[Natsu]

Causal has a specific meaning related to causal modeling, most studies can't show causality, a lot only show correlation[1].

And the third one seems to be about effect sizes. But a lot of this is still concerning, even if they appear to be trying to say technically true but misleading things.

[1] Yes, newer methods can show causation, not just correlation. See The Book of Why, by Judea Pearl for an introduction to how that works.

[pinkmuffinere]

Wow, the claim in your footnote is absolutely fascinating to me. I just bought the book, but in the meantime could you give a tl;dr? No worries if not

[Natsu]

Causation has a direction, but the equals sign doesn't, is probably an overly-pithy summary of the first section of the book. And it was hard to represent the simple observation that effects come after causes, not before.

So they introduce do calculus to intentionally break that symmetry to test causal models, which themselves are basically directed graphs going from cause to effect. These also help you see what you need to test to try to falsify your model and to show you how to measure how much of the variance in an effect is explained by variance in the cause. And it helps keep track of interventions, like opening doors in the Monty Hall problem[1].

There's a more detailed summary here that looks pretty good which probably does a better job than my quick summary. I skimmed it and it matches what I recall from the book:

https://www.tosummarise.com/book-summary-the-book-of-why-by-...

[1] https://en.wikipedia.org/wiki/Monty_Hall_problem