| This manuscript includes three topics in causal inference, all of which are under the randomization inference framework (Neyman, 1923; Fisher, 1935a; Rubin, 1978). This manuscript contains three self-contained chapters.;Chapter 1. Under the potential outcomes framework, causal effects are defined as comparisons between potential outcomes under treatment and control. To infer causal effects from randomized experiments, Neyman proposed to test the null hypothesis of zero average causal effect (Neyman's null), and Fisher proposed to test the null hypothesis of zero individual causal effect (Fisher's null). Although the subtle difference between Neyman's null and Fisher's null has caused lots of controversies and confusions for both theoretical and practical statisticians, a careful comparison between the two approaches has been lacking in the literature for more than eighty years. I fill in this historical gap by making a theoretical comparison between them and highlighting an intriguing paradox that has not been recognized by previous re- searchers. Logically, Fisher's null implies Neyman's null. It is therefore surprising that, in actual completely randomized experiments, rejection of Neyman's null does not imply rejection of Fisher's null for many realistic situations, including the case with constant causal effect. Furthermore, I show that this paradox also exists in other commonly-used experiments, such as stratified experiments, matched-pair experiments, and factorial experiments. Asymptotic analyses, numerical examples, and real data examples all support this surprising phenomenon. Besides its historical and theoretical importance, this paradox also leads to useful practical implications for modern researchers.;Chapter 2. Causal inference in completely randomized treatment-control studies with binary outcomes is discussed from Fisherian, Neymanian and Bayesian perspectives, using the potential outcomes framework. A randomization-based justification of Fisher's exact test is provided. Arguing that the crucial assumption of constant causal effect is often unrealistic, and holds only for extreme cases, some new asymptotic and Bayesian inferential procedures are proposed. The proposed procedures exploit the intrinsic non-additivity of unit-level causal effects, can be applied to linear and non- linear estimands, and dominate the existing methods, as verified theoretically and also through simulation studies.;Chapter 3. Recent literature has underscored the critical role of treatment effect variation in estimating and understanding causal effects. This approach, however, is in contrast to much of the foundational research on causal inference; Neyman, for example, avoided such variation through his focus on the average treatment effect and his definition of the confidence interval. In this chapter, I extend the Ney- manian framework to explicitly allow both for treatment effect variation explained by covariates, known as the systematic component, and for unexplained treatment effect variation, known as the idiosyncratic component. This perspective enables es- timation and testing of impact variation without imposing a model on the marginal distributions of potential outcomes, with the workhorse approach of regression with interaction terms being a special case. My approach leads to two practical results. First, I combine estimates of systematic impact variation with sharp bounds on over- all treatment variation to obtain bounds on the proportion of total impact variation explained by a given model---this is essentially an R2 for treatment effect variation. Second, by using covariates to partially account for the correlation of potential out- comes problem, I exploit this perspective to sharpen the bounds on the variance of the average treatment effect estimate itself. As long as the treatment effect varies across observed covariates, the resulting bounds are sharper than the current sharp bounds in the literature. I apply these ideas to a large randomized evaluation in educational research, showing that these results are meaningful in practice. |
