  Fisher randomization tests for Neyman's null hypothesis of no average
treatment effects are considered in a finite population setting associated with
completely randomized experiments with more than two treatments. The
consequences of using the $F$ statistic to conduct such a test are examined
both theoretically and computationally, and it is argued that under treatment
effect heterogeneity, use of the $F$ statistic in the Fisher randomization test
can severely inflate the type I error under Neyman's null hypothesis. An
alternative test statistic is proposed, its asymptotic distributions under
Fisher's and Neyman's null hypotheses are derived, and its advantages
demonstrated.
