One thing I teach is: when evaluating the performance of regression models you should not use correlation as your score.
This is because correlation tells you if a re-scaling of your result is useful, but you want to know if the result in your hand is in fact useful. For example: the Mars Climate Orbiter software issued thrust commands in pound-seconds units to an engine expecting the commands to be in newton-seconds units. The two quantities are related by a constant ratio of 1.4881639, and therefore anything measured in pound-seconds units will have a correlation of 1.0 with the same measurement in newton-seconds units. However, one is not the other and the difference is why the Mars Climate Orbiter “encountered Mars at a lower than anticipated altitude and disintegrated due to atmospheric stresses.”
The need for a convenient direct F-test without accidentally triggering the implicit re-scaling that is associated with calculating a correlation is one of the reasons we supply the sigr R library. However, even then things can become confusing.
Please read on for a nasty little example.
Consider the following “harmless data frame.”
d <- data.frame(prediction=c(0,0,-1,-2,0,0,-1,-2), actual=c(2,3,1,2,2,3,1,2))
The recommended test for checking the quality of “
prediction” related to “
actual” is an F-test (this is the test
stats::lm uses). We can directly run this test with
sigr (assuming we have installed the package) as follows:
F Test summary: (R2=-16, F(1,6)=-5.6, p=n.s.).
sigr reports an R-squared of -16 (please see here for some discussion of R-squared). This may be confusing, but it correctly communicates we have no model and in fact “
prediction” is worse than just using the average (a very traditional null-model).
cor.test appears to think “
prediction” is a usable model:
cor.test(d$prediction,d$actual) Pearson's product-moment correlation data: d$prediction and d$actual t = 1.1547, df = 6, p-value = 0.2921 alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval: -0.3977998 0.8697404 sample estimates: cor 0.4264014
This is all for a prediction where
sum((d$actual-d$prediction)^2)==66 which is larger than
sum((d$actual-mean(d$actual))^2)==4. We concentrate on effects measures (such as R-squared) as we can drive the p-values wherever we want just by adding more data rows. Our point is: you are worse off using this model than using the mean-value of the actual (2) as your constant predictor. To my mind that is not a good prediction. And
lm seems similarly excited about “
summary(lm(actual~prediction,data=d)) Call: lm(formula = actual ~ prediction, data = d) Residuals: Min 1Q Median 3Q Max -0.90909 -0.43182 0.09091 0.52273 0.72727 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 2.2727 0.3521 6.455 0.000655 *** prediction 0.3636 0.3149 1.155 0.292121 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.7385 on 6 degrees of freedom Multiple R-squared: 0.1818, Adjusted R-squared: 0.04545 F-statistic: 1.333 on 1 and 6 DF, p-value: 0.2921
One reason to not trust the
lm result is it didn’t score the quality of “
prediction“. It scored the quality of “
0.3636*prediction + 2.2727.” It can be the case that “
0.3636*prediction + 2.2727” is in fact a good predictor. But that doesn’t help us if it is “
prediction” we are showing to our boss or putting into production. We can try to mitigate this by insisting
lm try to stay closer to the original by turning off the intercept or offset with the “
0+” notation. That looks like the following.
summary(lm(actual~0+prediction,data=d)) Call: lm(formula = actual ~ 0 + prediction, data = d) Residuals: Min 1Q Median 3Q Max 0.00 0.00 1.00 2.25 3.00 Coefficients: Estimate Std. Error t value Pr(>|t|) prediction -1.0000 0.6094 -1.641 0.145 Residual standard error: 1.927 on 7 degrees of freedom Multiple R-squared: 0.2778, Adjusted R-squared: 0.1746 F-statistic: 2.692 on 1 and 7 DF, p-value: 0.1448
lm(0+)‘s adjusted prediction is bad as we see below:
d$lmPred <- predict(lm(actual~0+prediction,data=d)) sum((d$actual-d$lmPred)^2)  26
lm(0+) found a way to improve the prediction; but the improved prediction is still very bad (worse than using a well chosen constant). And it is hard to argue that “
-prediction” is the same model as “
sigr is fairly new code, so it is a bit bold saying it is right when it disagrees with the standard methods. However
sigr is right in this case. The standard methods are not so much wrong as different, for two reasons:
- They are answering different questions. The F-test is designed to check if the predictions in-hand are good or not; “
cor.test” and “
lm %>% summary” are designed to check if any re-scaling of the prediction is in fact good. These are different questions. Using “
cor.test” or “
lm %>% summary” to test the utility of a potential variable is a good idea. The reprocessing hidden in these tests is consistent with the later use of a variable in a model. Using them to score model results that are supposed be directly used is wrong.
- From the standard R code’s point of view it isn’t obvious what the right “null model” is. Remember our original point: the quality measures on
lm(0+)are designed to see how well
lm(0+)is working. This means the
lm(0+)scores the quality of its output (not its inputs) so it gets credit for flipping the sign on the prediction. Also it considers the natural null-model to be one it can form where there are no variable driven effects. Since there is no intercept or “dc-term” in these models (caused by the “
0+” notation) the grand average is not considered a plausible null-model as it isn’t in the concept space of the modeling situation the
lmwas presented with. Or from
R^2, the ‘fraction of variance explained by the model’,
R^2 = 1 – Sum(R[i]^2) / Sum((y[i]- y*)^2),
where y* is the mean of y[i] if there is an intercept and zero otherwise.
I admit, this is very confusing. But it corresponds to documentation, and makes sense from a modeling perspective. It is correct. The silent switching of null model from average to zero make sense in the context it is defined in. It does not make sense for testing our prediction, but that is just one more reason to use the proper F-test directly instead of trying to hack “
cor.test” or “
lm(0+) %>% summary” to calculate it for you.
And that is what
sigr is about: standard tests (using
R supplied implementations) with a slightly different calling convention to better document intent (which in our case is almost always measuring the quality of a model separate from model construction). It is a new library, so it doesn’t yet have the documentation needed to achieve its goal, but we will eventually get there.