I have assessed many manuscripts submitted to a medical journal. I see that three replications are used in ANOVA commonly.

OK, I know that the analyses are expensive in terms of time and/or money. However, I am surprised and believe that the results are unreliable. The problem refers to the assumptions: normal data distribution in the compared groups and variance homogeneity. When n=3, the power of the test used in verifying those assumptions (Shapiro-Wilk's, Levene's etc.) is close to zero. To conclude, when the number of replications is very small, verifying the ANOVA'a assumption is extremely unreliable. Therefore, ANOVA should not be used when n=3, for example. However, even such journal as Nature publishes papers that present parametric ANOVA results based on n=3.

In my opinion, when reliable verifying parametric test assumption is impossible, the given test should be replaced with an appropriate nonparametric test (Kruskal-Wallis test, or PERMANOVA, etc. in the case of ANOVA).

Am I right or wrong?