Conditional average treatment effect estimation with marginally constrained models

Treatment effect estimates are often available from randomized controlled trials as a single average treatment effect for a certain patient population. Estimates of the conditional average treatment effect (CATE) are more useful for individualized treatment decision-making, but randomized trials are often too small to estimate the CATE. Examples in medical literature make use of the relative treatment effect (e.g. an odds ratio) reported by randomized trials to estimate the CATE using large observational datasets. One approach to estimating these CATE models is by using the relative treatment effect as an offset, while estimating the covariate-specific untreated risk. We observe that the odds ratios reported in randomized controlled trials are not the odds ratios that are needed in offset models because trials often report the marginal odds ratio. We introduce a constraint or a regularizer to better use marginal odds ratios from randomized controlled trials and find that under the standard observational causal inference assumptions, this approach provides a consistent estimate of the CATE. Next, we show that the offset approach is not valid for CATE estimation in the presence of unobserved confounding. We study if the offset assumption and the marginal constraint lead to better approximations of the CATE relative to the alternative of using the average treatment effect estimate from the randomized trial. We empirically show that when the underlying CATE has sufficient variation, the constraint and offset approaches lead to closer approximations to the CATE.