Multivariate Non-Linear Regression: Fitted Surfaces and Associated Errors

#1
Hello Talk Stats Community,

Currently, I am working on a project where I have a 3D set of data. This data has 2 independent variables and 1 dependent variable. I have figured out how to create a nonlinear fit surface for this set of data however, I am now trying to understand the assumptions and limitations associated with creating sed surface. Understanding the this assumptions and limitations will help me understand the overall accuracy of my fit surface to this particular set of data. Here are a few questions I have about multivariable regression:

We will call the 2 independent variables X and Y and the dependent variable Z.

(1) If I have 9 data points (combinations of 3 different X and Y values), what is the max order fit I can for the X and Y variable? Is it a second order by second order equation?

(2) How do you calculate the RMSE and Residal standard error of a nonlinear multivariate regression?

If anyone has any insight into this please let me know at your convenience.

Thanks,
Randall Berdon