  Many scientific questions in biomedical, environmental, and psychological
research involve understanding the impact of multiple factors on outcomes.
While randomized factorial experiments are ideal for this purpose,
randomization is infeasible in many empirical studies. Therefore, investigators
often rely on observational data, where drawing reliable causal inferences for
multiple factors remains challenging. As the number of treatment combinations
grows exponentially with the number of factors, some treatment combinations can
be rare or even missing by chance in observed data, further complicating
factorial effects estimation. To address these challenges, we propose a novel
weighting method tailored to observational studies with multiple factors. Our
approach uses weighted observational data to emulate a randomized factorial
experiment, enabling simultaneous estimation of the effects of multiple factors
and their interactions. Our investigations reveal a crucial nuance: achieving
balance among covariates, as in single-factor scenarios, is necessary but
insufficient for unbiasedly estimating factorial effects. Our findings suggest
that balancing the factors is also essential in multi-factor settings.
Moreover, we extend our weighting method to handle missing treatment
combinations in observed data. Finally, we study the asymptotic behavior of the
new weighting estimators and propose a consistent variance estimator, providing
reliable inferences on factorial effects in observational studies.
