The following is not so much a modeling problem, the brms procedure works very well as such, it is about interpretation of results. This is what I study: the concentration of a chemical © as a function of time (t) is modeled by its initial concentration (c0), a rate constant ( k) and an order ( n) , according to the following equation:
I have a dataset available with repetitions of the experiment studying exactly the same reaction; concentration is measured as a function of time. Even though the conditions are the same, there is variation in the results, obviously and, consequently I want to know how the parameters vary between the repetitions. The data within one repetition are correlated because samples are taken from the same reactor as a function of time. The idea was therefore to apply multilevel modeling to investigate how variable the three parameters are over the various repetitions. The additional benefit of multilevel modeling is that it takes the experimental correlation within each repetition into account. Multilevel modeling using brms does give seemingly very reasonable output, showing nicely how the three parameters vary per repetition, part of the code I used for the model in brms is:
nlform<-bf(conc ~ (c0^(1-n)+(n-1)*k*time)^(1/(1-n)), c0 ~1+(1|ID|serial), n~1+(1|ID|serial), k ~ 1+(1|ID|serial), nl=TRUE)
The output is as such very nice and instructive but the problem I face is that the rate constant parameter (k) has dimensional units that contains the order parameter (n) in it:
This dimensional construction is needed in order to arrive at the right units for the concentration c (should be in mol/L) as a function of time (the order n is dimensionless). So, as soon as parameter n changes numerically (which it does because of the varying effect), the unit of k changes with it, and next to that, k itself is also subject to a varying effect. This makes it not well possible to compare the varying values of parameter k, it seems.
The multilevel modeling results indicate that k and n are correlated but not that strong ($\rho$=-0.07, SE=0.32): see attached figure made from brms output with GGally .
My question is: how can I interpret the varying k-values ? Of course, I also get an overall population value for k and n. Those can be used in conjunction without problem because they are “fixed” by definition. But I am struggling with the varying effects: can I compare them and how?