Sample size planning for conditional counterfactual mean estimation with
  a K-armed randomized experiment
Gabriel Ruiz
We cover how to determine a sufficiently large sample size for a $K$-armed
randomized experiment in order to estimate conditional counterfactual
expectations in data-driven subgroups. The sub-groups can be output by any
feature space partitioning algorithm, including as defined by binning users
having similar predictive scores or as defined by a learned policy tree. After
carefully specifying the inference target, a minimum confidence level, and a
maximum margin of error, the key is to turn the original goal into a
simultaneous inference problem where the recommended sample size to offset an
increased possibility of estimation error is directly related to the number of
inferences to be conducted. Given a fixed sample size budget, our result allows
us to invert the question to one about the feasible number of treatment arms or
partition complexity (e.g. number of decision tree leaves). Using policy trees
to learn sub-groups, we evaluate our nominal guarantees on a large
publicly-available randomized experiment test data set.