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Abstract Linear regression is a classic method of data analysis. In recent years, sketching—a method of dimension reduction using random sampling, random projections or both—has gained popularity as an effective computational approximation when the number of observations greatly exceeds the number of variables. In this paper, we address the following question: how does sketching affect the statistical properties of the solution and key quantities derived from it? To answer this question, we present a projector-based approach to sketched linear regression that is exact and that requires minimal assumptions on the sketching matrix. Therefore, downstream analyses hold exactly and generally for all sketching schemes. Additionally, a projector-based approach enables derivation of key quantities from classic linear regression that account for the combined model- and algorithm-induced uncertainties. We demonstrate the usefulness of a projector-based approach in quantifying and enabling insight on excess uncertainties and bias-variance decompositions for sketched linear regression. Finally, we demonstrate how the insights from our projector-based analyses can be used to produce practical sketching diagnostics to aid the design of judicious sketching schemes. 

The explosion of biobank data offers unprecedented opportunities for gene-environment interaction (GxE) studies of complex diseases because of the large sample sizes and the rich collection in genetic and non-genetic information. However, the extremely large sample size also introduces new computational challenges in G×E assessment, especially for set-based G×E variance component (VC) tests, which are a widely used strategy to boost overall G×E signals and to evaluate the joint G×E effect of multiple variants from a biologically meaningful unit (e.g., gene). In this work, we focus on continuous traits and present SEAGLE, a S calable E xact A l G orithm for L arge-scale set-based G× E tests, to permit G×E VC tests for biobank-scale data. SEAGLE employs modern matrix computations to calculate the test statistic and p -value of the GxE VC test in a computationally efficient fashion, without imposing additional assumptions or relying on approximations. SEAGLE can easily accommodate sample sizes in the order of 10 5 , is implementable on standard laptops, and does not require specialized computing equipment. We demonstrate the performance of SEAGLE using extensive simulations. We illustrate its utility by conducting genome-wide gene-based G×E analysis on the Taiwan Biobank data to explore the interaction of gene and physical activity status on body mass index.