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I have a model with predictor variables x1, x2, and x3. I have another model with predictor variable x1.

My understanding is that when you have multiple predictors, you use adjusted R-squared, but with only one predictor, you use x1.

So when I am trying to compare the goodness of fit of the two different models I mentioned above, should I be comparing R-squared or adjusted R-squared?

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    $\begingroup$ Depending on what the ultimate modelling goal is (descriptive, explanatory or predictive) the relevant comparison may not be $R^2_{adj}$ but rather AIC, BIC or yet some other criterion. There must be a role for $R^2_{adj}$, too, but it ( = the role) might be fairly small. Meanwhile, comparing $R^2$s is not a good idea, as the answer by @FelipeGerard indicates. $\endgroup$ – Richard Hardy Dec 2 '15 at 19:59
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In linear regression, $R^2$ can't decrease when you add a new regressor (another "x" variable), since the fit will be equal or better. It remains the same only if the new "x" is a linear combination of the previous ones you already had. But then is it better to have more variables? $R^2_{adj}$ takes this into account by penalizing models with more variables. In other words, the increase in $R^2$ (i.e. the improvement of the fitting) must be reasonably large for the inclusion of a new variable to cause an increase in $R^2_{adj}$. In conclusion, you should always use $R^2_{adj}$ when comparing models with a different number of regressors. In the case where your models have the same number, it doesn't really matter which one you use. So it is just better to use the adjusted one all the time.

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