Advances in Experimental Design and Causal Inference: Rerandomization, Sequential Experiments, and Incremental Effects

Experimental design and causal inference play key roles in modern statistical analysis. Developing new methods in each field is crucial for researchers to draw robust conclusions from empirical research. This dissertation advances these fields through three interconnected papers, developing novel methods that enhance the precision, flexibility, and practical applicability of experimental and causal analyses.

In chapter one, I introduce a theoretical framework for rerandomization in treatment-vs-control experiments for the class of ellipsoidal constraints on the covariate mean differences. This framework generalizes many existing methods and allows me to derive optimal decision rules both for variance reduction of the difference-in-means estimator and covariate balance between treatment and control groups. I find that the covariates’ eigenstructure plays a key role in determining the most precise estimator for the average treatment effect. My theoretical findings are validated by both simulation and an empirical application.

Chapter two extends and generalizes the framework established in chapter one to sequential experiments, where I now consider the class of balance metrics that are symmetric with respect to the covariate mean differences. Here, I establish novel updating strategies that dynamically adapt to the results of the previous experimental group. Additionally, I derive asymptotically valid confidence intervals so practitioners can quickly and easily conduct statistical inference after the experiment. Simulation studies underscore the improvements in precision and flexibility offered by these sequential strategies.

In chapter three I shift my focus to causal inference in observational settings. Here, I develop statistical methodology for incremental effects under continuous exposures; i.e., causal effects in which the treatment density has been tilted by some user-specified parameter. First, I derive the efficient influence function and semiparametric efficiency bound for the incremental effect estimator. I then derive novel minimax lower bounds that demonstrate the best possible estimation error scales with the size of the tilt parameter. Finally, I establish convergence rates that depend on the magnitude of the tilt. These results reveal key insights into the fundamental relationship between sample size, the tilt, and estimator precision.