Binary classifier high overall ROC AUC but low in different bins

Question

I'm trying to analyze the performance of a binary classifier on the test set on different ranges of the predictions. the classifier has a .97 ROC AUC on the test. Then I binarize the test set predictions into bins to check the ROC AUC on every bucket but in the bins, it has very low performance.

Reproducible example:

import numpy as np
import pandas as pd

from sklearn.ensemble import RandomForestClassifier
from sklearn.metrics import roc_auc_score
from sklearn.datasets import make_classification
from sklearn.model_selection import train_test_split


n = 50
X, y = make_classification(n_samples  = 10000,n_features = n, n_informative=27,n_classes = 2, random_state = 42)

X = pd.DataFrame(X, columns = [f"x_{i}" for i in range(n)])
y = pd.DataFrame(y, columns = ["target"])

frame = pd.merge(X,y, left_index = True, right_index = True)
frame.loc[:,"split"] = np.random.choice(a = ["train","test"], p = [.7,.3], size = frame.shape[0])
train_df = frame.query("split == 'train'").drop("split", axis = 1)
test_df = frame.query("split == 'test'").drop("split", axis = 1)



clf = RandomForestClassifier(random_state = 42).fit(train_df.drop("target", axis = 1),train_df["target"])

preds = clf.predict_proba(test_df.drop("target", axis = 1))[:,1]

roc_ = roc_auc_score(y_true = test_df["target"], y_score = preds)

print(f"ROC AUC: {roc_}")

test_df.loc[:,"prediction"] = clf.predict_proba(test_df.drop("target", axis = 1))[:,1]

test_df.loc[:,"band"] = pd.qcut(q = 10, x = test_df.prediction, duplicates = "drop")

def get_auc_(y_true, y_score):

  try:
    return roc_auc_score(y_true = y_true, y_score = y_score)
  except:
    return np.NaN

test_df.groupby("band").apply(lambda x: get_auc_(y_true = x["target"], y_score = x["prediction"]))


band
(0.009000000000000001, 0.16]         NaN
(0.16, 0.23]                    0.051780
(0.23, 0.3]                     0.592401
(0.3, 0.39]                     0.633804
(0.39, 0.5]                     0.626548
(0.5, 0.61]                     0.629141
(0.61, 0.7]                     0.633596
(0.7, 0.77]                     0.702138
(0.77, 0.84]                    0.477372
(0.84, 0.98]                    0.480072
dtype: float64

My question is what explains this low score in the different bins?

Answer 1

My two cents:

• If the goal is to predict the different bins corresponding to the probability of predicting the positive class, then this seems a strange design: why use binary classification if the main outcome of interest is not the binary class? This might be better framed as a regression problem where the goal is to predict some score instead (multiclass classification could be considered but it's not great for an ordinal target).
• I think it's wrong to study the bands of predicted probability with ROC. ROC makes sense for studying the whole range of probabilities on a dataset, not for an interval. This is because ROC is by definition about measuring how good the classifier is at placing the positive instances at one end of the range and the negative ones at the other (actually the AUC represents exactly this). Additionally it's very likely that some bands have very few instances,likely causing serious approximations in the resulting AUC score. So imho these scores are hardly interpretable, possibly even meaningless.
• A simple way to observe what happens with these bins would be to plot them as a histogram, with colours showing the proportion/number of positive/negative instances.

Answer 2

One interpretation of the AUROC is "the probability that a randomly selected positive instance is given a higher probability by the model than a randomly selected negative instance." With that in mind, it's clear that a good model will tend to have a much better AUROC over all instances than the AUROC for each bin: the former includes all the inter-bin pairs of instances, and the larger differences in their probabilities indicates the model's relative certainty, so those inter-bin pairs are much more likely to be put in the correct order.

Indeed, you could imagine shuffling the scores of the instances within each bin; as long as the model has done a decent job in creating the bins, the overall AUROC will still be reasonably good, but the AUROC inside these now-shuffled bins will all be approximately 0.5.

All that said, it of course is preferable that the AUROC inside the bins is useful. If you intend to sell the scores within the first few bins to one business, all they will care about is the rank-ordering within those bins. It's possible that a new model focusing only on those cases can do better (not being "distracted" by getting the other instances right), although then there's a big question about whether your first model is doing well enough at creating that population in the first place.

Here's a quick example notebook demonstrating the phenomenon with a synthetic dataset.