Fig. 4. Adjusting for covariates provided to DeepNull during association testing is necessary to avoid bias.

The unadjusted model regresses y_i on ${ḡ}_i$ and h(x_i), the prediction of y_i based on x_i, omitting x_i from the association model. This approach results in biased estimation of the genetic effect. The linear adjustment model regresses y_i on ${ḡ}_i$, x_i, and h(x_i). This approach is unbiased. The generative model regresses y_i on ${ḡ}_i$, x_i, and $x_i^2$. This represents the best possible performance. Each box plot summarizes results from n = 10³ independent simulation replicates. The box demarcates, from top to bottom, the 75th, 50th, and 25th percentiles of the corresponding distribution. The whiskers extend between the largest and smallest values within 1.5 times the interquartile range. Any values outside the whiskers are marked by points. Source data are provided as a Source Data file.