  There is a growing literature on design-based methods to estimate average
treatment effects (ATEs) for randomized controlled trials (RCTs) for full
sample analyses. This article extends these methods to estimate ATEs for
discrete subgroups defined by pre-treatment variables, with an application to
an RCT testing subgroup effects for a school voucher experiment in New York
City. We consider ratio estimators for subgroup effects using regression
methods, allowing for model covariates to improve precision, and prove a finite
population central limit theorem. We discuss extensions to blocked and
clustered RCT designs, and to other common estimators with random
treatment-control sample sizes (or weights): post-stratification estimators,
weighted estimators that adjust for data nonresponse, and estimators for
Bernoulli trials. We also develop simple variance estimators that share
features with robust estimators. Simulations show that the design-based
subgroup estimators yield confidence interval coverage near nominal levels,
even for small subgroups.
