Deriving Score Weights from GWAS Regression Effects

From a GWAS Regression to a Score Weight

A polygenic score combines genotype information across variants, but each contribution needs a weight. A common source is a genome-wide association study (GWAS): for each variant, a regression estimates how the phenotype changes with the number of copies of a specified allele. That estimated coefficient can supply the weight for the dosage of that same allele in a score.

The link is direct only when the allele counted in the score matches the allele to which the GWAS coefficient refers. Effect-allele harmonization establishes that correspondence. Once alleles are aligned, the coefficient’s sign and magnitude determine the direction and size of that variant’s contribution. The coefficient is still an estimate from a particular regression model, however; it is not automatically the variant’s isolated biological effect.

Definition: For individual $i$ and variant $j$, let $G_{ij}$ be the dosage of the designated effect allele, usually a count from 0 to 2. In a single-variant GWAS, $\hat{\beta}_j$ is the estimated regression coefficient for that dosage. Using these estimates as weights gives the score $$ \mathrm{PGS}_i=\sum_{j=1}^{m}\hat{\beta}_jG_{ij}, $$ where $m$ is the number of included variants. Each dosage must count the allele corresponding to its coefficient.

For a quantitative phenotype, a positive $\hat{\beta}_j$ means that, in the fitted regression, one additional copy of the counted allele is associated with a higher mean phenotype, conditional on any included covariates. A negative coefficient means a lower mean. The coefficient’s units are phenotype units per additional allele copy. It is the estimated regression slope, not a guarantee that every individual carrying the allele has a higher or lower outcome.

A score formed from GWAS coefficients is a weighted sum, not necessarily a complete prediction of the phenotype. An absolute prediction may also require an intercept, covariate effects, and a model fitted for prediction in the relevant population. Here the focus is how the variant regression coefficients supply the genetic weights.

What a Single-Variant Coefficient Estimates

In a quantitative-trait GWAS, a common model for variant $j$ is

Y_i=\alpha+\beta_jG_{ij}+C_i^\mathsf{T}\gamma+\varepsilon_i.

Here $Y_i$ is the phenotype, $G_{ij}$ is the effect-allele dosage, $C_i$ is a vector of covariates, and $\varepsilon_i$ is the residual. The fitted estimate $\hat{\beta}_j$ is the slope for dosage after accounting for the covariates in this regression. Standard covariates may include age, sex, or ancestry-related variables, depending on the study design. For the usual interpretation as a conditional mean slope, the model assumes a linear mean relationship in dosage, an appropriately specified covariate adjustment, and errors with conditional mean zero given the included predictors. The regression also requires enough variation in dosage to estimate a slope.

“Single-variant” describes which genetic predictor is fitted, not necessarily which genetic factor is responsible for the observed association. The coefficient is usually marginal with respect to other variants: the regression includes this variant and the specified covariates, but not every other genetic variant. If nearby or otherwise correlated variants affect the phenotype, the coefficient can partly reflect their associations as well. The precise relationship between marginal effects and correlated genetic effects is important, but the central point here is that a GWAS coefficient is defined by its regression model.

Key distinction: A single-variant GWAS coefficient is the dosage slope in a regression that tests that variant, with the stated covariates. A joint additive-model coefficient is the effect assigned to that variant when multiple variant dosages are included together. Those coefficients need not be equal when genotypes are correlated.

A joint additive model for a quantitative trait can be written as

Y_i=\alpha+C_i^\mathsf{T}\gamma+\sum_{k=1}^{m}G_{ik}\beta_k+\varepsilon_i.

In this model, $\beta_k$ is the coefficient for variant $k$ while holding the other included genotypes and covariates fixed. By contrast, a single-variant regression estimates the slope associated with $G_{ij}$ without conditioning on the other genotypes. Thus, using a single-variant estimate as a score weight is a practical way to weight alleles by their estimated GWAS associations; it does not establish that the estimate equals a joint or causal effect.

