I set up a kinetic model consisting of a system of ODE's. These ODE's are solved with an ode-solver (ode45) for varying parameters $k_1$ & $k_2$.
The parameters are estimated using lsqnonlin (for nonlinear least-squares (nonlinear data-fitting) problems) which minimizes the "difference" between experimental and model data. The dataset consists of 180 observations from 6 experiments.
To evaluate model quality with the resulting $k_1$ & $k_2$, I would like to calculate the Akaike Information criterion (AIC) (to compare to other models with more or less parameters). For AIC I use the following formula:
$$AIC = 2p - 2\ln(L)$$
where $p$ is the number of parameters and $\ln(L)$ is the maximum log-likelihood of estimated model.
From Spiess & Neumeyer: An evaluation of R² as an inadequate measure for nonlinear models in pharmacological and biochemical research I got the following formula for $\ln(L)$:
lnL = 0.5 * (-nRes * (log(2*pi) + 1 - log(nRes) + log(sum((yExp(:)-yModel(:)).^2))));
nRes - number of observations,
yExp - experimantal data,
yModel - model data.
And here are my questions (especially Q3-Q5 in EDITs unanswered):
Is this formula for max log-likelihood okay to use in this context? I feel like the "punishment" for more parameters is quite weak if I'm getting AIC values of e.g. -2343 and -2199 for a model with 3 and 2 parameters, respectively.
Should I look at a different parameter than AIC to compare the models?
Edit 1: Trying to give a bit more context on the data.
6 Experiments were conducted. During each, 10 samples at different time points were taken with the first one being the starting conditions. In each of the 60 samples, the concentration of 3 species were measured, leading to the total of 180 observations. Different Kinetic models for the concentrations of these species are fitted to the data for different kinetic parameters.
Question 3: Is the 180 correct to use as
Regarding the discussion with measure_theory on why
log(nRes) is in the formula of $\ln(L)$.
The original formula would be:
lnL = 0.5 * (-nRes * (log(2*pi) + 1 + log(σ_e²)));
with Maximum likelihood estimator
log(RSS/nRes) = log(RSS) - log(nRes)
So I think the formula should be okay, if the normality assumption for the residuals is correct, which I hope it is...
Question 4: Does someone have any input for question 3 in Edit 1?
Question 5: I'm still wondering about the number of parameters that I have to use for $p$. Of course all parameters estimated in the model are counted, but I read I have to count σ². Is this correct? (number of estimated parameters +1?)