Where are the exceptional responders?

Let me follow up now with some back-of-the-envelope calculations.

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} 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?