Why Marginal and Joint Effects Can Differ

The distinction can be made precise for linear regression. Suppose the joint additive model above is correct and the error has conditional mean zero given the genotypes and covariates. First remove the linear associations with the covariates from the phenotype and each genotype. Denote the resulting residuals by $\widetilde{Y}$ and $\widetilde{G}_k$. The coefficient for variant $j$ in the single-variant regression of $Y$ on $G_j$ and the covariates is

\beta_j^{\mathrm{marg}} =\frac{\operatorname{Cov}(\widetilde{G}_j,\widetilde{Y})} {\operatorname{Var}(\widetilde{G}_j)} =\beta_j+\sum_{k\ne j} \frac{\operatorname{Cov}(\widetilde{G}_j,\widetilde{G}_k)} {\operatorname{Var}(\widetilde{G}_j)}\beta_k.

This expression assumes $\operatorname{Var}(\widetilde{G}_j)>0$, so the dosage retains variation after covariate adjustment. It also uses the population regression coefficient; a sample GWAS reports an estimate of that coefficient. The sum shows why a marginal coefficient can include contributions associated with correlated genotypes. If the residualized genotypes are uncorrelated with all other causal genotypes in the model, the added terms are zero and the marginal and joint coefficients agree. Otherwise they may differ.

Residualize on covariates.
Regress the phenotype and each genotype on the covariates, and retain their residuals. This isolates the part not linearly explained by those covariates.

Use the single-predictor slope.
The adjusted marginal slope is the covariance of the residualized genotype and phenotype divided by the residualized genotype variance.

Substitute the joint model.
The residualized phenotype contains contributions from all residualized genotypes. Taking its covariance with genotype $j$ yields its own effect plus terms for correlated genotypes.

This result is for a linear quantitative-trait model. It explains what “marginal” means without requiring that the single-variant estimate be a pure, isolated effect. The covariance terms are related to linkage disequilibrium after covariate adjustment; their detailed implications for estimating score weights require additional treatment.

Worked Examples

Worked Example: Turning Three GWAS Effects Into Score Contributions

Suppose a hypothetical quantitative-trait GWAS reports coefficients of $0.12$, $-0.08$, and $0.05$ phenotype units per effect-allele copy for three variants. For one hypothetical individual, the aligned effect-allele dosages are 2, 1, and 0, respectively. Calculate the score.

The score is the sum of each coefficient multiplied by the dosage of its corresponding effect allele:

\mathrm{PGS} =(0.12)(2)+(-0.08)(1)+(0.05)(0) =0.24-0.08+0 =0.16.

The first variant contributes $0.24$, the second contributes $-0.08$, and the third contributes zero because this individual has no copies of that effect allele. The resulting score is $0.16$ on the weighted-sum scale. It is not, by itself, an absolute predicted phenotype: no intercept or non-genetic covariate contributions were included. The calculation also relies on the reported dosages counting the same effect alleles that define the GWAS coefficients.

Worked Example: Calculating a Single-Variant Regression Effect

Consider six hypothetical individuals in a GWAS of a quantitative phenotype. Their effect-allele dosages $G$ and observed phenotype values $Y$ are:

Individual	Dosage $G$	Phenotype $Y$
1	0	10
2	0	12
3	1	13
4	1	13
5	2	15
6	2	17

For this illustration, assume no covariates are included and fit the ordinary least-squares regression $Y_i=\alpha+\beta G_i+\varepsilon_i$. The sample means are $\bar{G}=1$ and $\bar{Y}=13.333$. The estimated slope is

\hat{\beta} =\frac{\sum_i(G_i-\bar{G})(Y_i-\bar{Y})} {\sum_i(G_i-\bar{G})^2} =\frac{10}{4} =2.5.

For the numerator, the two dosage-zero observations contribute $(-1)(-3.333)+(-1)(-1.333)$, the dosage-one observations contribute zero, and the dosage-two observations contribute $(1)(1.667)+(1)(3.667)$. Their sum is $10$. The denominator is $1+1+0+0+1+1=4$. Thus the fitted coefficient is 2.5 phenotype units per additional effect-allele copy. The corresponding fitted intercept is $\bar{Y}-\hat{\beta}\bar{G}=13.333-2.5=10.833$. This slope describes the linear association in this sample; the example does not establish a causal effect or account for possible covariates or other variants.

Worked Example: A Marginal Slope That Includes a Correlated Variant

Suppose a hypothetical, covariate-adjusted joint additive model has two residualized genotypes with variances of 1 and covariance of 0.6. The joint coefficients are $\beta_1=0.4$ and $\beta_2=0.3$, and the residual error is uncorrelated with both genotypes. Calculate the single-variant marginal slope for variant 1.

