Extended Data Fig. 2: Analytical estimator of \({{{\mathrm{sd}}}}(\widehat {{{{\mathrm{PRS}}}}}_{{{\mathrm{i}}}})\) provides an approximately unbiased estimates of average sd(PRSi) of testing individuals.
The x-axis is the average \({{{\mathrm{sd}}}}(\widehat {{{{\mathrm{PRS}}}}}_{{{\mathrm{i}}}})\) in testing individuals within each simulation replicate. The y-axis is the expected \({{{\mathrm{sd}}}}(\widehat {{{{\mathrm{PRS}}}}}_{{{\mathrm{i}}}})\) computed with equation (1), replacing M and \(h_g^2\) with estimates of the number of causal variants and SNP-heritability, respectively, from LDpred2. Each dot is an average of 10 simulation replicates for each \({{{\mathrm{p}}}}_{{{{\mathrm{causal}}}}} \in \left\{ {0.001,0.01,0.1,1} \right\}\). The horizontal whiskers represent ±1.96 standard deviations of average \({{{\mathrm{sd}}}}(\widehat {{{{\mathrm{PRS}}}}}_{{{\mathrm{i}}}})\). The vertical whiskers represent ±1.96 standard deviations of expected \({{{\mathrm{sd}}}}(\widehat {{{{\mathrm{PRS}}}}}_{{{\mathrm{i}}}})\). Note that when pcausal = 1, the independent LD assumption is violated but the analytical form still provides approximately unbiased estimates. When \(p_{{{{\mathrm{causal}}}}} \ne 1\), the infinitesimal assumption is violated, leading to downward bias in the analytical estimator. In these scenarios, since we simply replace M with \(M \times p_{{{{\mathrm{causal}}}}}\), the uncertainty identifying the causal variants is ignored by equation (1).
