Dimension reduction summaries for balanced contrasts

We discuss the covariate dimension reduction properties of conditional density ratios in the estimation of balanced contrasts of expectations. Conditional density ratios, as well as related sufficient summaries, can be used to replace the covariates with a smaller number of variables. For example, for comparisons among k populations the covariates can be replaced with k-1 conditional density ratios. The dimension reduction properties of conditional density ratios are directly connected with sufficiency, the dimension reduction concepts considered in regression theory, and propensity theory. The theory presented here extends the ideas in propensity theory to situations in which propensities do not exist and develops an approach to dimension reduction outside of the potential outcomes or counterfactual framework. Under general conditions, we show that a principal components transformation of the estimated conditional density ratios can be used to investigate whether a sufficient summary of dimension lower than k-1 exists and to identify such a lower dimensional summary.