  Factorial designs are widely used due to their ability to accommodate
multiple factors simultaneously. The factor-based regression with main effects
and some interactions is the dominant strategy for downstream data analysis,
delivering point estimators and standard errors via one single regression.
Justification of these convenient estimators from the design-based perspective
requires quantifying their sampling properties under the assignment mechanism
conditioning on the potential outcomes. To this end, we derive the sampling
properties of the factor-based regression estimators from both saturated and
unsaturated models, and demonstrate the appropriateness of the robust standard
errors for the Wald-type inference. We then quantify the bias-variance
trade-off between the saturated and unsaturated models from the design-based
perspective, and establish a novel design-based Gauss--Markov theorem that
ensures the latter's gain in efficiency when the nuisance effects omitted
indeed do not exist. As a byproduct of the process, we unify the definitions of
factorial effects in various literatures and propose a location-shift strategy
for their direct estimation from factor-based regressions. Our theory and
simulation suggest using factor-based inference for general factorial effects,
preferably with parsimonious specifications in accordance with the prior
knowledge of zero nuisance effects.
