Is R square useful at all?


Omega Contributor
That is a nice link. The take home message is as long as you know the limitations and interpretations of R^2, it is a fine parameter. And yes, if doesn't tell you about goodness of fit or if the model is valid, but it is not trying to do that, people are trying to do that. An example is that you can easily overfit a model and get a very high R^2, but in some circumstances you want a model that is generalizable, so the overfit model does not work as a great metric as well as say using MSE and cross-validation.


Fortran must die
The take home message is as long as you know the limitations and interpretations of R^2, it is a fine parameter.
I didn't get that sense from the article at all :p To me they were saying it was pretty much useless. The author they are citing argues that R square does not really tell you about explained variance in Y given X, it just as logically tells you about the explained variance in X given Y [the results are the same]. That is very surprising to me. I assume you have to know which way the influence is going, and accept the influence does not go both ways, for the statistic to work.

What is a good measure of fit for linear regression? I know what they are for logistic regression not linear regression. The truth is I don't pay much attention to overall model fit when I run models, I focus on the effect size of variables as long as the various model indicators show the model meets the minimum requirement. :p Of which r square is not one.


TS Contributor
I never use R-squared by itself. At a minimum, use R-squared (adjusted), or even better, use R-squared (predicted). See The Minitab Blog. I have seen regression analyses with R-squared and R-squared (adjusted) in the 80s, with an R-squared (predicted) = 0.


Fortran must die
Given that I rarely use R square as much as I do I use adjusted R square. R square predicted is something I have not heard of before.

I find AIC much more useful for what I do than R square. Beyond the limitations mentioned above it is rarely clear what a "good" r square should be. You can look in the literature, if you have access to it which I commonly don't, and if it exists, which commonly it does not in my field. But other than that too many things can influence it to know whether what you found or not is a good result.

My own theory, I have not seen this addressed, is that for complex phenomenon you will have lower R square than less complex simply because so many things can influence the results with complex realities. So you will be less likely to have useful variables in the model.


TS Contributor
For me, it boils down to the Prediction limits. It doesn't matter if R-squared (adjusted) is 98% if the prediction limits are extremely wide. If R-squared is lower than I am comfortable with, I can compared the standard deviation of the model to the measurement variation. If the former is larger than the measurement variation, I suspect a missing factor. However, I do have the luxury of being able to quantify my measurement variation. I know this is difficult to impossible in many disciplines.


Fortran must die
What is strange to me is that most of the regression I see is not tied to prediction or relative impact. It is tied to determining whether some factor is or is not statistically valid. I think this because most analysis I see is academic and prediction is commonly not what they are interested in. Theory building is, or discovering some variable is important which is commonly the same thing. Unfortunately practitioner results are not easy to find (or I have not found them anyway)?

What is a good way to measure fit in a linear regression (or a non-linear regression) model?

I explain the importance of confidence intervals over effect size by saying I can predict something as five plus or minus a million with great confidence :)
Yes, it is pretty meaningless. But you know that I have been saying that for some time.

The usefulness of it is that you can brag about it for your friends - if it is big (especially males are interested in this). If it is small, don't talk about it.
You use the delta to talk about percent additional explained variance in the outcome. Is this wrong?
No, it is not wrong. It is just an other way or rewriting the t-test (or F-test) for the extra included parameter. So it doesn't add any information.

But is the delta-R^2 more meaningful? It can be high or low depending on how it is with the multicolinearity, even if the model beta parameters are the same (in two hypothetical) layout. Remember that the multicolinearity is a problem in the sample, not the population.

The R^2 is quite meaningless. But what influences it is not meaningless - the standard deviation in the residuals and the "spread" in the X-values. They matter.

By the way, it is not strange that you will get the same R^2 with regression y on x as with x on y, since R^2 is the correlation (r) squared and:

r = Cov(x,y)/(sd(x)*sd(y))

where Cov() is the covariance and sd() is the standard deviation. They stay the same.


TS Contributor
I think the biggest problem with R-sq(adj) is that it leads to overfitting. It can easily happen that one gets a fine R-sq value on the training set and abysmal performance on a test-set . The R-sq(pred) is alleviating this somewhat, but it still has a pretty weak link to the prediction performance imho.

IIRC BIC has a proven link to the test set performance, so, theoretically it would be a better measure (so, AIC also?)

AIC, BIC, log likliehood, etc all make better fit measure than R-squared. What r-square (or its derivatives) is useful for is determining simple effect size. In which case it does exactly what it claims to do. Now, does that mean that you specified your model correctly? No. Its just a useful piece of evidence along with fit measures, p-values, etc.

I really wonder who is publishing without reporting the whole suite of statistics: p-values, effect sizes, fit indices, random effect correlations, confidence intervals, etc. What are these journals and are they indexed?