Power loss due to testing association between covariate-adjusted traits and genetic variants

Multiple linear regression is commonly used to test for association between genetic variants and continuous traits and estimate genetic effect sizes. Confounding variables are controlled for by including them as additional covariates. An alternative technique that is increasingly used is to regress out covariates from the raw trait and then perform regression analysis with only the genetic variants included as predictors. In the case of single-variant analysis, this adjusted trait regression (ATR) technique is known to be less powerful than the traditional technique when the genetic variant is correlated with the covariates We extend previous results for single-variant tests by deriving exact relationships between the single-variant score, Wald, likelihood-ratio, and F test statistics and their ATR analogs. We also derive the asymptotic power of ATR analogs of the multiple-variant score and burden tests. We show that the maximum power loss of the ATR analog of the multiple-variant score test is completely characterized by the canonical correlations between the set of genetic variants and the set of covariates. Further, we show that for both single- and multiple-variant tests, the power loss for ATR analogs increases with increasing stringency of Type 1 error control ( $$ ) and increasing correlation (or canonical correlations) between the genetic variant (or multiple variants) and covariates. We recommend using ATR only when maximum canonical correlation between variants and covariates is low, as is typically true.