Its a meta observation of how we view p-values. He's saying that the same logic behind p-values and declaring something "statistically significant" when p < X (

Its a meta observation of how we view p-values. He's saying that the same logic behind p-values and declaring something "statistically significant" when p < X (often 0.05), falls apart when applied to differences between two observed p-values. Essentially, you would say there is no "statistically significant" difference between an experiment that resulted in a p-value of 0.2 and one that resulted in a p-value of 0.05. And the argument will hold no matter what threshold you choose for rejecting the null hypothesis. You can choose to reject if p > X, and for any X, there will be values greater than X that, applying this meta-logic, are not statistically different from X. > At what point would this author say something is not consistent with the null hypothesis? Gelman's argument, I presume, is against the idea of significance testing as a whole. Declaring something "statistically significant" is in itself a very problematic thing, as it distills the entire phenomenon, the uncertainty surrounding the experiment, and the uncertainty surrounding the researcher's decisions to a single, binary conclusion. Gelman is a Bayesian (perhaps the most famous modern Bayesian), and the Bayesian philosophy is to focus on producing a posterior distribution of the phenomenon being studied. I presume the alternative to significance and null hypothesis testing that he was suggest would be something closer to a model where people are reporting their priors/data/posteriors, and the discussion focuses around the implications and replication of those.