A randomization-based perspective of analysis of variance: a test   statistic robust to treatment effect heterogeneity
  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. 
Peng Ding, Tirthankar Dasgupta
math.ST stat.ME stat.TH