  Social distancing is widely acknowledged as an effective public health policy
combating the novel coronavirus. But extreme social distancing has costs and it
is not clear how much social distancing is needed to achieve public health
effects. In this article, we develop a design-based framework to make inference
about the dose-response relationship between social distancing and COVID-19
related death toll and case numbers. We first discuss how to embed
observational data with a time-independent, continuous treatment dose into an
approximate randomized experiment, and develop a randomization-based procedure
that tests if a structured dose-response relationship fits the data. We then
generalize the design and testing procedure to accommodate a time-dependent,
treatment dose trajectory, and generalize a dose-response relationship to a
longitudinal setting. Finally, we apply the proposed design and testing
procedures to investigate the effect of social distancing during the phased
reopening in the United States on public health outcomes using data compiled
from sources including Unacast, the United States Census Bureau, and the County
Health Rankings and Roadmaps Program. We rejected a primary analysis null
hypothesis that stated the social distancing from April 27, 2020, to June 28,
2020, had no effect on the COVID-19-related death toll from June 29, 2020, to
August 2, 2020 (p-value < 0.001), and found that it took more reduction in
mobility to prevent exponential growth in case numbers for non-rural counties
compared to rural counties.
