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R squared, also known as coefficient of determination, is a popular measure of quality of fit in regression. However, it does not offer any significant insights into how well our regression model can predict future values. Instead, the PRESS statistic (the predicted residual sum of squares) can be used as a measure of predictive power. The PRESS statistic can be computed in the leave-one-out cross validation process, by adding the square of the residuals for the case that is left out. As a reminder, in the leave-one-out cross validation, one case of the data set is used as the testing set and the remaining are used as the testing set. We iterate this process, until all cases have served as the testing set.
Here is an example implemented in R, on the gala dataset in the faraway package:
> gala[1:3,]
Species Endemics Area Elevation Nearest Scruz Adjacent
Baltra 58 23 25.09 346 0.6 0.6 1.84
Bartolome 31 21 1.24 109 0.6 26.3 572.33
Caldwell 3 3 0.21 114 2.8 58.7 0.78
Model1:
>model1<-lm(Species~Endemics+Area+Elevation)
>summary(model1)
....
Residual standard error: 27.29 on 26 degrees of freedom
Multiple R-squared: 0.9492, Adjusted R-squared: 0.9433
F-statistic: 161.8 on 3 and 26 DF, p-value: < 2.2e-16
Model2:
> model2<-lm(Species~I(Endemics^2))
> summary(model2)
...
Residual standard error: 27.1 on 28 degrees of freedom
Multiple R-squared: 0.946, Adjusted R-squared: 0.9441
F-statistic: 491 on 1 and 28 DF, p-value: < 2.2e-16
Model3:
> model3<-lm(Species~Endemics+I(Endemics^2))
> summary(model3)
.....
Residual standard error: 22.94 on 27 degrees of freedom
Multiple R-squared: 0.9627, Adjusted R-squared: 0.9599
F-statistic: 348.5 on 2 and 27 DF, p-value: < 2.2e-16
Here are now the AIC (Akaike test criterion), BIC (Bayesian information criterion), and PRESS statistic of the three models:
Model 1:
>AIC(model1)
289.243
> BIC(model1)
296.249
PRESS(model1)=259520.5
Model 2:
> AIC(model2)
287.0325
> BIC(model2)
291.2361
PRESS(model2)=26382.22
Model 3:
> AIC(model3)
277.9558
> BIC(model3)
283.5606
PRESS(model3)=22567.03
As we can see, the PRESS statistic is significantly smaller (better) for models 2 and 3, while R squared has a trivial improvement for model 3. So, according to PRESS, model 3 has the highest predictive power. It is interesting to note that the AIC and BIC also get their best values for model 3.
If you are interested in how I computed the PRESS statistic doing cross-validation in R, please check my next blog post.
Comment
This is an amazing post. Thanks so much. R-Squared discussions tend to launch many bar fights.
The ability to predict the future performance, rather than goodness of fit on existing data, is a great advantage. This can be achieved using cross-validation, which your method does in some way, through the leaving-one-out procedure. It would be nice to see a metric that simultaneously addresses
One of our readers wrote:
Vincent, to normalize Rsquared, use Fisher Transform and then apply the T test to the results. It takes care of the data variability and the data size. Outliers are a problem, but they will mess up the quality of the least squares model, anyway, regardless of the criteria by which you judge the quality of your model. if you don't want to worry about them, use quantile regression.
Great reading for statisticians and data scientists. R^2 has many flaws: it is sensitive to outliers and size-sensitive:an R^2 of 0.65 does not have the same meaning for a data set with 20 observations, than for a dataset with 10,000 observations. How do you normalize this?
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