.30(1)=2.30\) trait units. Under the new coding \(G^*=2\), the fitted value is \(2.0+0.15(2)=2.30\) trait units. The recoded genotype is twice as large, so its coefficient is half as large; the prediction is unchanged.

If the recoding had instead counted the other allele, \(G^*=2-G\), then \(a=2\) and \(b=-1\). The transformed slope would be \(-0.30\), and the intercept would be \(2.0+0.60=2.60\). For this person \(G^*=1\), giving \(2.60-0.30(1)=2.30\) trait units, again the same fitted value. This sign reversal is why allele orientation must be harmonized before score weights are applied.

Phenotype Units and the Meaning of a Score

Genotype scaling is only one source of units. The phenotype may also be reported on a raw or standardized scale. If \(Y^*=(Y-\mu_Y)/\sigma_Y\), then a regression coefficient for \(Y^*\) is measured in phenotype standard deviations per genotype unit. Multiplying that coefficient by \(\sigma_Y\) converts it to raw phenotype units per genotype unit, as long as the same phenotype standard deviation is applicable.

A score made from linear-regression effects on a raw quantitative phenotype is expressed in that phenotype's units, relative to the model's intercept or baseline. For example, a score contribution of 1.2 could represent 1.2 mmHg when the GWAS outcome was blood pressure measured in mmHg. If the GWAS outcome was standardized, the corresponding score is in standard-deviation units instead. The score is not automatically a calibrated individual prediction: calibration also depends on the fitted intercept, the model specification, and whether the effects transport to the people being scored.

For a binary phenotype analyzed with logistic regression, a coefficient is on the log-odds scale. A per-allele coefficient \(\beta\) corresponds to an odds ratio \(\exp(\beta)\) per additional effect allele. The weighted sum of dosages and such coefficients is a contribution to a log-odds linear predictor, not a probability. Converting that linear predictor to a probability requires the intercept and the logistic transformation \(1/(1+\exp(-\eta))\). A log-odds score should not be described as a risk difference or as a probability without that conversion and the relevant model parameters.

Worked Example: Interpreting Weights on Different Phenotype Scales

Suppose a hypothetical quantitative-trait GWAS reports an effect of \(0.08\) phenotype standard deviations per effect allele. The trait's standard deviation in the stated reference population is \(15\) units. The genotype dosage has standard deviation \(0.80\). Convert the effect first to raw phenotype units per allele, then to raw phenotype units per genotype standard deviation.

The raw-scale per-allele effect is \(0.08(15)=1.20\) trait units per allele. The standardized-genotype effect is \(0.08(0.80)=0.064\) phenotype standard deviations per genotype standard deviation. Converting that to raw phenotype units gives \(0.064(15)=0.960\) trait units per genotype standard deviation. This also follows directly from \(1.20(0.80)=0.960\).

Thus the three numerical effects refer to different contrasts: \(0.08\) phenotype standard deviations per allele, \(1.20\) trait units per allele, and \(0.960\) trait units per genotype standard deviation. They are consistent after their units are stated. A standardized effect is not itself a raw-unit change, and a per-standard-deviation genotype effect is not a per-allele effect.

Practical Checks Before Applying Weights

Before multiplying genotypes by reported effects, establish the coding and scale used for both. The effect allele named by the GWAS must match the allele counted in the score data. The weight's phenotype scale and genotype scale must also be identified. A coefficient reported per allele can be applied to allele-count dosage; a coefficient per standardized dosage requires standardized genotypes or conversion back to per-allele units.

Standardized genotype weights can depend on the genotype standard deviation used to define the standardized variable. Since allele frequencies differ among samples, the standard deviation may differ too. A weight expressed per genotype standard deviation is therefore tied to a specified scaling convention or reference population. If genotypes are standardized with a different standard deviation, the coefficient must be converted consistently to preserve the same contribution. In contrast, an allele-count effect remains interpretable per additional effect allele, subject to the usual requirements that the allele and phenotype definitions match.

These scaling identities do not establish that a set of GWAS effects are jointly valid prediction weights. They only show how to express the same given effects under different genotype or phenotype codings. Questions about how correlated variants affect marginal GWAS effects are a separate issue.

Common Mistakes

Keeping the same weight after changing genotype scale. If \(G\) is multiplied by \(b\), its equivalent coefficient is divided by \(b\). Otherwise the score contribution is altered rather than merely recoded.
Assuming centering changes the genetic contrast. Centering subtracts the same dosage mean for everyone at a variant. It shifts the score baseline but preserves differences between individuals when weights and means are fixed.
Calling a standardized-genotype effect a per-allele effect. A coefficient on \(Z=(G-\mu)/\sigma\) is per genotype standard deviation. Multiply it by \(1/\sigma\) to express it per allele count.
Interpreting a logistic score as a probability. Logistic-regression weights add on the log-odds scale. A probability requires the intercept and inverse-logit transformation.
Ignoring allele direction. Counting the opposite allele reverses the sign of the per-allele coefficient. Harmonization must precede scoring.

Key takeaway: To preserve a score under a genotype recoding \(G^*=a+bG\), transform the weight to \(w^*=w/b\) and adjust the intercept for the shift. Always state whether weights are per allele or per genotype standard deviation, and whether their phenotype units are raw, standardized, or log odds.

Check Your Understanding

Use the definitions and transformations above to answer the following questions.

A genotype dosage is centered by subtracting its mean. What happens to its regression slope, and what happens to the intercept?
A GWAS reports a coefficient per standardized genotype. What information is needed to convert it to a per-allele coefficient?
If a genotype is recoded as twice its original value, by what factor must its weight change to preserve the contribution?
Why can two scores based on centered and uncentered dosages rank individuals identically but have different numerical values?
A logistic-regression weight is reported per effect allele. On what scale does its weighted contribution lie, and what additional model component is needed to obtain a probability?