Bias-Variance tradeoff for single-site estimation with equal estimation noise and equal heterozygozity across contexts. The x-axis shows the difference in context specific effects, while the y-axis shows the standard deviation of the context-specific estimators—both in raw measurement units. The color on the plot indicates the difference between the additive and GxE estimators in bias (A), variance (B) or MSE (C). (A) Only the additive estimator is potentially biased. The bias is proportional to the difference in context-specific effects and independent of the estimation noise. (B) The difference in variance is is proportional to context-specific estimation noise and independent of the difference of context-specific effects. (C) The decision boundary is linear in both the estimation noise and the difference between context-specific effects.

The decision boundary with different ratios of context-specific estimation noises. In all panels, the heterozygozity of the variant is assumed to be equal across contexts. The x and y axes are the same as in Fig. 1. Estimation noise in the focal context, A, is half that of the other context, B. (B) Estimation noise is equal in both contexts. (C) Estimation noise in focal context is double that of the other context.

Applying the decision rule to sex-dependent effects on human physiological traits. (A-B) The x-axis shows the estimated absolute difference between the effect of variants in males and females. The y-axis shows the measured standard error for each variant in males, the focal context here. The dashed line shows the decision boundary for effect estimation in males. The difference in MSE between estimation methods increases linearly with distance from the dashed line, as in Fig. 2. If a variant falls above (below) the line, the additive (GxE) estimator has a lower MSE. (A) shows a random sample of 15K single nucleotide variants whereas (B) shows only variants with a marginal p-value less than 5× 10−8 in males. (C-D) The percent of effects in males which would be better estimated by the GxE estimator, across conutinuous physiological traits. To estimate these percentages, one single nucleotide variant is sampled from each of 1,700 approximately independent auotosomal linkage blocks, and this procedure is repeated 10 times. Shown are average percentages across the 10 iterations.

A focus on top hits may be lead to mischaracterization of polygenic GxE. (A) Data from an experiment measuring allelic effects on longevity in caged flies given one of two diets, “control” and “high sugar”. Shown are allelic effect estimates under each diet for a random sample of approximately 12K variants. (B) Simulated data where all true allelic effects are exactly 1.4 times larger under a high-sugar diet. The effects are estimated with sampling noise mimicking the Pallares et al. data. (C) Allelic effect estimates of variants ascertained as significant and classified as “diet-specific” or “shared” by Pallares et al. (D) Simulated effects ascertained as significant and classified using a similar procedure to that applied in (C). While the generative mode of GxE we used in our simulations was not considered by Pallares et al., the simulation results (left panels) closely match the patterns observed in their data (right panels) across all effects (top panels) and as reflected via their classification approach (bottom panels).

Polygenic score performance for context-dependent prediction models.

In each simulation, a GWAS is performed on 5, 000 biallelic variants, half of which have no effect in either context. Of the other half, some percent of the variants (indicated on the x-axis) had effects 1.4× larger in one of contexts and the remaining SNPs had equal effects in both contexts. The broad sense heritability was set to 0.4 in all simulations. The y-axis shows the average, over 11, 000 simulations, of the out-of-sample Pearson correlation between polygenic score and trait value. (A) Results with a GWAS sample size of 1, 000 individuals. (B) Results with a GWAS size of 50, 000 individuals.