LD Mismatch and Tagging Transfer

Why Tagging Can Transfer Poorly

A genome-wide association study may identify a variant because it is correlated with a nearby causal variant, rather than because the associated variant itself changes the trait. Such an associated variant is a tag: it carries information about the causal genotype through linkage disequilibrium (LD). If the correlation between tag and causal variants differs between a discovery population and a target population, the tag's predictive value can change even when the causal effect is identical in both populations.

This tutorial isolates that mechanism with one causal variant and one tag variant. It compares the population-level performance of the discovery tag weight when transferred to the target. The calculations assume standardized genotypes, so differences in genotype variance due to allele frequencies are not part of the comparison. They also assume a shared causal effect; the question here is how LD mismatch alone affects tagging.

A Single-Causal-Variant Model

Under this model, \(r_D\) and \(r_T\) can differ in both magnitude and sign. The sign depends on allele coding, so the tag and causal alleles must be oriented consistently when comparing correlations. A negative correlation is not automatically evidence of weaker tagging: its magnitude can be large, but it implies an association in the opposite direction for the same allele coding.

The tag's discovery association is obtained from its covariance with the outcome. Since the residual is uncorrelated with the tag, the population regression coefficient of \(Y_D\) on \(G_D\) is \(\operatorname{Cov}(G_D,Y_D)\), because the tag variance is one. The discovery weight is therefore

The transferred score is \(S_T=b_DG_T\): it uses the discovery weight unchanged, applied to the target tag genotype. Its performance depends on the target correlation between \(G_T\) and \(X_T\), not just on the association estimated in discovery.

Deriving the Transfer Result

For a nonconstant score, its squared correlation with the outcome is the squared covariance divided by the product of the two variances. In the target, the transferred score has variance \(b_D^2\), while its covariance with the outcome is \(b_D\beta r_T\). The outcome variance is \(\beta^2+\sigma^2\). Thus, whenever \(b_D\ne0\),

The same calculation in discovery, using \(S_D=b_DG_D\), gives \(R_D^2=\beta^2r_D^2/(\beta^2+\sigma^2)\), when the discovery score is nonconstant. Therefore the ratio is

These conditions are substantive. Since \(b_D=\beta r_D\), both are needed to ensure the discovery score has nonzero variance and \(R_D^2\) is nonzero. If \(\beta=0\), the discovery weight is zero and the score is constant; its correlation with the outcome is undefined, not zero, so the ratio cannot be formed. If \(r_D=0\), the tag has no discovery association in this model and the same problem occurs. If \(r_T=0\) while \(\beta r_D\ne0\), the target score is nonconstant but uncorrelated with the outcome, so \(R_T^2=0\) and the ratio is defined as zero.

The derivation cancels the squared discovery weight \(b_D^2\). That cancellation explains why the relative squared-correlation performance depends on the squared LD correlations, not on the numerical size of the nonzero transferred weight. The result applies to this one-tag model under the stated shared-effect and equal-variance assumptions; it is not a general formula for a multi-variant score or for settings where other model features differ.

What “Explained Variance” Means Here

In this model, \(R^2\) means the squared correlation between the score and the outcome. Equivalently, it is the fraction of outcome variance accounted for by the best linear projection of the outcome on that score, with an intercept. It allows the projection slope to be optimal in the population being evaluated. It is not necessarily the performance of every prediction rule that applies the discovery coefficient without adjustment.

For example, with centered outcome and score, a frozen prediction rule that predicts \(S_T\) has mean squared error \(\operatorname{Var}(Y_T)-2\operatorname{Cov}(S_T,Y_T)+\operatorname{Var}(S_T)\). A negative target covariance can make that error exceed the error from predicting the mean, even though the score's squared correlation is nonnegative. Consequently, state which metric is being compared: squared correlation, an optimally recalibrated linear prediction, or error from a fixed score rule. The LD-ratio result above is specifically for squared-correlation performance.

Common Mistakes and Practical Interpretation

• Using only the LD magnitude informally. A smaller \(|r_T|\) than \(|r_D|\) implies lower squared-correlation performance under this model, but the sign also determines whether the target association direction agrees with the discovery weight.
• Claiming the ratio when the discovery performance is zero. A zero \(\beta\) or zero \(r_D\) makes \(b_D=0\); the score is constant and its correlation is undefined. The ratio formula requires both \(\beta\ne0\) and \(r_D\ne0\).
• Calling the result a universal prediction-loss formula. The ratio compares squared correlations under a single causal variant, standardized genotypes, and shared effects and residual variance. It does not automatically describe fixed-rule prediction error or multi-variant scores.
• Confusing LD mismatch with a causal effect change. Here \(\beta\) is fixed across populations by assumption. The transfer difference arises because the tag captures the causal genotype differently; effect-size heterogeneity is a distinct source of portability change.

The practical lesson is that a GWAS tag is predictive only to the extent that it tracks the causal variant in the population where the score is used. Under the stated single-tag model, squared-correlation transfer is proportional to \(r_T^2\); comparing it with discovery performance requires a nonzero discovery association. For real scores, this calculation is a useful mechanism-level result, not a substitute for evaluating the score in the intended target population.