# Why is AUC higher for a classifier that is less accurate than for one that is more accurate?

I have two classifiers

• A: naive Bayesian network
• B: tree (singly-connected) Bayesian network

In terms of accuracy and other measures, A performs comparatively worse than B. However, when I use the R packages ROCR and AUC to perform ROC analysis, it turns out that the AUC for A is higher than the AUC for B. Why is this happening?

The true positive (tp), false positive (fp), false negative (fn), true negative (tn), sensitivity (sen), specificity (spec), positive predictive value (ppv), negative predictive value (npv), and accuracy (acc) for A and B are as follows.

+------+---------+---------+
|      |    A    |    B    |
+------+---------+---------+
| tp   | 3601    | 769     |
| fp   | 0       | 0       |
| fn   | 6569    | 5918    |
| tn   | 15655   | 19138   |
| sens | 0.35408 | 0.11500 |
| spec | 1.00000 | 1.00000 |
| ppv  | 1.00000 | 1.00000 |
| npv  | 0.70442 | 0.76381 |
| acc  | 0.74563 | 0.77084 |
+------+---------+---------+


With the exception of sens and ties (spec and ppv) on the marginals (excluding tp, fn, fn, and tn), B seems to perform better than A.

When I compute the AUC for sens (y-axis) vs 1-spec (x-axis)

aucroc <- auc(roc(data$prediction,data$labels));


here is the AUC comparison.

+----------------+---------+---------+
|                |    A    |    B    |
+----------------+---------+---------+
| sens vs 1-spec | 0.77540 | 0.64590 |
| sens vs spec   | 0.70770 | 0.61000 |
+----------------+---------+---------+


So here are my questions:

• Why is the AUC for A better than B, when B "seems" to outperform A with respect to accuracy?
• So, how do I really judge / compare the classification performances of A and B? I mean, do I use the AUC value? Do I use the acc value, and if so why?
• Furthermore, when I apply proper scoring rules to A and B, B outperforms A in terms of log loss, quadratic loss, and spherical loss (p < 0.001). How do these weigh in on judging classification performance with respect to AUC?
• The ROC graph for A looks very smooth (it is a curved arc), but the ROC graph for B looks like a set of connected lines. Why is this?

As requested, here are the plots for model A.

Here are the plots for model B.

Here are the histogram plots of the distribution of the probabilities for A and B. (breaks are set to 20).

Here is the scatter plot of the probabilities of B vs A.

• Your tables don't make sense: how did you choose the point at which you compute those performance values? – Calimo Mar 20 '14 at 7:32
• Remember AUC measures the performance over all possible thresholds. It would help (you as well) if you could show the curves (ideally on the same plot). – Calimo Mar 20 '14 at 7:32
• @Calimo sorry, i forgot to include that information, but the threshold used to create that confusion matrix was 50%. – Jane Wayne Mar 20 '14 at 17:02
• You mean 0.5? The predicted values of A and B look clearly different, and if you haven't got the hint yet, you should definitely plot the histograms side by side... – Calimo Mar 20 '14 at 17:37
• @Calimo could you please clarify, the histograms of what side-by-side? – Jane Wayne Mar 20 '14 at 17:49

Improper scoring rules such as proportion classified correctly, sensitivity, and specificity are not only arbitrary (in choice of threshold) but are improper, i.e., they have the property that maximizing them leads to a bogus model, inaccurate predictions, and selecting the wrong features. It is good that they disagree with proper scoring (log-likelihood; logarithmic scoring rule; Brier score) rules and the $c$-index (a semi-proper scoring rule - area under ROC curve; concordance probability; Wilcoxon statistic; Somers' $D_{xy}$ rank correlation coefficient); this gives us more confidence in proper scoring rules.

• I wish I had a good reference for that, but briefly any measure based solely on ranks such as $c$ (AUROC) cannot give enough credit to extreme predictions that are "correct". Brier, and even more so the logarithmic scoring rule (log likelihood) give such credit. This is also an explanation why comparing two $c$-indexes is not competitive with other approaches power-wise. – Frank Harrell Mar 20 '14 at 16:35
• @alto, 0.5 is extremely arbitrary, coinciding with a most unusual utility/loss/cost function in which two kinds of errors are equally bad. This is seldom the case. Thinking probabilistically, which is the way I happen believe the way nature works, there is no such thing as a "mistake", but rather a degree of badness of a risk prediction. For example, predicting a probability of 0.6 then observing an event is worse than predicting a probability of 0.9 and then observing an event. But neither prediction is "wrong". You can use probability accuracy scores that require no thresholds. – Frank Harrell Mar 20 '14 at 22:20
• An unbounded model such as the logistic does not lead to any more overfitting than any other approach. The logistic transformation ensures that probability estimates are well behaved. The only downside to a logarithmic scoring rule is if you predict a probability extremely close to 0 or 1 and you are "wrong". It is true that one ultimately makes a decision but it does not follow at all that the analyst should make the decision by using a threshold. The decision should be deferred to the decision maker. Nate Silver's book Signal and Noise documents great benefits of probabilistic thinking. – Frank Harrell Mar 21 '14 at 16:43
• @FrankHarrell, it is frustrating that you keep misconstruing my opinion. I never advocated a black box approach. I simply think your statement "x is useless, only use y" is too strong. – alto Mar 22 '14 at 14:51
• @alto that is perceptive. I think that real-time pattern recognition does not have time for utilities. This is not the world I work in. But still there are cases in real time where you would rather have a black box tell you "uncertain" than force a choice between "that is a tank coming at you" vs. "that is a passenger car". – Frank Harrell Mar 24 '14 at 18:47
1. Why is the AUC for A better than B, when B "seems" to outperform A with respect to accuracy?

