  The increasing availability of passively observed data has yielded a growing
methodological interest in "data fusion." These methods involve merging data
from observational and experimental sources to draw causal conclusions -- and
they typically require a precarious tradeoff between the unknown bias in the
observational dataset and the often-large variance in the experimental dataset.
We propose an alternative approach to leveraging observational data, which
avoids this tradeoff: rather than using observational data for inference, we
use it to design a more efficient experiment. We consider the case of a
stratified experiment with a binary outcome, and suppose pilot estimates for
the stratum potential outcome variances can be obtained from the observational
study. We extend results from Zhao et al. (2019) in order to generate
confidence sets for these variances, while accounting for the possibility of
unmeasured confounding. Then, we pose the experimental design problem as one of
regret minimization, subject to the constraints imposed by our confidence sets.
We show that this problem can be converted into a convex minimization and
solved using conventional methods. Lastly, we demonstrate the practical utility
of our methods using data from the Women's Health Initiative.
