  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.
