# How to choose an error metric when evaluating a classifier?

I've seen different error metrics used in the Kaggle competitions: RMS, mean-square, AUC, amongst others. What's the general rule of thumb on choosing an error metric, i.e. how do you know which error metric to use for a given problem? Are there any guidelines?

The pool of error metrics you can choose from is different between classification and regression. In the latter you try to predict one continuous value, and with classification you predict discrete classes such as "healthy" or "not healthy". From the examples you mentioned, root mean square error would be applicable for regression and AUC for classification with two classes.

Let me give you a little bit more detail on classification. You mentioned AUC as a measure, which is the area under the ROC curve, which usually is only applied to binary classification problems with two classes. Although, there are ways to construct a ROC curve for more than two classes, they loose the simplicity of the ROC curve for two classes. In addition, ROC curves can only be constructed if the classifier of choice outputs some kind of score associated with each prediction. For instance, logistic regression will give you probabilities for each of the two classes. In addition to their simplicity ROC curves have the advantage that they are not affected by the ratio between positively and negatively labelled instances in your datasets and don't force you to choice a threshold. Nevertheless, it is recommended to not only look at the ROC curve alone but other visualizations as well. I'd recommend having a look at precision-recall curves and cost-curves. There is not one true error measurement, they all have their strength and weaknesses.

Literature I found helpful in this regard are:

If your classifier does not provide some kind of score, you have to fall back to the basic measures that can be obtained from a confusion matrix containing the number of true positives, false positives, true negatives and false negatives. The visualizations mentioned above (ROC, precision-recall, cost curve) are all based on these tables obtained by using a different threshold of the classifier's score. The most popular measure in this case is probably the F1-Measure. In addition, there is a long list of measurements you can retrieve from a confusion matrix: sensitivity, specificity, positive predictive value, negative predictive value, accuracy, Matthews correlation coefficient, … Similar to ROC curves, confusion matrices are very easy to understand in the binary classification problem, but get more complicated with multiple classes, because for $N$ classes you have to consider either a single $N \times N$ table or $N$ $2 \times 2$ tables each of them comparing one of the classes ($A$) against all other classes (not $A$).

• The last sentence is wrong: confusion tables for $N$ classes are usually of size $N \times N$. From that you can derive $N$ $2\times2$ tables for "class a" vs. "not-class-a", but that is not so common although sometimes done implicitly (e.g. specificity calculations). – cbeleites Aug 13 '12 at 15:30
• Thanks a lot for pointing out this mistake, I corrected it in the answer above. – sebp Aug 13 '12 at 18:31

• most classifiers do in fact have an intermediate continuous score, on which usually a threshold for assigning hard classes (below t: class a, above: class b) is applied. Varying this threshold yields the ROC.
• In general, it is not a good idea to compress such a curve into one number. see e.g. The Case Against Accuracy Estimation for Comparing Induction Algorithms
There are lots of different ROC that have the same AUC, and the usefulness may vary widely for a given application.
• the other way round: the choice of the threshold may be pretty much determined by the application you have.
• You don't need to look at the classifier performance outside these boundaries and if you choose one metric, that should at least summarize only the relevant range of the bounded other metrics.
• depending on your study design, overall fraction of correctly or misclassified samples may be an appropriate summary or not, and the conclusions you can draw from that will also depend on the study design: Does your test data reflect the prior probabilities (prevalence) of the classes? For the population that your classifier is supposed to be used on? Was it collected in a stratified manner? This is closely connected to the fact that most users of a classifier are more interested in the predictive values, but sensitivity and specificity are much more easy to measure.

• You ask about general guidelines. One general guideline is that you need to know

• what kind of performance you need (sensitivity, specificity, predictive values etc. answer specific questions about the behaviour of your classifier, see what I wrote here).
• What acceptable working ranges for these performance characteristice for your application?.
These can vary widely: you may be willing to accept quite some false-negatives in spam detection, but that would not be an acceptable set-up for HIV diagnosis...

I think you won't be able to find a useful metric unless you can answer these questions.

It's a little bit like there's no free lunch in classifier validation, either.

Expected misclassification error rate is the method I have used and seen most often. The AUC of the ROC is a measure of a set of classification rules. If the idea is to compare a specific classifier with another then the AUC is not appropriate. Some form of classification error makes the most sense as it represents most directly the performance of the classification rule.

Much work has gone into finding good estimates of the classification error rate because of the large bias of the resubstitution estimate and the high variance of leave-one-out. Bootstrap and smooth estimators have been conisdered. See for example Efron's paper in JASA 1983 about bootstrap improvements over cross validation.

Here is a 1995 Stanford University technical report by Efron and Tibshirami summing up the literature including some of my own work.

• Comparing the performance of two classifiers on the same dataset is another topic to argue about. Especially, in the case of ROC and AUC there are a couple of methods to compare either ROC curves as a whole or the AUC estimates. Those are essentially statistical tests with the null hypothesis that the ROC/AUC does not differ. Cross-validation vs. bootstrap is another interesting topic, I recently saw a paper (dx.doi.org/10.1016/j.csda.2010.03.004) about that. I guess if you consider all the aspects at once, it can get pretty intimidating. – sebp Aug 13 '12 at 11:19