# Checking if two sum of squared residuals are significantly different

I have a system of non linear differential equations and in order to find model parameters I have fitted the model to some data using non linear least squares method and have calculated the sum of squared residuals. But for some reason I had to deviate from this best fit parameters and use slightly different values for the model. Now I want to check whether when these adjusted parameter values are used and check with data if the sum of squared errors that I obtain now is significantly different to than that of the previous one (as a way to indicate that although that I have used slightly different values form the best fit parameters it doesn't affect significantly).

So, basically I want to check if the difference of two sum of squared errors is significantly different from one another.

If $e_1$ and $e_2$ are the two sum of squared errors what type of test can I use? can I use t test and can you please explain how to construct it.

• Do you care about the magnitude of the squared errors or the pattern of the square errors? Apr 16, 2018 at 2:39
• @ReneBt magnitude of the squared errors Apr 16, 2018 at 5:52

## 1 Answer

The T-test works on normal data and the assumption under many fitted models is that the residual variation (error) is normally distributed. You can test this assumption using a test such as Kolmogorov-Smirnov and if it fails you can use non-parametric equivalents of the T-test (although it would probably also lead to questions about the appropriateness of your model if one assumption was normality in the residuals).

To work with a T-test you would take the magnitude of the errors as vectors for each model (not the square errors, the T-test is based on variation not variance) and compare them using a paired T-test. What this will allow you to assess is whether the differences between the models can be explained by random chance reasonably well.

However, if what you want is to demonstrate equivalence of the models, then you have a more stringent test to perform, a test of equivalence: http://homepage.stat.uiowa.edu/~rdecook/stat6220/Class_notes/equivalence_testing.pdf https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5502906/