Test and Measure for Partial Mean Dependence Based on Machine Learning Methods

It is of importance to investigate the significance of a subset of covariates $W$ for the response $Y$ given covariates $Z$ in regression modeling. To this end, we propose a significance test for the partial mean independence problem based on machine learning methods and data splitting. The test statistic converges to the standard chi-squared distribution under the null hypothesis while it converges to a normal distribution under the fixed alternative hypothesis. Power enhancement and algorithm stability are also discussed. If the null hypothesis is rejected, we propose a partial Generalized Measure of Correlation (pGMC) to measure the partial mean dependence of $Y$ given $W$ after controlling for the nonlinear effect of $Z$. We present the appealing theoretical properties of the pGMC and establish the asymptotic normality of its estimator with the optimal root-$N$ convergence rate. Furthermore, the valid confidence interval for the pGMC is also derived. As an important special case when there are no conditional covariates $Z$, we introduce a new test of overall significance of covariates for the response in a model-free setting. Numerical studies and real data analysis are also conducted to compare with existing approaches and to demonstrate the validity and flexibility of our proposed procedures.