{
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  "path": "/t/where-are-the-exceptional-responders/17961#post_8",
  "publishedAt": "2026-03-08T06:26:37.000Z",
  "site": "https://discourse.datamethods.org",
  "tags": [
    "back-of-the-envelope calculations.",
    "\\text{Inv-Gamma}"
  ],
  "textContent": "Let me follow up now with some back-of-the-envelope calculations.\n\nWriting \\theta_p for the p-quantile of \\theta, we have\n\np = P(\\theta < \\theta_p) = \\theta_p^\\lambda,\n\nfrom which we obtain\n\n\\frac{\\ln(1/\\theta_p)}{\\ln(1/p)} = \\frac{1}{\\lambda}.\n\nNow, since \\lambda \\sim \\text{Gamma}(\\alpha,\\beta_n), we know that\n\n\\frac{\\ln(1/\\theta_p)}{\\ln(1/p)} = \\frac{1}{\\lambda} \\sim \\text{Inv-Gamma}(\\alpha,\\beta_n).\n\nBecause 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.\n\nGiven 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):\n\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}\n\nLooking 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:\n\n\\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},\n\ncorresponding to \\theta_{0.9} < 0.1 — a quite dismal bound on efficacy.\n\nNow 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\n\n\\text{median}\\left(\\ln\\frac{1}{\\theta_p}\\right) > \\text{mode}\\left(\\ln\\frac{1}{\\theta_p}\\right),\n\nwe can at least state that **median** \\theta_{0.9} < 0.1.\n\nThus, 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.\n\nDoes anybody see a flaw in the argument, or a mistake in my math?",
  "title": "Where are the exceptional responders?"
}