INTRODUCTION

When a researcher wishes to proceed to estimate causal effects under the assumption of unconfoundedness, there are various statistical methods that can be used to attempt to adjust for differences in covariate distributions. These methods include simple linear regressions, which is adequate in simple situations. They also include more sophisticated methods involving subclassification on the propensity score and matching, the latter two possibly in combination with model-based imputation methods, which can work well even in complicated situations. In order to decide on the appropriate methods, it is important first to assess the severity of the statistical challenge to adjust for the differences in covariates. In other words, it is useful to assess how different the covariate distributions are in the treatment and control groups. If the covariate distributions are similar, as they would be, in expectation, in the setting of a completely randomized experiment, there is less reason to be concerned about the sensitivity of estimates to the specific method choosen than if these distributions are substantially different. On the other hand, even if unconfoundedness holds, it may be that there are regions of the covariate space with relatively few treated units or relatively few control units, and, as a result, inferences for such regions rely largely on extrapolation and are therefore less credible than inferences for regions with substantial overlap in covariate distributions.

In this chapter we address the problem of assessing the degree of overlap in the covariate distributions – or, in other words, the covariate balance between the treated and control samples prior to any analyses to adjust for these differences. These assessments do not involve the outcome data and therefore do not introduce any systematic biases in subsequent analyses. In principle we are interested in the comparison of two multivariate distributions, the distributions of the covariates in the treated and control subsamples. We wish to explore how different the measures of central tendency are, and how much overlap there is in the tails of the distributions.