Where are the exceptional responders?

Intriguing article! (It’s available here BTW for anyone immoral enough to use SciHub.) My hot take would be that I’m actually dealing with a case of zero denominator — drawing inferences from the unreportedness of results. And indeed this corresponds to the ‘inversion’ you suggest.

But I can see this piece deserves to be read with some care.

In terms of uses for this model, then certainly its prime target would be the sponsor’s rhetoric — and specifically the shifting goalposts aspect. (The shifting mode of the Gamma distribution indeed looks like a moving goalpost.) Initially, of course, there’s lots of hope that patients will benefit. But as time goes on, this morphs implicitly into the argument that we can’t be certain that patients aren’t benefiting to some [subclinical] degree. This kind of model may serve to mark approximately where that goalpost has moved at any point in time.

Beyond this, however the \theta_c parameter in this model might help focus attention on improving clinical assessment.

Addendum: I’ve read the Hanley & Lippman-Hand article properly, and enjoyed being reminded of that hoary old Rule of three (statistics) - Wikipedia. The psychologistic dimensions of their argument are also worth attending to. But in contrast to what I’m doing here, they confine themselves to inferences about a simple rate parameter. In the model above, I draw inferences about the whole distribution of therapeutic effect sizes. That is, rather than confining myself to estimating just the prevalence of effects above the \theta_c threshold, I’m trying to learn something about even clinically undetectable \theta_i \ll \theta_c effect sizes. Of course, such an effort rests on invoking some kind of Power law - Wikipedia principle (see also Manfred Schroeder’s lovely Fractals, Chaos, Power Laws), which I’ve done implicitly in setting up my model. I conjecture that other reasonable model set-ups would effectively recapitulate the linear growth of \beta(n).