[Optimization – Solver] Good model fit, but wrong standard deviation coefficients

cgmac

New Member
#1
Dear reader,
I am facing an optimization problem and, at this point, I need help from someone with more knowledge than me.

My goal:
For a non-linear regression method, I use the solver add-in function of Microsoft Excel to obtain the equation parameters by minimizing the sum of the squares of the errors (ERRSQ) between theoretical and experimental data.
ERRSQ = \( \sum_{i = 0}^n ( q_{calc} - q_{exp})^2_i \)
I follow a trial an error procedure to accomplish the task. My solution is obtained after trying different sets of initial estimates to test the validity of my answer (global vs local minimum).
Through the Solvstat macro, I calculate the standard deviation of the coefficients, the correlation coefficient (\(R^2\)) and the standard error of the y-estimate (the root-mean-square deviation, RMSD).

Input data:
>>> Sample size = 8 measurements
>>> Coefficients to be determined: 5
>>> Solver Options: Convergence = 0.0001; Derivatives= Central; Automating scaling activated; Solving method=GRG Nonlinear; No constrains used.

Output data example:
The error between predicted and experimental data is low and the fit is good based on \(R^2\) value. Visually the fitting looks fine but the standard deviation of coefficients (std) does not seems appropriated.
>>> Coefficient value: Q = 1.139 ; std = 4.650
>>> Coefficient value: Ks = 143.860 ; std = 433.971
>>> Coefficient value: KL = 0.0033 ; std = 0.0195
>>> Coefficient value: m = 0.596 ; std = 1.456
>>> Coefficient value: p = 1.714 ; std = 35.048
>>> \(R^2\) = 0.995
>>> RMSD = 7.551

My concern:
I have read all relevant documentation I was able to find, that includes books, blogs/websites, forums and scientific articles, related with the application of Solver, as well as, the Solvstat macro to my dataset, as far as I know, my set up is right.
In a different case, when only two coefficient determination is required, I noticed that the std of the coefficients are suitable, could it be a sample size problem?
It might be required for you to know how exactly the std is computed by this solvstat macro, I initially will not explain it to make this query as short as possible, but I can provide the information.
Since nobody in the publications seems to provide these std for the coefficients, and they only base their good results in the \(R^2\) and RMSD, should I be worried about the std of the coefficients?
If you have found time to read these lines, thank you. Could you please let me know about your opinion?

Regards.
 

Dason

Ambassador to the humans
#2
You have 8 measurements and 5 parameters. It doesn't surprise me that the standard errors for the coefficients would be fairly large.