The role of Somers's D in propensity modeling

The Rubin method of confounder adjustment, in its 21st-century version, is a two-phase method for using observational data to estimate a causal treatment effect on an outcome variable. It involves first finding a propensity model in the joint distribution of a treatment variable and its confounders (the design phase), and then estimating the treatment effect from the conditional distribution of the outcome, given the treatments and confounders (the analysis phase). In the design phase, we want to limit the level of spurious treatment effect that might be caused by any residual imbalance between treatment and confounders that may remain, after adjusting for the propensity score by propensity matching and weighting and/or stratification. A good measure of this is Somers's D(W|X), where W is a confounder or a propensity score and X is the treatment variable. The SSC package somersd calculates Somers's D for a wide range of sampling schemes, allowing matching and weighting and restriction to comparisons within strata. Somers's D has the feature that if Y is an outcome, then a higher-magnitude D(Y|X) cannot be secondary to a lower-magnitude D(W|X), implying that D(W|X) can be used to set an upper bound to the size of a spurious treatment effect on an outcome. For a binary treatment variable X, D(W|X) gives an upper bound to the size of a difference between the proportions, in the two treatment groups, that can be caused for a binary outcome. If D(W|X) is less than 0.5, then it can be doubled to give an upper bound to the size of a difference between the means, in the two treatment groups, that can be caused for an equal-variance normal outcome, expressed in units of the common standard deviation for the two treatment groups. We illustrate this method using a familiar dataset, with examples using propensity matching, weighting, and stratification. We use the SSC package haif in the design phase to check for variance inflation caused by propensity adjustment and use the SSC package scenttest (an addition to the punaf family) to estimate the treatment effect in the analysis phase.