  It often happens that the same problem presents itself to different
communities and the solutions proposed or adopted by those communities are
different. We take the case of the variance estimation of the population
average treatment effect in cluster-randomized experiments. The econometrics
literature promotes the cluster-robust variance estimator (Athey and Imbens,
2017), which can be dated back to the study of linear regression with clustered
residuals (Liang and Zeger, 1986). The A/B testing or online experimentation
literature promotes the delta method (Kohavi et al., 2010, Deng et al., 2017,
2018), which tackles the variance estimation of the ATE estimator directly
using large sample theory. The two methods are seemly different as the former
begins with a regression setting at the individual unit level and the latter is
semi-parametric with only i.i.d. assumptions on the clusters. Both methods are
widely used in practice. It begs the question for their connection and
comparison. In this paper we prove they are equivalent and in the canonical
implementation they should give exactly the same result.
