Comparison of two simulated curves for given experimental data Physicist here with a naive question,
I've got a data set (points $x_i,y(x_i),i\in \mathbb N$) and two different smooth curves $f(x),g(x)$ from my simulation. I can't quite judge with my eye which one fits better with my eyes only. What are some tools to make a judgement here?
My initial thought is to compute the quantities 
$$\sum_i(f(x_i)-y(x_i))^2,$$
$$\sum_i(g(x_i)-y(x_i))^2,$$
which comes from my familarity with least square fits. Here the situation is that I'm already given all data and curves.

What are standard ways to do this? What are generally interesting quantities which let me judge the simulated curves?

 A: If both methods have the same number of parameters, then pick the one that comes closest as assessed by the smallest sum-of-squares of the residuals. 
If the two models fit different numbers of parameters, then you need to look at more than the sum-of-squares of the residuals.  The equation with more parameters will almost always fit better, so you've got to ask if the improvement in sum-of-squares is large enough given the number of extra parameters. 
There are two approaches one can use to compare the fits of two models, accounting for the fact that you'll almost always get better fit if you fit more parameters:


*

*The extra sum of squares F test (it doesn't have a standard name). This will compute a P value testing the null hypothesis that the
model with fewer parameters is correct.

*Use AIC -Aikaike's Information Criteria - (or BIC, Bayesian Information Criteria) which are information criteria. These methods compute the relative likelihood that each model is correct (given that those are the only choices).
I've written about both approaches, without much math for biologists. 
A: Your intuition is correct, in that what you propose is a sensible measure for comparing two different models provided that


*

*the models have the same number of parameters, and

*that the errors are the same from data point to data point. 


You probably want to look into using the $\chi^2$ per degree of freedom for comparing different models.  The first hit on a google search yielded a reasonable tutorial (it even has physics content).
