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I just started to learn about the following statistical measures, r-squared and adjusted r-squared and was wondering why can't we use adjusted r-squared for every regression model considering the fact that it penalizes the model for useless variables, unlike the former. Is there any advantage of r-squared over adjusted r-squared in some conditions?

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Adjusted $R^2$ is the better model when you compare models that have a different amount of variables.

The logic behind it is, that $R^2$ always increases when the number of variables increases. Meaning that even if you add a useless variable to you model, your $R^2$ will still increase. To balance that out, you should always compare models with different number of independent variables with adjusted $R^2$.

Adjusted $R^2$ only increases if the new variable improves the model more than would be expected by chance.

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  • $\begingroup$ Yes. I know how both r2 and adjusted r2 are calculated. My question was: should I use adjusted r2 for every regression model, considering the fact that it is better than r2? $\endgroup$ – Ronith Jun 29 '18 at 18:39
  • $\begingroup$ It's not about being better or worse. When you compare models use adjusted R2. When you only look at one model report R2, as it is the not adjusted measure of how much variance is explained by your model. $\endgroup$ – LN_P Jun 29 '18 at 18:45
  • $\begingroup$ Considering I have only one model and I use R2, it won't give me an indication of the insignificant variables I am adding to my model, in such a case why should I prefer R2 over adjusted R2? $\endgroup$ – Ronith Jun 29 '18 at 18:47
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    $\begingroup$ Neither R2 nor adjusted R2 will give you information about the significance of your variables. you have to look at your model output. When you want to see if some of your variables are insignificant when adding or removing them, then again you are comparing two models. Then you should use adjusted R2, because you are comparing models with a different number of independent variables in it. But if you only have one model , adjusted R2 will not tell you anythign about significant variables, as you don't compare it to another model... $\endgroup$ – LN_P Jun 29 '18 at 19:24
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    $\begingroup$ My bad! I should've framed my question correctly. What I meant was "If I use R2, it does not account for useless variables whereas adjusted R2 does. So why shouldn't I use adjusted R2 even if I have one model?" $\endgroup$ – Ronith Jun 29 '18 at 20:31

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