  Many modern causal questions ask how treatments affect complex outcomes that
are measured using wearable devices and sensors. Current analysis approaches
require summarizing these data into scalar statistics (e.g., the mean), but
these summaries can be misleading. For example, disparate distributions can
have the same means, variances, and other statistics. Researchers can overcome
the loss of information by instead representing the data as distributions. We
develop an interpretable method for distributional data analysis that ensures
trustworthy and robust decision-making: Analyzing Distributional Data via
Matching After Learning to Stretch (ADD MALTS). We (i) provide analytical
guarantees of the correctness of our estimation strategy, (ii) demonstrate via
simulation that ADD MALTS outperforms other distributional data analysis
methods at estimating treatment effects, and (iii) illustrate ADD MALTS'
ability to verify whether there is enough cohesion between treatment and
control units within subpopulations to trustworthily estimate treatment
effects. We demonstrate ADD MALTS' utility by studying the effectiveness of
continuous glucose monitors in mitigating diabetes risks.
