# Where are the exceptional responders?

**URL:** <https://discourse.datamethods.org/t/where-are-the-exceptional-responders/17961>\
**Category:** research methods\
**Created:** [July 24, 2024, 10:38pm UTC](https://discourse.datamethods.org/t/where-are-the-exceptional-responders/17961 "2024-07-24T22:38:09Z")\
**Posts on this page:** 1\
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**Author:** ![davidcnorrismd](https://discourse.datamethods.org/user_avatar/discourse.datamethods.org/davidcnorrismd/32/3502_2.png) [@davidcnorrismd](https://discourse.datamethods.org/u/davidcnorrismd)\
**Post date:** [March 8, 2026, 6:26am UTC](https://discourse.datamethods.org/t/where-are-the-exceptional-responders/17961/8 "2026-03-08T06:26:37Z")

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Let me follow up now with some [back-of-the-envelope calculations.](https://en.wikipedia.org/wiki/Back-of-the-envelope_calculation)

Writing \theta\_p for the p-quantile of \theta, we have

p = P(\theta \< \theta\_p) = \theta\_p^\lambda,

from which we obtain

\frac{\ln(1/\theta\_p)}{\ln(1/p)} = \frac{1}{\lambda}.

Now, since \lambda \sim \text{Gamma}(\alpha,\beta\_n), we know that

\frac{\ln(1/\theta\_p)}{\ln(1/p)} = \frac{1}{\lambda} \sim \text{Inv-Gamma}(\alpha,\beta\_n).

Because the \beta parameter of [\text{Inv-Gamma}](https://en.wikipedia.org/wiki/Inverse-gamma_distribution) is a **scale** parameter (rather than an _inverse_-scale, as with the \text{Gamma} distribution), we now have that \beta\_n \rightarrow \infty shifts our distribution to the _right_. This drives \ln(1/\theta\_p) \rightarrow \infty and consequently \theta\_p \rightarrow 0.

Given our interest (as noted above) in _watching the goalposts move_, it makes sense to focus on the _mode_ of \text{Inv-Gamma}(\alpha,\beta\_n):

\begin{aligned} \text{mode}\left(\ln\frac{1}{\theta\_p}\right) = \ln\frac{1}{p}\cdot\frac{\beta\_n}{\alpha+1} & = \ln\frac{1}{p}\cdot\frac{\beta + n\ln\frac{1}{\theta\_c}}{\alpha+1} \\ & \> \ln\frac{1}{p}\cdot\frac{n\ln\frac{1}{\theta\_c}}{\alpha+1} = \frac{n}{\alpha+1}\cdot\ln(1/p)\cdot\ln(1/\theta\_c). \end{aligned}

Looking for some reasonable numbers to plug in here, consider first that a low-information prior will have small \alpha \sim \mathcal{O}(1). Accordingly, let’s suppose \alpha = 1. If we generously (to the sponsor) allow \theta\_c = 0.8 (requiring a 80% reduction in disease intensity to cross the threshold of clinical detectability), and choose p = 0.9 (so that we are asking for the therapeutic effect at the _top decile of responses_), then n\approx 200 enrolled to date in eteplirsen trials (noted in the top post) yields:

\text{mode}\left(\ln\frac{1}{\theta\_{0.9}}\right) \> \frac{200}{1+1}\cdot\ln\frac{1}{0.9}\cdot\ln\frac{1}{0.8} = 2.35 \doteq \ln\frac{1}{0.095},

corresponding to \theta\_{0.9} \< 0.1 — a quite dismal bound on efficacy.

Now it should be said that the mode (unlike the median) is _not_ invariant under transformations. So this ‘correspondence’ doesn’t directly bound the _modal_ \theta\_{0.9} (on the \theta scale). Still, since the rightward skew of the \text{Inv-Gamma} distribution guarantees that

\text{median}\left(\ln\frac{1}{\theta\_p}\right) \> \text{mode}\left(\ln\frac{1}{\theta\_p}\right),

we can at least state that **median** \theta\_{0.9} \< 0.1.

Thus, we conclude there’s a below-50% (Bayesian) chance the top decile of responses does better than a 10% reduction in disease activity — a strong suggestion that this is a truly marginal drug.

Does anybody see a flaw in the argument, or a mistake in my math?

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_[View the full topic](https://discourse.datamethods.org/t/where-are-the-exceptional-responders/17961)._