Applying the marginal-slope expression gives

\beta_1^{\mathrm{marg}} =\beta_1+ \frac{\operatorname{Cov}(\widetilde{G}_1,\widetilde{G}_2)} {\operatorname{Var}(\widetilde{G}_1)}\beta_2 =0.4+\frac{0.6}{1}(0.3) =0.58.

The joint coefficient for variant 1 is $0.4$, whereas its marginal coefficient is $0.58$ under these assumptions. The difference, $0.18$, comes from the covariance with variant 2 multiplied by variant 2’s joint coefficient. If the two residualized genotypes were uncorrelated, the covariance term would be zero and the marginal coefficient would equal $0.4$. A single-variant weight can therefore reflect more than the corresponding joint coefficient.

Binary Traits and the Meaning of the Weight

For a binary case-control phenotype, a GWAS commonly uses logistic regression rather than linear regression. In a single-variant model, the coefficient is on the log-odds scale. With dosage $G_j$, a coefficient $\beta_j$ means that the fitted odds ratio for one additional effect-allele copy is $\exp(\beta_j)$, conditional on included covariates under that model. For example, if $\hat{\beta}_j=0.20$, then $\exp(0.20)\approx1.221$: the fitted odds are multiplied by about 1.221 per additional copy, holding the model’s covariates fixed.

Such a coefficient may be used as a weight in a weighted score, but the resulting sum is on the model’s log-odds scale, not a probability scale. Converting a score into an individual’s absolute disease risk requires additional model information and appropriate calibration. Also, the simple covariance formula derived above is for linear regression; it should not be applied as an exact relationship between marginal and joint logistic-regression coefficients.

Assumptions and Practical Interpretation

Before using GWAS regression effects as score weights, establish what each coefficient represents. For a quantitative trait, check the phenotype scale, dosage coding, covariate adjustment, and the regression model used. For a binary trait, check whether the reported effect is a log-odds coefficient or another defined quantity. In either case, the score must use the same allele direction as the coefficient.

A coefficient is estimated with sampling uncertainty. A small estimate may reflect a genuinely small association, random error, or both; its numerical size alone does not establish precision. This tutorial’s score calculation uses the reported estimates as weights but does not address how to select variants or modify estimates to manage uncertainty. Those are separate choices in score construction.

Marginal estimates can also depend on the genotype correlations and covariate structure in the GWAS sample. Consequently, a coefficient need not transfer unchanged to a population with a different correlation structure. This is another reason to describe a coefficient by the model and population in which it was estimated, rather than treating it as a universal variant effect.

Common Mistake

Wrong reasoning: “The GWAS coefficient is the variant’s isolated causal effect, so multiplying it by dosage gives that individual’s causal contribution.”
Correction: A single-variant coefficient is a regression association conditional on the included covariates. Under a joint additive model, correlation with other genotypes can make it differ from the joint coefficient. The weighted product is a score contribution under the chosen model and coding; it does not by itself establish causation.
Wrong reasoning: “A positive coefficient can be multiplied by whichever allele dosage is convenient.”
Correction: The sign applies to the allele specified by the GWAS. If the score dosage counts the other allele, the direction must be harmonized before using the weight.

Key takeaway: In a polygenic score, each GWAS coefficient weights the dosage of its corresponding effect allele. For quantitative traits, a single-variant coefficient is a per-copy regression slope conditional on the stated covariates. It is generally marginal with respect to other variants, so it should not automatically be interpreted as a joint or causal effect.

Check Your Understanding

Use these questions to check your interpretation of GWAS effects as score weights.

A quantitative-trait GWAS estimates a coefficient of $-0.15$ phenotype units per effect-allele copy. What does the sign and scale mean in the fitted single-variant regression?
A score uses a coefficient for allele A, but its dosage counts allele G at the same variant. What must be checked before calculating the contribution?
Under what condition in the linear additive model does the marginal coefficient for a variant equal its joint coefficient?
Why is a single-variant coefficient not automatically the isolated causal effect of that variant?
If a logistic GWAS coefficient is $0.20$, what is the corresponding per-copy odds ratio, and is it a probability difference?