  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 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.
