Contrastive linear regression

Contrastive dimension reduction methods have been developed for case-control study data to identify variation that is enriched in the foreground (case) data <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math> relative to the background (control) data <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Y</mml:mi></mml:math> . Here we develop contrastive regression for the setting where there is a response variable <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>r</mml:mi></mml:math> associated with each foreground observation. This situation occurs frequently when, for example, the unaffected controls do not have a disease grade or intervention dosage, but the affected cases have a disease grade or intervention dosage, as in autism severity, solid tumors stages, polyp sizes, or warfarin dosages. Our contrastive regression model captures shared low-dimensional variation between the predictors in the case and control groups and then explains the case-specific response variables through the variance that remains in the predictors after shared variation is removed. We show that, in one single-cell RNA sequencing dataset on cellular differentiation in chronic rhinosinusitis with and without nasal polyps and in another single-nucleus RNA sequencing dataset on autism severity in postmortem brain samples from donors with and without autism, our contrastive linear regression performs feature ranking and identifies biologically-informative predictors associated with response that cannot be identified using other approaches.