I have two models, one lm(y ~ x1 + x2 + 0)
which gives me a close to 0.90 something $R^2$ and another model lm(y ~ x1 + x2)
which gives me a close to 0.0003 ish $R^2$. Based on $R^2$ one might choose the first model. I have domain (prior) knowledge that a 0 value of $x_1$ and a 0 value of $x_2$ should produce a 0 value of $y$. So, is my explicit setting of 0 a fair thing? Based on this knowledge and the obtained value of $R^2$ can I go with the first model?
Edit: From the threads Removal of statistically significant intercept term increases $R^2$ in linear model and When is it ok to remove the intercept in a linear regression model? I have learned that it is okay to do it as long as the domain knowledge above holds true. From Is $R^2$ useful or dangerous? I learned that $R^2$ alone is insufficient to make a decision about how good the model is. I need to look at AIC and BIC to make further decisions. However, because I my background is not in statistics, I am having trouble understanding the following in the first link:
- Why does the first equality occur only when intercept is included?
- why does $\bar{y}$ become 0 when intercept disappears?
- Why in the second case, $R_0^2$ uses a reference model corresponding to noise only?
- What is the $\bar{y_1}$ in the notation?
- Why is $\|\ y \|_2^2 = \| y - \bar{y_1}\|_2^2 + n \bar y^2$ true?