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Social distancing remains an important strategy to combat the COVID-19 pandemic in the United States. However, the impacts of specific state-level policies on mobility and subsequent COVID-19 case trajectories have not been completely quantified. Using anonymized and aggregated mobility data from opted-in Google users, we found that state-level emergency declarations resulted in a 9.9% reduction in time spent away from places of residence. Implementation of one or more social distancing policies resulted in an additional 24.5% reduction in mobility the following week, and subsequent shelter-in-place mandates yielded an additional 29.0% reduction. Decreases in mobility were associated with substantial reductions in case growth 2 to 4 weeks later. For example, a 10% reduction in mobility was associated with a 17.5% reduction in case growth 2 weeks later. Given the continued reliance on social distancing policies to limit the spread of COVID-19, these results may be helpful to public health officials trying to balance infection control with the economic and social consequences of these policies., Comment: Co-first Authors: GAW, SV, VE, and AF contributed equally. Corresponding Author: Dr. Evgeniy Gabrilovich, gabr@google.com 32 pages (including supplemental material), 4 figures in the main text, additional figures in the supplemental material

We consider the conditional randomization test as a way to account for covariate imbalance in randomized experiments. The test accounts for covariate imbalance by comparing the observed test statistic to the null distribution of the test statistic conditional on the observed covariate imbalance. We prove that the conditional randomization test has the correct significance level and introduce original notation to describe covariate balance more formally. Through simulation, we verify that conditional randomization tests behave like more traditional forms of covariate adjustmet but have the added benefit of having the correct conditional significance level. Finally, we apply the approach to a randomized product marketing experiment where covariate information was collected after randomization.