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  "path": "/t/where-are-the-exceptional-responders/17961#post_5",
  "publishedAt": "2026-03-02T14:37:57.000Z",
  "site": "https://discourse.datamethods.org",
  "tags": [
    "_conceptually_ , at least — I acknowledge [doubts",
    "inverse scale parameter",
    "“limited in extent, significance or stature”"
  ],
  "textContent": "I’ve been working toward a formulation that crystallizes the intuition here, in something like the way economists tend to do with their ‘toy models’. Here’s what I have thus far:\n\nLet’s scope the discussion specifically to **degenerative disease** , in which we have a concept of _disease activity_ A(t) as a function of time, which is the time-derivative of a cumulative state. In DMD, one could think of this as the rate of fibrofatty replacement (FFR) of inflamed muscle tissue. In Alzheimer dementia, this could be _conceptually_ , at least — I acknowledge [doubts about the amyloid cascade hypothesis] the rate of amyloid β deposition. Lysosomal storage diseases may lend themselves to a similar concept, etc.\n\nI think it important that this picture could be elaborated substantively in terms of stochastic processes, with activity A_t modeled as (say) a mean-reverting Ornstein-Uhlenbeck process, and disease progression captured in its time-integral S_t = \\int_0^t A_u \\text{d}u . But I won’t pursue that here.\n\nWhat I’d like to focus on is characterizing the putative effect of a therapy as _reduction of disease activity_. Certainly, this accords with the concept of steroid use in DMD, which targets the inflammation that [presumably] lies immediately upstream of the FFR process.\n\nAs a first approximation, let’s treat this reduction as multiplication by a factor 1-\\theta for \\theta \\in [0,1]. (_Th_ eta for _th_ erapy.) Thus, \\theta = 0 is a null therapy, while \\theta =1 corresponds to a cure. In a heterogeneous degenerative disease — muscular dystrophy makes an excellent example here, with its many different mutations generating a wide spectrum of severity — we have to expect substantial HTE. The _population distribution_ of \\theta then needs to be considered. For simplicity, I’ll posit that a randomly selected individual i has \\theta_i with the 1-parameter distribution,\n\n\\text{P}(\\theta_i < \\theta) = \\theta^\\lambda.\n\nThus, for \\lambda=1 we have \\theta \\sim \\text{U}[0,1], while for \\lambda \\rightarrow \\infty we get increasing concentration of mass near \\theta \\approx 1 (a cure), and for \\lambda \\rightarrow 0 we concentrate the mass near \\theta \\approx 0 (a marginal therapy).\n\nIn a situation where a sponsor thrashes thru a bunch of trials that never get reported , outsiders can nevertheless draw some inferences about \\lambda by using the **lack** of any case-report of exceptional response as a _censored_ observation of \\theta.\n\nSuppose that any activity reduction exceeding \\theta_c would be _clinically evident_ (_c_ for _c_ linical or _c_ ritical, say). Then every participant enrolled in one of these hidden trials and never highlighted in favorable a case report contributes a factor \\theta_c^\\lambda to the likelihood, and the likelihood for n such participants is (\\theta_c^\\lambda)^n = e^{n \\ln\\theta_c \\cdot \\lambda}.\n\nNow \\lambda \\sim \\text{Gamma}(\\alpha,\\beta) would yield a conjugate prior. That is, if we chose the prior\n\np(\\lambda) = \\frac{\\beta^\\alpha}{\\Gamma(\\alpha)} \\lambda^{\\alpha-1} e^{-\\beta \\lambda},\n\nthen our posterior is also Gamma-distributed:\n\n\\begin{aligned} p(\\lambda \\mid n) = p(\\lambda)\\cdot\\theta_c^{n\\lambda} & \\propto \\lambda^{\\alpha-1}e^{-\\beta \\lambda}\\cdot e^{-n \\ln(1/\\theta_c)\\lambda} \\\\\\ & = \\lambda^{\\alpha-1} e^{-[\\beta +n \\ln(1/\\theta_c)]\\lambda} \\end{aligned}\n\nIndeed, we see \\lambda \\sim \\text{Gamma}(\\alpha,\\beta_n) with \\beta_n = \\beta+n\\ln(1/\\theta_c).\n\nObserve that \\theta_c < 1 \\implies \\ln(1/\\theta_c) > 0, so that we have here a \\beta_n parameter increasing linearly with the number n of participants enrolled quietly in these never-reported trials. Moreover, the constant \\ln(1/\\theta_c) will be on the order of \\frac{1}{2}, if e.g. a 60% reduction in disease activity would be clinically evident: \\ln (1/0.6)\\doteq 0.51.\n\nBecause the Gamma distribution’s \\beta is an inverse scale parameter, \\beta\\rightarrow\\infty concentrates the posterior mass toward \\lambda \\rightarrow 0 (marginality).\n\nIncidentally, this term ‘marginal’ seems appropriate here on the dual grounds of its colloquial meaning “limited in extent, significance or stature” and its more formal statistical meaning — that any efficacy will only ever be detected in a _marginal_ analysis averaging over many patients, and never _clinically_ in any given individual.",
  "title": "Where are the exceptional responders?"
}