It is straightforward to prove that if a decisionmaker has information on treatment response for a population, and covariates describing attributes of that population, optimal treatment must condition on all available attributes. For instance, a doctor with information on treatment outcomes for a new drug on a population, with information on the ages, sexes and races of the previously treated, should decide whether or not to prescribe the drug to a new patient conditional on all of the available attributes. The situation when only a sample of the population is available is less straightforward: conditioning on all attributes will lower the sample size of each strata, and therefore lower the accuracy the decisionmaker has about the true conditional effect of the drug. How should low sample size and more information from covariates be balanced? The author assumes a Hurwicz minimax-regret social welfare function (where the decisionmaker wishes to minimize his worst-case scenario if his treatment is wrong), and through a clever application of the Hoeffding’s Large Deviations Theorem is able to give a numerical formula for calculating when it is better (in the sense of sufficiency) to condition on an extra variable versus leaving it out. A numerical exercise hints that treatment decisions may be ignoring useful – in the welfare sense – covariates when deciding on treatment because of their low statistical significance even though they ought be used given a minimax-regret decisionmaker.
“Statistical Treatment Rules for Heterogeneous Populations,” C. Manski (2004)