Why Combine Ancestry-Specific Scores?
A polygenic score built using one discovery population may predict less well in a target population with a different ancestry composition. Methods such as PRS-CSx produce scores informed by multiple ancestry-specific discovery analyses. In a target cohort, a further question is how to combine the available scores for prediction. The answer depends on the target population, the outcome, and the prediction objective; there is not generally one set of weights that is optimal everywhere.
The weights are fitted from target-cohort data, so their estimates can be unstable when the cohort is small or the scores are strongly correlated. They can also reflect which subgroups are most represented in the cohort. A fitted combination should therefore be treated as a prediction rule for a specified target population, not as a universal biological decomposition.
Scores should be put on a clearly specified scale before interpreting their coefficients. A common choice is to center and scale each score using means and standard deviations estimated in the target training data. Then a coefficient describes the outcome change associated with a one-standard-deviation increase in that score, conditional on the other included scores and covariates. If scores are left on different scales, their coefficients also reflect those scale differences. Apply the training-data transformations unchanged to held-out data.
Fitting Weights for a Continuous Outcome
For a continuous outcome \(Y_i\), one direct approach is a multiple linear regression of the outcome on all ancestry-specific scores and relevant covariates. For example, with covariate vector \(Z_i\), the model is
The fitted score coefficients \(\hat{\beta}_k\) are the weights in the resulting linear predictor. This model is a useful choice when the target metric is squared prediction error. Its standard model assumptions include a correctly specified conditional mean for coefficient interpretation, errors with mean zero conditional on the included predictors, and independent observations for the usual standard errors. Finite error variance is also needed for standard large-sample inference. Exact collinearity among predictors prevents unique coefficient estimates. Even when a linear mean model is not exactly correct, least squares still defines the best linear predictor in the training data; that fact does not guarantee good performance in a new sample.
For prediction, the main safeguard is separation of fitting from evaluation. Use a target training or validation sample to estimate weights and all preprocessing choices, then evaluate the resulting rule on held-out target data that did not influence those choices. If the same data are used both to optimize and report performance, the reported performance is optimistic on average. Nested validation for PRS tuning addresses this reuse problem when score construction or other choices are also being selected.
Worked Example: Least-Squares Weights for Two Scores
Suppose a hypothetical target training cohort has a continuous phenotype \(Y\), standardized to variance 1, and two centered ancestry-specific scores \(S_1\) and \(S_2\), each with variance 1. The sample correlations are \(r(S_1,S_2)=0.40\), \(r(Y,S_1)=0.30\), and \(r(Y,S_2)=0.20\). Assume the regression has an intercept, the scores are not perfectly collinear, and the observations are independent for this illustration.
With centered, standardized variables, the least-squares coefficients solve \(R_{SS}\hat{\beta}=r_{SY}\), where \(R_{SS}\) is the score correlation matrix and \(r_{SY}\) is the vector of score-outcome correlations:
The determinant is \(1-(0.40)^2=0.84\), which is positive, so the matrix is invertible. Thus
The fitted linear prediction is \(0.2619S_1+0.09524S_2\), apart from the intercept. The coefficients do not sum to one; they are not mixture proportions. In the training data, the fitted proportion of outcome variance explained by this linear regression is \(\hat{\beta}^\mathsf{T}r_{SY}=(0.2619)(0.30)+(0.09524)(0.20)=0.09762\), or about 9.76% for this standardized outcome. This is a training estimate, not an independent evaluation. Its predictive value must be checked in held-out target individuals.
What the Weights Optimize—and What They Do Not
Least squares chooses coefficients to minimize the sum of squared residuals in the fitting sample. It does not require that weights be positive or sum to one. Negative coefficients can occur, for example, when one score contributes little information after accounting for a correlated score. Such coefficients may be numerically valid but unstable, especially when the scores are highly correlated. In that setting, small changes in the sample can produce large changes in individual coefficients while leaving predictions similar.
A convex combination instead imposes \(w_k\geq 0\) and \(\sum_k w_k=1\). This can be attractive when the desired rule is literally an average of scores, but it is a different optimization problem and may predict worse than unconstrained regression. Equal weights are another simple option. They reduce estimation variability because they do not use target outcomes to estimate each coefficient, but they may be inefficient when scores have different predictive value or redundancy. For a binary outcome, weights can be estimated in a logistic regression if the target objective is probabilistic prediction; held-out evaluation should then use a metric suited to that objective, such as log loss, alongside discrimination measures where appropriate.
Whatever the outcome type, score weights do not by themselves establish biological effect sizes or causal effects. With correlated scores, each regression coefficient describes a conditional association given the other included predictors. Changing the score scaling, covariates, cohort composition, or outcome definition can change the weights.
Subgroup Composition Changes the Pooled Combination
A pooled model fits one set of coefficients to the target cohort as a whole. If score-outcome relationships differ across subgroups, the pooled coefficients depend partly on how many individuals from each subgroup were sampled. Even when those relationships are stable within each subgroup, changing the subgroup proportions can change the pooled prediction rule. This is one reason to specify the intended target population before fitting and evaluating weights.
The following derivation makes the composition effect explicit in a simple setting. Suppose there are two subgroups, \(G=1,2\), two scores centered within subgroup, and subgroup indicators included as covariates. Assume that within each subgroup the score covariance matrix is the identity matrix, so both scores have variance 1 and have covariance 0. Write \(\beta_g\) for the vector of subgroup-specific score slopes, and let \(\pi_g\) be the subgroup's proportion in the target sample. The pooled common-slope least-squares coefficient solves the weighted normal equations:
This equality relies on the stated equal within-subgroup score covariance matrices and subgroup-centered predictors; with different covariance matrices, the pooled coefficient is instead weighted through those matrices as well as through subgroup proportions. Thus subgroup mix is not the only determinant of pooled weights, but it can be decisive.
