An important goal in causal inference is to achieve balance in the covariates among the treatment groups. In this article, we introduce the concept of distributional balance preserving which requires the distribution of the covariates to be the same in different treatment groups. We also introduce a new balance measure called kernel distance, which is the empirical estimate of the probability metric defined in the reproducing kernel Hilbert spaces. Compared to the traditional balance metrics, the kernel distance measures the difference in the two multivariate distributions instead of the difference in the finite moments of the distributions. Simulation results show that the kernel distance is the best indicator of bias in the estimated casual effect compared to several commonly used balance measures. We then incorporate kernel distance into genetic matching, the state-of-the-art matching procedure and apply the proposed approach to analyze the Early Dieting in Girls study. The study indicates that mothers' overall weight concern increases the likelihood of daughters' early dieting behavior, but the causal effect is not significant.
Keywords: Causal effect; Distributional covariate balance; Probability metric; Reproducing kernel Hilbert space.