# Richard Royall

## How likelihoodists exaggerate evidence from statistical tests

I insist on point against point, no matter how much it hurts

Have you ever noticed that some leading advocates of a statistical account, say a testing account A, upon discovering account A is unable to handle a certain kind of important testing problem that a rival testing account, account B, has no trouble at all with, will mount an argument that being able to handle that kind of problem is actually a bad thing? In fact, they might argue that testing account B is not a  “real” testing account because it can handle such a problem? You have? Sure you have, if you read this blog. But that’s only a subliminal point of this post.

I’ve had three posts recently on the Law of Likelihood (LL): Breaking the [LL](a)(b)[c], and [LL] is bankrupt. Please read at least one of them for background. All deal with Royall’s comparative likelihoodist account, which some will say only a few people even use, but I promise you that these same points come up again and again in foundational criticisms from entirely other quarters.[i]

An example from Royall is typical: He makes it clear that an account based on the (LL) is unable to handle composite tests, even simple one-sided tests for which account B supplies uniformly most powerful (UMP) tests. He concludes, not that his test comes up short, but that any genuine test or ‘rule of rejection’ must have a point alternative!  Here’s the case (Royall, 1997, pp. 19-20):

[M]edical researchers are interested in the success probability, θ, associated with a new treatment. They are particularly interested in how θ relates to the old treatment’s success probability, believed to be about 0.2. They have reason to hope θ is considerably greater, perhaps 0.8 or even greater. To obtain evidence about θ, they carry out a study in which the new treatment is given to 17 subjects, and find that it is successful in nine.

Let me interject at this point that of all of Stephen Senn’s posts on this blog, my favorite is the one where he zeroes in on the proper way to think about the discrepancy we hope to find (the .8 in this example). (See note [ii]) Continue reading

Categories: law of likelihood, Richard Royall, Statistics | Tags:

## Why the Law of Likelihood is bankrupt–as an account of evidence

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There was a session at the Philosophy of Science Association meeting last week where two of the speakers, Greg Gandenberger and Jiji Zhang had insightful things to say about the “Law of Likelihood” (LL)[i]. Recall from recent posts here and here that the (LL) regards data x as evidence supporting H1 over H0   iff

Pr(x; H1) > Pr(x; H0).

On many accounts, the likelihood ratio also measures the strength of that comparative evidence. (Royall 1997, p.3). [ii]

H0 and H1 are statistical hypothesis that assign probabilities to the random variable X taking value x.  As I recall, the speakers limited  H1 and H0  to simple statistical hypotheses (as Richard Royall generally does)–already restricting the account to rather artificial cases, but I put that to one side. Remember, with likelihoods, the data x are fixed, the hypotheses vary.

1. Maximally likely alternatives. I didn’t really disagree with anything the speakers said. I welcomed their recognition that a central problem facing the (LL) is the ease of constructing maximally likely alternatives: so long as Pr(x; H0) < 1, a maximum likely alternative H1 would be evidentially “favored”. There is no onus on the likelihoodist to predesignate the rival, you are free to search, hunt, post-designate and construct a best (or better) fitting rival. If you’re bothered by this, says Royall, then this just means the evidence disagrees with your prior beliefs.

After all, Royall famously distinguishes between evidence and belief (recall the evidence-belief-action distinction), and these problematic cases, he thinks, do not vitiate his account as an account of evidence. But I think they do! In fact, I think they render the (LL) utterly bankrupt as an account of evidence. Here are a few reasons. (Let me be clear that I am not pinning Royall’s defense on the speakers[iii], so much as saying it came up in the general discussion[iv].) Continue reading

## BREAKING THE (Royall) LAW! (of likelihood) (C)

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With this post, I finally get back to the promised sequel to “Breaking the Law! (of likelihood) (A) and (B)” from a few weeks ago. You might wish to read that one first.* A relevant paper by Royall is here.

Richard Royall is a statistician1 who has had a deep impact on recent philosophy of statistics by giving a neat proposal that appears to settle disagreements about statistical philosophy! He distinguishes three questions:

• What should I believe?
• How should I act?
• Is this data evidence of some claim? (or How should I interpret this body of observations as evidence?)

It all sounds quite sensible– at first–and, impressively, many statisticians and philosophers of different persuasions have bought into it. At least they appear willing to go this far with him on the 3 questions.

How is each question to be answered? According to Royall’s commandments writings, what to believe is captured by Bayesian posteriors; how to act, by a behavioristic, N-P long-run performance. And what method answers the evidential question? A comparative likelihood approach. You may want to reject all of them (as I do),2 but just focus on the last.

Remember with likelihoods, the data x are fixed, the hypotheses vary. A great many critical discussions of frequentist error statistical inference (significance tests, confidence intervals, p- values, power, etc.) start with “the law”. But I fail to see why we should obey it.

To begin with, a report of comparative likelihoods isn’t very useful: H might be less likely than H’, given x, but so what? What do I do with that information? It doesn’t tell me I have evidence against or for either.3 Recall, as well, Hacking’s points here about the variability in the meanings of a likelihood ratio across problems. Continue reading

Categories: law of likelihood, Richard Royall, Statistics