The Difference Between “Significant” and “Not Significant” is not Itself Statistically Significant

It is common to summarize statistical comparisons by declarations of statistical significance or nonsignificance. Here we discuss one problem with such declarations, namely that changes in statistical significance are often not themselves statistically significant. By this, we are not merely making the commonplace observation that any particular threshold is arbitrary—for example, only a small change is required to move an estimate from a 5.1% significance level to 4.9%, thus moving it into statistical significance. Rather, we are pointing out that even large changes in significance levels can correspond to small, nonsignificant changes in the underlying quantities.