Comments on the Neyman-Fisher Controversy and its Consequences

The Neyman-Fisher controversy considered here originated with the 1935 presentation of Jerzy Neyman’s Statistical Problems in Agricultural Experimentation to the Royal Statistical Society. Neyman asserted that the standard ANOVA F-test for randomized complete block designs is valid, whereas the analogous test for Latin squares is invalid in the sense of detecting differentiation among the treatments, when none existed on average, more often than desired (i.e., having a higher Type I error than advertised). However, Neyman’s expressions for the expected mean residual sum of squares, for both designs, are generally incorrect. Furthermore, Neyman’s belief that the Type I error (when testing the null hypothesis of zero average treatment effects) is higher than desired whenever the expected mean treatment sum of squares is greater than the expected mean residual sum of squares, is generally incorrect. Simple examples show that, without further assumptions on the potential outcomes, one cannot determine the Type I error of the F-test from expected sums of squares. Ultimately, we believe that the Neyman-Fisher controversy had a deleterious impact on the development of statistics, with a major consequence being that potential outcomes were ignored in favor of linear models and classical statistical procedures that are imprecise without applied contexts. AMS 2000 subject classifications: analysis of variance, Latin squares, non-additivity, randomization tests, randomized complete blocks. 1. CONFLICT AND CONTROVERSY Prior to the presentation of Statistical Problems in Agricultural Experimentation to the Royal Statistical Society in 1935 (Neyman, 1935), Jerzy Neyman and Ronald Aylmer Fisher were on fairly good terms, both professionally and personally. Joan Fisher Box’s biography of her father (Box, 1978: p. 262-263, 451) and Neyman’s oral autobiography (Reid, 1982: p. 102, 114-117) describe two scientists Department of Statistics, Harvard University, 1 Oxford Street Fl. 7, Cambridge, MA 02138, USA (e-mail: [email protected]), (e-mail: [email protected]). ∗Research supported by the United States National Science Foundation Graduate Research Fellowship under Grant No. DGE-1144152.