Where are the exceptional responders?

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:

[CONTENT WARNING — Rated NC17 for full frontal HTE]

Let’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.

I think it important that this picture could be elaborated substantively in terms of stochastic processes, with activity A_t modeled as (e.g.) 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.

What 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.

As a first approximation, let’s treat this reduction as multiplication by a factor \theta \in [0,1]. (Th eta for th erapy.) 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 any given individual i has \theta_i with the 1-parameter distribution,

\text{P}(\theta_i < \theta) = \theta^\lambda.

Thus, for \lambda=1 we have \theta \sim \text{U}[0,1], while for \lambda \rightarrow 0 we get increasing concentration of mass near \theta \approx 0 (a cure), and for \lambda \rightarrow \infty we concentrate the mass near \theta \approx 1 (a marginal therapy).

In 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.

Suppose that a certain level of activity reduction \theta_c would be clinically evident (c for c linical or c ritical, say). Then every participant enrolled in one of these hidden trials contributes a factor \theta_c^\lambda to the likelihood. The likelihood for n participants enrolled and not subsequently published as case-reports of exceptional response is (\theta_c^\lambda)^n = e^{n \ln \theta_c \cdot \lambda}.

Now \lambda \sim \text{Gamma}(\alpha,\beta) would yield a conjugate prior. That is, if we chose the prior

p(\lambda) = \frac{\beta^\alpha}{\Gamma(\alpha)} \lambda^{\alpha-1} e^{-\beta \lambda},

then our posterior is also Gamma-distributed:

\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}

Indeed, we see \lambda \sim \text{Gamma}(\alpha,\beta_n) with \beta_n = \beta+n\ln(1/\theta_c).

Observe 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 unreported trials.

Because the Gamma distribution’s \beta is an inverse scale parameter, this drives the posterior mass away from zero (cure) toward \lambda \rightarrow \infty (marginality).

Incidentally, 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 every be detected in a marginal analysis averaging over many patients, and never clinically in any given individual.