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what is a good adjusted R square value for any multiple regression model or it depends? If it depends then what are the conditions?

Arvinder
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At the least, there is no universally $\textit{good}$ $R_{a}^2$ value ; it can vary greatly depending on what type of data you are modeling. In certain social science fields one might find that an $R_{a}^2$ as low as 0.300 or so will suffice. On the other hand, if you are physicist testing a model or mathematical equation derived from physical laws, you are probably looking for something on the order of 0.900 or higher, where the remainder would account for measurement error or the like (I'm not a physicist but this is what I've heard from those in natural science fields).

Any undergraduate regression textbook worth its salt will surely contain a more in-depth examination of the issues related to using $R_{a}^2$ as a measure of goodness of fit, such as multicollinearity, etc...; or you could read the Wikipedia page. In short, $R_{a}^2$ will likely be one of the first things you glance at when you check the output of your model summary, but there are many other (arguably more robust) criteria you should be checking after this.

nrussell
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  • Did you notice that the question concerns the *adjusted* $R^2$? These values eventually *decrease* as new variables are entered into the model. – whuber Jul 01 '14 at 15:07
  • @whuber Wow I completely missed that; I edited the post and left what should still be relevant for $R_{a}^2$. Thank you. – nrussell Jul 01 '14 at 15:14
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Note that $R^2$ and $R^2_a$ can be highly influenced by the range and structure of the predictor variables. This is one reason that many people are placing less importance on this measure (it can still be useful for a quick idea, but using it as a cutoff for decisions is probably not a good idea).

You can see this by simulating a dataset with a reasonable relationship and good sample size and computing $R^2_a$, then drop the half of the points that are closes to the mean(s) of the predictor variable(s) and recomputed $R^2_a$. I did a quick simulation with $n=100$ and 1 predictor variable. Removing the central 50 data points (I just kept 25 on each end) increased $R^2_a$ from 0.51 to 0.70.

The F-statistic for testing for an overall relationship can be computed from $R^2$ and the degrees of freedom and therefore (if assumptions are reasonable) the statistical significance can be computed. $R^2_a$ would be similar. So if "good" means statistically significant then you could work out what is good based on your d.f. and significance level, but you could then also look at the p-value on the computer output to see the same information.

Greg Snow
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    This interesting answer, which seems more like a comment to the question, might be more appropriate at [Is R2 useful or dangerous?](http://stats.stackexchange.com/questions/13314). – whuber Jul 01 '14 at 15:42