Logical Contradictions in the One-Way ANOVA and Tukey–Kramer Multiple Comparisons Tests with More Than Two Groups of Observations

We show that the one-way ANOVA and Tukey–Kramer (TK) tests agree on any sample with two groups. This result is based a simple identity connecting Fisher–Snedecor studentized probabilistic distributions proven without additional assumptions; in particular, standard assumptions (independence, normality, homoscedasticity (INAH)) are not needed. In contrast, it known for k>2 groups of observations, even under INAH assumptions, same significance level ?, above may give opposite results: (i) rejects its null hypothesis H0A:?1=…=?k, while TK one, H0TK(i,j):?i=?j, rejected pair i,j?{1,…,k}; (ii) test H0TK(i,j) (i,j) (with i?j), does reject H0A. construct large infinite pseudo-random families samples both types satisfying INAH: case k?3 some larger k. Furthermore, ANOVA, being restricted to (i,j), equality ?i=?j ?. an obvious contradiction, since ?1=…=?k implies all i,j?{1,…,k}. Such contradictions appear already symmetric k=3, or other words, three d,d, c observations means +1,?1, 0, respectively. outline conditions necessary sufficient this phenomenon. Similar contradictory examples constructed multivariable linear regression (MLR). However, these constructions, seems difficult verify Gauss–Markov which standardly required MLR. Mathematics Subject Classification: 62 Statistics.