# Probabilistic interpretation of AUC

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Unfortunately this was not taught in any of my statistics or data analysis classes at university (wtf it so needs to be :scream_cat:). So it took me some until I learned that the AUC has a nice probabilistic meaning.

## What’s AUC anyway?

Consider:

- A dataset : , where
- is a vector of features collected for the th subject,
- is the th subject’s label (binary outcome variable of interest, like a disease status, class membership, or whatever binary label).

- A classification algorithm (like logistic regression, SVM, deep neural net, or whatever you like), trained on , that assigns a score (or probability) to any new observation signifying how likely its label is .

Then:

- A
*decision threshold*(or*operating point*) can be chosen to assign a class label ( or ) to based on the value of . The chosen threshold determines the balance between how many*false positives*and*false negatives*will result from this classification. - Plotting the
*true positive rate*(TPR) against the*false positive rate*(FPR)*as the operating point changes from its minimum to its maximum value*yields the*receiver operating characteristic (ROC) curve*. Check the confusion matrix if you are not sure what TPR and FPR refer to. - The area under the ROC curve, or AUC, is used as a measure of classifier performance.

Here is some R code for clarification (not even using `tidyverse`

:stuck_out_tongue:):

# load some data, fit a logistic regression classifier data(iris) versicolor_virginica <- iris[iris$Species != "setosa", ] logistic_reg_fit <- glm(Species ~ Sepal.Width + Sepal.Length, data = versicolor_virginica, family = "binomial") y <- ifelse(versicolor_virginica$Species == "versicolor", 0, 1) y_pred <- logistic_reg_fit$fitted.values # get TPR and FPR at different values of the decision threshold threshold <- seq(0, 1, length = 100) FPR <- sapply(threshold, function(thresh) { sum(y_pred >= thresh & y != 1) / sum(y != 1) }) TPR <- sapply(threshold, function(thresh) { sum(y_pred >= thresh & y == 1) / sum(y == 1) }) # plot an ROC curve plot(FPR, TPR) lines(FPR, TPR)

A rather ugly ROC curve emerges:

The area under the ROC curve, or AUC, seem like a nice heuristic to evaluate and compare the overall performance of classification models independent of the exact decision threshold chosen. But there’s more to it.

## Probabilistic interpretation

As above, assume that we are looking at a dataset where we want to distinguish data points of *type 0* from those of *type 1*. Consider a classification algorithm that assigns to a random observation a score (or probability) signifying membership in *class 1*. If the final classification between *class 1* and *class 0* is determined by a decision threshold , then the *true positive rate* (a.k.a. *sensitivity* or *recall*) can be written as a conditional probability