The Projected Covariance Measure for assumption-lean variable significance testing
30th September 2022, 2:00 pm – 3:00 pm
Fry Building, TBA
Testing the significance of a variable or group of variables X for predicting a response Y given additional covariates Z, is a ubiquitous task in statistics. A simple but common approach is to specify a linear model, and then test whether the regression coefficient for X is non-zero. However, when the model is misspecified, as will invariably be the case, the test may have poor power, for example when X is involved in complex interactions, or lead to many false rejections.
In this work we study the problem of testing the model-free null of conditional mean independence, i.e. that the conditional mean of Y given X and Z does not depend on X. We propose a simple and general framework that can leverage flexible nonparametric or machine learning methods, such as additive models or boosted trees, to yield both robust error control and high power. The procedure involves using these methods to perform regressions, first to estimate a form of projection of Y on X and Z using one half of the data, and then to estimate the expected conditional covariance between this projection and Y on the remaining half of the data. While the approach is general, we show that a version of our procedure using spline regression achieves what we show is the minimax optimal rate in this nonparametric testing problem.