Goodness of fit for exponential model in lmfit I'm having a dataset where I measured the growth of a given molecule over 7 timepoints.
The growth can be described by the formula:
$(P_{0} - plateau) * e^{-k t} + plateau$ , where k is the growth constant and $t$ the time.
I want to estimate $P_{0}$, $plateau$ and $k$
All values are normalized to be between 0 and 1.
Due to the cost of the experiment, I only have one (depending on the perspective maybe 2) measurements per timepoint, however I have measurements for several thousands of molecules.
From what I have found out, the $R^{2}$ is not a good measure of fit in this case, despite it being used in many other studies I found.
I also considered using the chi square or chi square reduced value, which I obtain when I fit my model using the python lmfit library with Levenberg-Marquardt or Least square minimization (https://lmfit.github.io/lmfit-py/fitting.html).
However, I'm not sure how to interprete this value in terms what would be a good fit and what not.
Also, according to wikipedia, I need to scale my chi square value, by a variance, which is not done in lmfit as far as I understand:
$\chi^{2}=\sum_{i}{\frac{(O_{i} - C_{i})^{2}}{\sigma^{2}_{i}}}$
I assume in my case, O would be the values I measured, and C the values from my fit.
However I'm unsure what my variance would be in this case?
Here is an example plot of the data with a fit using the model and method described above:

Can somebody tell me if I'm on the right path and or if there are any alternatives for estimating the quality of my fits relative to the other fits using the same model (maybe BIC or AIC)?
Also, if there is any information missing I'm happy to add these.
I'm happy about any hints which would put me in the right direction!
 A: I'd leave $R^2$ out since your nonlinear models under comparison may have a different number of parameters. I would, instead, rely on one of these three approaches:
(a) use of information criteria such as BIC or AIC to select the best model or
(b) goodness of fit (GOF) test
(c)  paired two-sample test.
In (a) you compute the AIC (or BIC) for each model and, then you choose as the 'best model' that with the lowest AIC (or BIC).
In (b) essentially you have to choose a statistic that measures either the fit of the model or the difference in fit between two models and compare it's observed value against a distribution. There is huge literature here, see Goodnessof-Fit Techniques, edited by Ralph B. D'Agostmo and Michael A, Stephens.
(c) is more heuristic, still kind of GOF, and goes as this. Suppose you want to compare two models $M_1, M_2$. Use your preferred estimation method and get the fitted values under both models, say $\hat y_{M_1}$ and $\hat y_{M_2}$. Now, we expect $\hat y_{M_1},\hat y_{M_2}$ to be correlated, so take
$$
\hat{d} = \hat y_{M_1}-\hat y_{M_2},
$$
and apply a $t$-test or $z$-test to $\hat d$. If you reject the null then, the two fits have different population averages, i.e. the fits are different. Alternatively to $t$ and $z$, you may use non-parametric tests for differences in location for two paired samples.
If you want to test if the distributions of the fitted values under $M_1$ and $M_2$ are the same, you may try something like the Anderson-Darling test or the Kolmogorov-Smirnov test.
A: 
I only have one (depending on the perspective maybe 2) measurements per timepoint

Having only one measurement per timepoint brings you in a difficult situation. This makes it difficult to analyze whether the discrepancy between your fit and the data points is due to noise or due to a wrong model.
You need some independent measure of the expected noise.

*

*If you would have multiple measurements of the same situation (maybe you have at least a few cases with multiple measurements per time point?) then you can use the method described here: Linear regression: F-test for lack of fit (using ANOVA to test regression model) - intuition?
A technical/practical problem when applying this to your model might be that the source of error might be misrepresented. In your case it might be that you have errors that are heteroskedastic, or measurements errors in the "independent" x-variable besides errors in the y-variable. Such complex cases will make the analysis more difficult.


*Possibly you can have some theoretic considerations about the expected noise.
For instance, if it is a reasonable assumption that the distribution is Poisson distributed (this assumption can be tricky because one might have over or under dispersion) then you can estimate an expected noise level based on the estimated mean. If your observed noise level is larger, then you do not have a good fit.
In your case there might also have a priori models about the noise level. E.g. some physical models for growth dynamics and expected noise/variations in the growth curves.
