Modeling environmental effects on an inferred parameter

Modeling
1

Hi all,

I would kindly like to inquire about modeling environmental effects in an exponential (sigmoidal) recovery function.

The function is as follows: X_t = α + (ω - α) * (1 - exp(-λ * t))^δ

I would, however, like to get something like: “the absolute recovery after 20 years is positively affected by precipitation,” with a corresponding effect size and 95% CI.

I can add an environmental effect on λ (e.g. λ = λ_i + θ_λprecipitation), as well as on ω or (ω - α) (e.g. ω = ω_i + θ_ωprecipitation).

However, these effects remain specific to the corresponding parameter, so I could say “precipitation has a positive effect on λ and a negative effect on ω” with corresponding uncertainties, but what about the inferred quantity:

abs_rec_20y = [α + (ω - α) * (1 - exp(-λ * 20))^δ] - [α] = (ω - α) * (1 - exp(-λ * 20))^δ

==> This depends on both the intrinsic rate λ and the absolute difference (ω - α).

Thanks in advance!
Viktor

2

Unless I’m missing something here, you could simply put something like

abs_rec_20y = (alpha + (omega - alpha)*pow((1 - exp(-lambda*20)), delta)

in generated_quantities (where alpha, omega, lambda, and delta are the parameters in your statistical model.

3

When you say:

it seems like you could answer this more directly with a simple regression style model:

X_{20} | \mu, \beta, \sigma^2, z_{precipitation} \sim \mathcal{N}(\mu + \beta z_{precipitation}, \sigma^2)

and examining posterior inferences on \beta?

When you talk about including precipitation effects in your growth model, where you include it in the model really depends on your domain expertise. Including a precipitation effect on \omega would model the effect of precipitation on the terminal growth size; including it on \lambda would model the effect on the growth rate; and including it on \delta would model how it affects the bend of the curve/how sharp growth is.

Any of those seem plausible, so it’d be up to your expertise to determine how it should be included in the model. If you’re happy with the inferences, then you could pick a fixed time window like X_{20}, as you suggest, and compare how varying precipitation changes the posterior values of X_{20}.

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