Worked Example: A Different Subgroup Mix, Different Pooled Weights
Consider hypothetical subgroup-specific slopes for two scores: \(\beta_1=(0.60,0.10)\) in subgroup 1 and \(\beta_2=(0.10,0.50)\) in subgroup 2. Assume the within-subgroup score covariance matrices are both the identity matrix, as in the derivation. If the target sample is 80% subgroup 1 and 20% subgroup 2, then
If the sample proportions are reversed, the result is
The subgroup-specific slopes did not change, but the pooled coefficients did. The first pooled model is better aligned with a target population resembling the first sample; the second reflects a different target mix. Neither is automatically the right choice for a third population. If subgroup-specific performance is important, report it and consider whether a common-weight model is adequate rather than relying only on an overall average metric.
Validation Size, Stability, and Diagnostics
The number of target individuals used to fit weights affects how much sampling noise enters the fitted combination. Small validation samples provide imprecise estimates, especially when many scores or covariates are included, when outcome variation is high, or when some subgroups are sparsely represented. Correlated scores make it harder to distinguish their separate coefficients. A model may therefore have unstable weights even when its combined predictions vary little.
Worked Example: How Validation Size Affects Weight Precision
Suppose, hypothetically, two standardized scores are uncorrelated, each has variance 1, and a continuous outcome model has independent homoskedastic errors with standard deviation \(\sigma=0.80\). Under these assumptions, the approximate standard error of either least-squares coefficient is \(SE(\hat{\beta}_k)=\sigma/\sqrt{n}\). For a validation sample of \(n=25\),
For \(n=400\), the corresponding standard error is \(0.80/\sqrt{400}=0.04\). If the estimated coefficient is \(0.26\) in either case, a rough 95% interval using \(1.96SE\) is \(0.26\pm1.96(0.16)=[-0.054,0.574]\) at \(n=25\), compared with \(0.26\pm1.96(0.04)=[0.182,0.338]\) at \(n=400\). These intervals are illustrative and rely on the stated model assumptions and approximate normality. The smaller sample provides much weaker evidence about the coefficient's magnitude. In a real analysis, correlated scores and uneven subgroup sizes can further increase uncertainty.
Useful diagnostics include checking score correlations and the condition of the score covariance matrix, inspecting whether weights change substantially across resamples or folds, and comparing overall and subgroup-specific performance on held-out data. Also check that centering and scaling, covariate adjustment, and any score selection were performed within the training data. For binary outcomes, examine whether predicted probabilities and discrimination are adequate in the held-out target sample; a strong ranking metric alone does not guarantee well-calibrated probabilities.
A common mistake is to treat the largest fitted coefficient as proof that its ancestry-specific score is intrinsically the most informative. That conclusion ignores score scale, correlation with the other scores, sample size, covariate choices, and subgroup composition. The correction is to describe weights as conditional, target-sample estimates; evaluate the combined rule on held-out data; and report uncertainty and subgroup performance where feasible.
Choosing and Reporting a Combination
There is no universally best fitting strategy. The appropriate comparison depends on the intended use and the available target data.
| Approach | Strength | Main caution |
|---|---|---|
| Equal weights | Simple and avoids fitting several outcome-dependent weights. | May ignore differences in predictive value and redundancy. |
| Unconstrained regression weights | Directly optimizes the training-sample linear prediction for squared error. | Can be unstable or have negative coefficients when scores are correlated. |
| Nonnegative weights summing to one | Produces an interpretable convex average. | The constraints can reduce predictive performance if the best linear rule lies outside that set. |
| Subgroup-specific weights | Can represent different score-outcome relationships across groups. | Requires enough data in each subgroup and careful validation; estimates can be noisy. |
A defensible report states which scores were available, how they were scaled, which covariates and outcome model were used, how weights were estimated, and which data were reserved for evaluation. Give the target population and subgroup composition, the validation sample sizes, and held-out predictive performance. If the weights are unstable across folds or resamples, report that instability rather than presenting a single coefficient vector as precise. A fitted combination is limited by the quality of its component scores, the target sample, and the prediction metric used to select it.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- Two scores are standardized and a regression gives coefficients 0.30 and 0.20. Why should these not automatically be reported as 60% and 40% contributions?
- In the subgroup example, calculate the pooled coefficients when subgroup 1 makes up 50% of the target sample.
- Under the assumptions in the validation-size example, what is the approximate coefficient standard error when \(n=100\)?
- Why can highly correlated ancestry-specific scores yield unstable individual weights even if the combined prediction is relatively stable?
- What is wrong with reporting performance on the same target sample used to choose the weights?
Solutions. (1) The coefficients sum to 0.50, not 1, and regression coefficients are conditional prediction coefficients rather than mixture proportions; their interpretation also depends on score scaling and correlation. (2) Equal subgroup proportions give \(\hat{\beta}=0.50(0.60,0.10)+0.50(0.10,0.50)=(0.35,0.30)\). (3) \(SE=0.80/\sqrt{100}=0.08\). (4) Correlated scores contain overlapping information, so multiple coefficient combinations can produce similar predictions; small data changes can shift the conditional coefficients substantially. (5) The fitting procedure has adapted to random features of that sample, so its reported performance is optimistic on average; use held-out target data or a properly nested validation design for evaluation.