Nonlinear Regression - How to choose and how to evaluate

I have a dataset with several variables. The dependent variable is income, and the most important independent is age. I wish to model the relation between income and age, while taking into account other covariates. I have two questions: 1. How do I choose the right nonlinear model? I ask about nonlinear for two reasons: Theoretically, the relation should be quadratic. In my data, it looks exponential (image attached). 2. The sample is large, so small P-Values are likely. How do I evaluate the correctness of the model? In a t-test, I can always calculate the effect size. What is the effect size here? Should I use predictions? How do I know if a prediction is good or not? Thank you !

• Just to clarify, do you have multiple income curves like the one you show or does this curve is your final object you try to model? In addition is this income curve samples densely and regularly? (Say, at 20+ equally spaced points) Aug 10 '16 at 13:48
• I do have multiple, but I did not mention it. I wanted an answer in the simple case, where there is only one. Let's say I have y, x1, x2, x3,x4, where x1 is age and y is income. In reality, I have 10 dataset like this, one for each year, but this is a different problem. Aug 10 '16 at 13:50
• A different problem warrants a different solution... Anyway, if you are interested in just a single curve, just follow Peter's advice (+1). It is a standard linear regression problem. If you feel really inclined you could also fit a spline or even use a local linear regression routine across time but that's probably an overkill for this task. Aug 10 '16 at 13:54

As to how you know about how good your model is - all the standard methods. You can plot predicted values vs. actual; you can examine $R^2$ and its variants; you can compare AIC and other similar methods across models; you can say how reasonable it is based on substantive concerns; etc.