In the current model, we take this parameterization of the logistic growth curve.
\[\begin{align*} \mu &= c+\frac{A}{1+e^{-k_(d-x_{0})}} \end{align*}\]
where \(A\) is the asymptote, \(x_0\) is the midpoint, and \(k\) is the rate.
With this parameterization, we can derive the following parameters analytically.
\[\begin{align*} v &= \frac {A k } {4} \\ D &= \frac {2 log(19) } { k} \\ d_{5} &= x_0 - \frac{D}{2} \\ d_{95} &= x_0 + \frac{D}{2} \end{align*}\]
where \(v\) is the maximum speed or speed at the midpoint, and \(D\) is the time it takes from 5% to 95% of the asymptote.
The concern is that speed and duration have build-in correlation through \(k\). As we want to investigate potential trade-off between duration and speed, we might need them to be inferred as two independent parameters.
When fitting this reparameterized Bayesian model, we will calculate \(k\) and \(A\) from \(v\) and \(D\) in the model.
\[\begin{align*} k &= \frac{2 log(19)} {D} \\ A &= \frac{4 v} {k} = \frac{4 v D} {2 log(19)} \end{align*}\]
Currently, we used a uniform prior for \(A\) and a uniform model for \(log(k)\). If we have to reparameterize, we might use a uniform prior for \(log(v)\) and a uniform model for \(D\).
However, in a previous exploration, we addressed this by reparameterizing with \(d_{5}\), \(d_{95}\), and \(v\). That turned out to be very challenging.
I tried to reparameterize the logistic function with min, start, end, and speed. This was intended to make the inference more interpretable and to break the built-in correlations between treatment effects on the derived parameters e.g., start and end, duration and speed.
The reparameterization compiled much slower, had worse mixing, did not improve the model fit, and the inferred parameters did not look reasonable. I decided to keep the original parameterization with asym, xmid, and logk and proceed with post-hoc analyses.
I suspect that the \(v\) \(D\) parameterization will suffer the same problem as it takes away \(A\) which is probably the easiest to pin down.
I have tried to fit a nonlinear regression with \(v\) \(D\) parameterization using the nls function, but that did not remove the negative correlation. I suspect that there are identifiability issues, such that the measurements in this fast transition does not support us to tease apart duration and growth.