Fisher's Exact P-Values for Completely Randomized Experiments

Abstract

INTRODUCTION As discussed in Chapter 2, Fisher appears to have been the first to grasp fully the importance of physical randomization for credibly assessing causal effects (1925, 1936). A few years earlier, Neyman (1923) had introduced the language and the notation of potential outcomes, using this notation to define causal effects as if the assignments were determined by random draws from an urn, but he did not take the next logical step of appreciating the importance of actually randomizing. It was instead Fisher who made this leap. Given data from a completely randomized experiment, Fisher was intent on assessing the sharp null hypothesis (or exact null hypothesis , Fisher, 1935) of no effect of the active versus control treatment, that is, the null hypothesis under which, for each unit in the experiment, both values of the potential outcomes are identical. In this setting, Fisher developed methods for calculating “p-values.” We refer to them as Fisher Exact P-values (FEPs), although we use them more generally than Fisher originally proposed. Note that Fisher's null hypothesis of no effect of the treatment versus control whatsoever is distinct from the possibly more practical question of whether the typical (e.g., average) treatment effect across all units is zero. The latter is a weaker hypothesis, because the average treatment effect may be zero even when for some units the treatment effect is positive, as long as for some others the effect is negative. We discuss the testing of hypotheses on, and inference for, average treatment effects in Chapter 6. Under Fisher's null hypothesis, and under sharp null hypotheses more generally, for units with either potential outcome observed, the other potential outcome is known; and so, under such a sharp null hypothesis, both potential outcomes are “known” for each unit in the sample – being either directly observed or inferred through the sharp null hypothesis. Consider any test statistic T : a function of the stochastic assignment vector, W ; the observed outcomes, Y obs ; and any pre-treatment variables, X .