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Double blind randomized controlled trials are traditionally seen as the goldstandard for causal inferences as the difference-in-means estimator is anunbiased estimator of the average treatment effect in the experiment. The factthat this estimator is unbiased over all possible randomizations does not,however, mean that any given estimate is close to the true treatment effect.Similarly, while pre-determined covariates will be balanced between treatmentand control groups on average, large imbalances may be observed in a givenexperiment and the researcher may therefore want to condition on suchcovariates using linear regression. This paper studies the theoreticalproperties of both the difference-in-means and OLS estimators\emph{conditional} on observed differences in covariates. By deriving thestatistical properties of the conditional estimators, we can establish guidancefor how to deal with covariate imbalances. We study both inference with OLS, aswell as with a new version of Fisher's exact test, where the randomizationdistribution comes from a small subset of all possible assignment vectors.

Inference in experiments conditional on observed imbalances in covariates