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## Textbooks problem — teaching the wrong things all too well

from Lars Syll

It is well known that even experienced scientists routinely misinterpret p-values in all sorts of ways, including confusion of statistical and practical significance, treating non-rejection as acceptance of the null hypothesis, and interpreting the p-value as some sort of replication probability or as the posterior probability that the null hypothesis is true …

It is shocking that these errors seem so hard-wired into statisticians’ thinking, and this suggests that our profession really needs to look at how it teaches the interpretation of statistical inferences. The problem does not seem just to be technical misunderstandings; rather, statistical analysis is being asked to do something that it simply can’t do, to bring out a signal from any data, no matter how noisy. We suspect that, to make progress in pedagogy, statisticians will have to give up some of the claims we have implicitly been making about the effectiveness of our methods …

It would be nice if the statistics profession was offering a good solution to the significance testing problem and we just needed to convey it more clearly. But, no, … many statisticians misunderstand the core ideas too. It might be a good idea for other reasons to recommend that students take more statistics classes—but this won’t solve the problems if textbooks point in the wrong direction and instructors don’t understand what they are teaching. To put it another way, it’s not that we’re teaching the right thing poorly; unfortunately, we’ve been teaching the wrong thing all too well.

Andrew Gelman & John Carlin

Teaching both statistics and economics, yours truly can’t but notice that the statements “give up some of the claims we have implicitly been making about the effectiveness of our methods” and “it’s not that we’re teaching the right thing poorly; unfortunately, we’ve been teaching the wrong thing all too well” obviously apply not only to statistics …

And the solution? Certainly not — as Gelman and Carlin also underline — to reform p-values. Instead we have to accept that we live in a world permeated by genuine uncertainty and that it takes a lot of variation to make good inductive inferences.

Sounds familiar? It definitely should!

The standard view in statistics – and the axiomatic probability theory underlying it – is to a large extent based on the rather simplistic idea that ‘more is better.’ But as Keynes argues in his seminal A Treatise on Probability (1921), ‘more of the same’ is not what is important when making inductive inferences. It’s rather a question of ‘more but different’ — i.e., variation.

Variation, not replication, is at the core of induction. Finding that p(x|y) = p(x|y & w) doesn’t make w ‘irrelevant.’ Knowing that the probability is unchanged when w is present gives p(x|y & w) another evidential weight (‘weight of argument’). Running 10 replicative experiments do not make you as ‘sure’ of your inductions as when running 10 000 varied experiments – even if the probability values happen to be the same.

According to Keynes we live in a world permeated by unmeasurable uncertainty – not quantifiable stochastic risk – which often forces us to make decisions based on anything but ‘rational expectations.’ Keynes rather thinks that we base our expectations on the confidence or ‘weight’ we put on different events and alternatives. To Keynes expectations are a question of weighing probabilities by ‘degrees of belief,’ beliefs that often have preciously little to do with the kind of stochastic probabilistic calculations made by the rational agents as modeled by “modern” social sciences. And often we ‘simply do not know.’