Accuracy is computed at the threshold value of 0.5. While AUC is computed by adding all the "accuracies" computed for all the possible threshold values. ROC can be seen as an average (expected value) of those accuracies when are computed for all threshold values.

2. So, how do i really judge/compare the classification performances of A and B? I mean, do i use the AUC value? do i use the acc value? and why?

It depends. ROC curves tells you something about how well your model your model separates the two classes, no matter where the threshold value is. Accuracy is a measure which works well usually when classes keeps the same balance on train and test sets, and when scores are really probabilities. ROC gives you more hints on how model will behave if this assumption is violated (however is only an idea).

3. furthermore, when i apply proper scoring rules to A and B, B outperforms A in terms of log loss, quadratic loss, and spherical loss (p < 0.001). how do these weigh in on judging classification performance with respect to AUC?

I do not know. You have to understand better what you data is about. What each model is capable to understand from your data. And decide later which is the best compromise. The reason why that happens is that there is no universal metric about a classifier performance.

4. The ROC graph for A looks very smooth (it is a curved arc), but the ROC graph for B looks like a set of connected lines. why is this?

That is probably because the bayesian model gives you smooth transitions between those two classes. That is translated in many threshold values. Which means many points on ROC curve. The second model probably produce less values due to prediction with the same value on bigger regions of the input space. Basically, also the first ROC curve is made by lines, the only difference is that there are so many adjacent small lines, that you see it as a curve.

• Accuracy can be computed at threshold values other than 0.5. – Calimo Mar 20 '14 at 10:02
• Of course you are right. That is why I used "accuracies" in the next proposition. However, when one talks about accuracy, without other context information, the best guess for the threshold value is 0.5. – rapaio Mar 20 '14 at 10:23
• It is easy to see how arbitrary such a process is. Few estimators in statistics that require binning or arbitrary choices have survived without heavy criticism. And I would never call proportion classified correct as "accuracy". – Frank Harrell Mar 20 '14 at 16:36
• @unreasonablelearner you are right on your assumption.. the confusion matrix above was computed at the threshold 0.5. is there any advantage to a different threshold? – Jane Wayne Mar 20 '14 at 16:46
• @JaneWayne The formula is indeed for the proportion of classified correct. Accuracy is the most often used term for this. However accuracy means a lot more, and in the light of what Frank Harrell said, I think now that accuracy is by far not the best term for that. Now I think that its usage might harm, even if it is popular. This is how I was wrong. – rapaio Mar 20 '14 at 17:40

Why is the AUC for A better than B, when B "seems" to outperform A with respect to accuracy?

First, although the cut-off (0.5) is the same, it is not comparable at all between A and B. In fact, it looks pretty different from your histograms! Look at B: all your predictions are < 0.5.

Second, why is B so accurate? Because of class imbalance. In test B, you have 19138 negative examples, and 6687 positives (why the numbers are different in A is unclear to me: missing values maybe?). This means that by simply saying that everything is negative, I can already achieve a pretty good accuracy: precisely 19138 / (19138 + 6687) = 74%. Note that this requires absolutely no knowledge at all beyond the fact that there is an imbalance between the classes: even the dumbest model can do that!

And this is exactly what test B does at the 0.5 threshold... you get (nearly) only negative predictions.

A is more of a mixed bag with. Although it has a slightly lower accuracy, note that its sensitivity is much higher at this cut-off...

Finally, you cannot compare the accuracy (a performance at one threshold) with the AUC (an average performance on all possible thresholds). As these metrics measure different things, it is not surprising that they are different.

So, how do I really judge/compare the classification performances of A and B? i mean, do i use the AUC value? do i use the acc value? and why?

Furthermore, when I apply proper scoring rules to A and B, B outperforms A in terms of log loss, quadratic loss, and spherical loss (p < 0.001). How do these weigh in on judging classification performance with respect to AUC?

You have to think: what is it you really want to do? What is important? Ultimately, only you can answer this question based on your knowledge of the question. Maybe AUC makes sense (it rarely really does when you really think about it, except when you don't want to make a decision youself but let others do so - that's most likely if you are making a tool for others to use), maybe the accuracy (if you need a binary, go-no go answer), but maybe at different thresholds, maybe some other more continuous measures, maybe one of the measures suggested by Frank Harrell... as already stated, there is no universal question here.

The ROC graph for A looks very smooth (it is a curved arc), but the ROC graph for B looks like a set of connected lines. Why is this?

Back to the predictions that you showed on the histograms. A gives you a continuous, or nearly-continuous prediction. To the contrary, B returns mostly only a few different values (as you can see by the "spiky" histogram).

In a ROC curve, each point correspond to a threshold. In A, you have a lot of thresholds (because the predictions are continuous), so the curve is smooth. In B, you have only a few thresholds, so the curve looks "jumps" from a SN/SP to an other.

You see vertical jumps when the sensitivity only changes (the threshold makes differences only for positive cases), horizontal jumps when the specificity only changes (the threshold makes differences only for negative examples), and diagonal jumps when the change of threshold affects both classes.