adj. r-square

I got a negative adj. r-square. Since the effect size (Adj R-sq=-.016) was low, did it indicate that a time in treatment (the independent variable) explained -1.6% of the variance in anger responses (the dependent variable)?


Ambassador to the humans
Adjusted R-square doesn't have the same interpretation as R-square itself - you can't talk about it in terms of % of variation explained by the model.


Fortran must die
Adjusted R-square doesn't have the same interpretation as R-square itself - you can't talk about it in terms of % of variation explained by the model.
Really? I know the formula is different, but in terms of substantive intepretation all the books and articles I have seen it in treat it exactly as they do r squared (that is percent of variation explained - but penalizing for added variables).

*sigh* another assumption I learned in class wrong....
Thanks for your response. What does a low adjusted R-square indicate? Does a low adjusted R-square reflect that this finding may have limited practical significance?


TS Contributor
If your model is parsimonious, R^2 and R^2(adj.) should be fairly close in agreement with R^2(adj) being slightly lower. If your model contains extra terms that are not significant, the R^2(adj) formula penalizes you for the extra terms more than they contribute by explaining variation. And R^2(pred) guards against overfitting your model.


TS Contributor
Effectively if adjusted R square is below 0 it has no explanatory value.
Just to get semantic: it is possible to get a very low adj-R2 despite high explanatory value. Lets dissertate the following example:

We know that the variation in genes explains the variation in length to a great deal, lets say 95 percent. The rest is explained by various environmental factors. Now, the marginal effect of environmental factors on length is very low according to R2 and adj-R2, but you can still influence the length to a great deal by altering envionmental factors. For example, you can cut off your legs and voila, you're only 84 centimeters tall. Thus the variation in environmental factors does not affect the variation of length to a great deal (since not so many cut off their legs), but environmental factors can explain length to a great deal.

Please let me now if my logic contains any flaws =)


Fortran must die
R square gets at what explains reality as it actually occurs not what could occur in theory. Thus you could cut off the legs and that would have signficant impact, but this occurs rarely in fact so these environmental factors have limited explanatory power (and thus little impact on r squared) as you noted. If cutting off legs, or other environmental factors, actually influenced length in practice they would have a high r square value.

You have to consider what the purpose of a statistic is in using it. Rsquare looks at what explains variance in actual reality. Not the theoretical impact on variation something could have, if it does not occur often enough in fact to matter.