Dose-Response Model(s) in Stan

Dear community,
I am facing a new challenge, so I’m here to ask you for some advice.
Is there some publication, worksheet, manual… to understand better how to build a dose-response model?
I would like to work on some data that show hormesis, when exposure to an agent can be both a stimulus (at a lower dosage) and an inhibition (at a higher dosage).
I am very new to such a broad field. Is there a way to describe hormesis with differential equations?
I know that Stan can handle such non-linearity in the response in some way. Have you advice or tricks to try?

@martinmodrak , any hints here?

Dose-response models often appear within the field of (microbial) risk assessment, so the general literature is indeed pretty broad and includes a lot beyond the models themselves, so you’ll find very general references you may or may not want to get into (like this). The modeling itself varies from bottom-up formulations of the curves to simply fitting a sigmoidal curve to the data. The basic model would be something like p = 1 - e^{-pd}, and there are some considerations about why that and not just some logistic function.

Two references which I think address very well both the biological and modeling considerations are Furumoto and Mickey (1967). It’s not just a cute reference because it’s from the 60s, but they actually derive the expression above from theoretical considerations about the probability of infection of each infectious particle, and also consider non-homogeneous probabilities of infection, resulting in the model p_{het} = 1 - \int_0^1 e^{-xpd}q(x)dx, with q(x) \sim Beta(x|\alpha, \beta) , and go on to to experimental tests by fitting the model (crudely, after all it was the 60s). In my opinion a lot can be learned from those considerations alone, probably more than a much larger review of phenomenological approaches to dose-response models.

We have rescued some of these formulations and used bayesian inference with the beta distribution and underlying survival curves for each dose in Pessoa et al. (2014) and gamma distribution in King, Souto-Maior et al. (2018) – the implementation of those is not in Stan, but a recent matters arising comment is. Especially in the first paper there are also more references and detailed description of building the model.

To your goal, I don’t know how you identify hormesis (wasn’t familiar with the term) from the data, but one thing we did notice in our systems is that the heterogeneous model you could get higher risks for lower doses and lower risk for higher ones (compared to the homogeneous “exponential” model), and that this was a result of certain mean and variance combinations, i.e. the distribution of susceptibility. This sounds a bit like you are describing, though I’m not sure if this is similar to what you are trying to model or not. The point is, sometime it may seem like you need to explicitly model to some effect, but there may be a simple probabilistic property that explains the result without it – I find that pretty cool, and tend to think stochasticity doesn’t get as much credit as it deserves for things happening.

Hope this helps, let me know if it does (or does not).


So many things to read and study, so thanks @caesoma. I will start to tackle both theory and data analysis, so I hope we will talk here more about the risks and the heterogeneous models.

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I got sensible advances using the Cedergreen-Ritz-Streibig modified log-logistic model that is a 6 parameter model (\alpha, b, c, d, e, f) so that

\mu(x) = c + \frac{d-c+f exp(-1/(x^\alpha))}{1+exp(b (log(x) -log(e)))}\\ y \sim Normal(\mu, \sigma)

I tried The CRS model both on simulated data (Stan recovers the parameters well) and on measured data, doing inference.

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I am not familiar with that model and don’t know what processes the parameters are supposed to describe, but it looks like quite a few parameters for a dose-response model. I’d be interested in what shape is produced by that. If you can post some of the outputs with simulated and/or real data it would be nice to see.