I am running a simple linear regression $y\sim x$.
The $R^2$ is $0.002$, while the adjusted $R^2$ is $-0.001$.
I don't understand this. I had thought Adjusted $R^2$ can only be negative when we do multi-regression.
If there is only one variable, what is to be 'adjusted' in this case?
Does anyone know?
Best Answer
Adjusted $R^2$ is:
$${R}_\text{adj}^{2}={1-(1-R^{2}){n-1 \over n-p-1}}$$
where $p$ is the number of predictors (not counting the intercept) and $n$ is the number of observations.
This will be less than $0$ when
$$\frac{p}{n-1}>R^2\,.$$
$R^2$ can be as low as $0$, so this may happen any time $p>0$. This means that it can indeed happen with $p=1$.