Posts Tagged With: coin toss

A Highly Anomalous Event

The journey to San Francisco was smooth sailing with no plane delays; within two hours of landing I found myself in the E.R. of St. Francis Hospital (with the philosopher of science Ronald Giere), unable to walk.  I have just described an unexpected, “anomalous”, highly unusual event, but no one would suppose it was anomalous FOR, i.e., evidence against some theory, say, in molecular biology.  Yet I am  getting e-mails (from readers) saying, in effect, that since the improbable coin toss result is very unexpected/anomalous in its own right, it therefore is anomalous for any and all theories, which is patently absurd.  What had happened, in case you want to know, is that just as I lunged forward to grab my (bulging) suitcase off the airline baggage thingy, out of the corner of my eye I saw my computer bag being pulled away by someone on my left, and as I simultaneously yanked it back, I tumbled over—very gently it seemed– twisting my knee in a funny way.  To my surprise/alarm, much as a tried, I could put no weight on my right leg without succumbing to a Geppeto-puppet-like collapse.  The event, of course, could rightly be regarded as anomalous for hypotheses about my invulnerability to such mishaps, because it runs counter to them.  I will assume this issue is now settled for our discussions, yes?

Categories: Statistics | Tags: , , , , | 29 Comments

Getting It Right But for the Wrong Reason

Sitting in the airport . . . a temporary escape from Elba, which I’m becoming more and more loathe to leave.  I fear that some might agree, rightly, that Kadane’s “trivial test” is no indictment of significance tests and yet for the WRONG reason. I don’t want to beat a dead horse, but perhaps a certain confusion is going to obstruct understanding later on. Let us abbreviate “tails” on a coin toss that lands tails 5% of the time, as “a rare coin toss outcome”. Some seem to reason: since a rare coin toss outcome is an event with probability .05 REGARDLESS of the truth or falsity of a hypothesis H, then the test is still a legitimate significance test with significance level .05; it is just a lousy one, with no discriminating ability. I claim it is no significance test at all, and that there is an important equivocation going on (in some letters I’ve received)—one which I hoped would be skirted by the analogy with ordinary hypothesis testing in science. Heading off this confusion was the key rationale for my discussion in the Kuru post. Finding no nucleic acid in prions is inconsistent, or virtually so, under the hypothesis H: all pathogens are transmitted with nucleic acid. The observed results are anomalous for the central dogma H BECAUSE they are counter to what H says we would expect. If you maintain that the “rare coin toss outcome” is anomalous for a statistical null hypothesis H, then you would also have to say they are anomalous for H: all pathogens have nucleic acid. But it is obvious this is false in the case of the scientific hypothesis. It must also be rejected in the case of the statistical hypothesis (Rule #1).

A legitimate statistical test hypothesis must tell us (i.e., let us compute) how improbably far different experimental outcomes are from what would be expected under H. It is correct to regard experimental results as anomalous for a hypothesis H only if, and only because, they run counter to what H tells us would occur in a universe where H is correct. A hypothesis on pathogen transmission, say, does not tell us the improbability of the rare coin toss outcome. Thus it is no significance test at all. As I wrote in the Kuru post:  It is not that infectious protein events are “very improbable” in their own right (however one construes this); it is rather that these events are counter to, and forbidden under, the assumption of the hypothesis H.

Categories: Statistics | Tags: , , , | 4 Comments

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