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:

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 (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.

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 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,

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

Thus, 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).

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 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}.

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 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.

Because the Gamma distribution’s \beta is an inverse scale parameter, \beta\rightarrow\infty concentrates the posterior mass toward \lambda \rightarrow 0 (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 ever be detected in a marginal analysis averaging over many patients, and never clinically in any given individual.