# Useful properties of ROC curves, AUC scoring, and Gini Coefficients

Receiver Operating Characteristic (ROC) curves and AUC values are often used to score binary classification models in Kaggle and in papers. However, for a long time I found them fairly unintuitive and confusing. In this blog post, I will explain some basic properties of ROC curves that are useful to know for Kaggle competitions, and how you should interpret them.

Above: Example of a ROC curve

First, the definitions. A ROC curve plots the performance of a binary classifier under various threshold settings; this is measured by true positive rate and false positive rate. If your classifier predicts “true” more often, it will have more true positives (good) but also more false positives (bad). If your classifier is more conservative, predicting “true” less often, it will have fewer false positives but fewer true positives as well. The ROC curve is a graphical representation of this tradeoff.

A perfect classifier has a 100% true positive rate and 0% false positive rate, so its ROC curve passes through the upper left corner of the square. A completely random classifier (ie: predicting “true” with probability p and “false” with probability 1-p for all inputs) will by random chance correctly classify proportion p of the actual true values and incorrectly classify proportion p of the false values, so its true and false positive rates are both p. Therefore, a completely random classifier’s ROC curve is a straight line through the diagonal of the plot.

The AUC (Area Under Curve) is the area enclosed by the ROC curve. A perfect classifier has AUC = 1 and a completely random classifier has AUC = 0.5. Usually, your model will score somewhere in between. The range of possible AUC values is [0, 1]. However, if your AUC is below 0.5, that means you can invert all the outputs of your classifier and get a better score, so you did something wrong.

The Gini Coefficient is 2*AUC – 1, and its purpose is to normalize the AUC so that a random classifier scores 0, and a perfect classifier scores 1. The range of possible Gini coefficient scores is [-1, 1]. If you search for “Gini Coefficient” on Google, you will find a closely related concept from economics that measures wealth inequality within a country.

Why do we care about AUC, why not just score by percentage accuracy?

AUC is good for classification problems with a class imbalance. Suppose the task is to detect dementia from speech, and 99% of people don’t have dementia and only 1% do. Then you can submit a classifier that always outputs “no dementia”, and that would achieve 99% accuracy. It would seem like your 99% accurate classifier is pretty good, when in fact it is completely useless. Using AUC scoring, your classifier would score 0.5.

In many classification problems, the cost of a false positive is different from the cost of a false negative. For example, it is worse to falsely imprison an innocent person than to let a guilty criminal get away, which is why our justice system assumes you’re innocent until proven guilty, and not the other way around. In a classification system, we would use a threshold rule, where everything above a certain probability is treated as 1, and everything below is treated as 0. However, deciding on where to draw the line requires weighing the cost of a false positive versus a false negative — this depends on external factors and has nothing to do with the classification problem.

AUC scoring lets us evaluate models independently of the threshold. This is why AUC is so popular in Kaggle: it enables competitors to focus on developing a good classifier without worrying about choosing the threshold, and let the organizers choose the threshold later.

(Note: This isn’t quite true — a classifier can sometimes be better at certain thresholds and worse at other thresholds. Sometimes it’s necessary to combine classifiers to get the best one for a particular threshold. Details in the paper linked at the end of this post.)

Next, here’s a mix of useful properties to know when working with ROC curves and AUC scoring.

AUC is not directly comparable to accuracy, precision, recall, or F1-score. If your model is achieving 0.65 AUC, it’s incorrect to interpret that as “65% accurate”. The reason is that AUC exists independently of a threshold and is immune to class imbalance, whereas accuracy / precision / recall / F1-score do require you picking a threshold, so you’re measuring two different things.

Only relative order matters for AUC score. When computing ROC AUC, we predict a probability for each data point, sort the points by predicted probability, and evaluate how close is it from a perfect ordering of the points. Therefore, AUC is invariant under scaling, or any transformation that preserves relative order. For example, predicting [0.03, 0.99, 0.05, 0.06] is the same as predicting [0.15, 0.92, 0.89, 0.91] because the relative ordering for the 4 items is the same in both cases.

A corollary of this is we can’t treat outputs of an AUC-optimized model as the likelihood that it’s true. Some models may be poorly calibrated (eg: its output is always between 0.3 and 0.32) but still achieve a good AUC score because its relative ordering is correct. This is something to look out for when blending together predictions of different models.

That’s my summary of the most important properties to know about ROC curves. There’s more that I haven’t talked about, like how to compute AUC score. If you’d like to learn more, I’d recommend reading “An introduction to ROC analysis” by Tom Fawcett.