  In medical research, a scenario often entertained is randomized controlled
$2^2$ factorial design with a binary outcome. By utilizing the concept of
potential outcomes, Dasgupta et al. (2015) proposed a randomization-based
causal inference framework, allowing flexible and simultaneous estimations and
inferences of the factorial effects. However, a fundamental challenge that
Dasgupta et al. (2015)'s proposed methodology faces is that the sampling
variance of the randomization-based factorial effect estimator is
unidentifiable, rendering the corresponding classic "Neymanian" variance
estimator suffering from over-estimation. To address this issue, for randomized
controlled $2^2$ factorial designs with binary outcomes, we derive the sharp
lower bound of the sampling variance of the factorial effect estimator, which
leads to a new variance estimator that sharpens the finite-population Neymanian
causal inference. We demonstrate the advantages of the new variance estimator
through a series of simulation studies, and apply our newly proposed
methodology to two real-life datasets from randomized clinical trials, where we
gain new insights.
