Hal Stern updated our paper, ‘The difference between “significant” and “not significant” is not itself statistically significant,’ to include this example of sexual preference and birth order. Here’s the abstract of our paper:

It is common to summarize statistical comparisons by declarations of statistical significance or non-significance. 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, non-significant changes in the underlying variables.

The error we describe is conceptually different from other oft-cited problems—that statistical significance is not the same as practical importance, that dichotomization into significant and non-significant results encourages the dismissal of observed differences in favor of the usually less interesting null hypothesis of no difference, and that any particular threshold for declaring significance is arbitrary. We are troubled by all of these concerns and do not intend to minimize their importance. Rather, our goal is to bring attention to what we have found is an important but much less discussed point. We illustrate with a theoretical example and two applied examples.

The full paper is here, and here are some more of my thoughts on statistical significance.