Eberhardt's Statistic

TL;DR

• A performance measure for binary classification models based on TPR and FPR.
• Computed as the difference between TPR and FPR divided by TPR.
• Useful in settings that prioritize minimizing false positives (e.g., medical diagnosis, fraud detection).

Definition

Eberhardt’s statistic is a measure of the effectiveness of a classification model, specifically a binary classification model (a model that assigns data points to one of two classes). The statistic is calculated by dividing the difference between the true positive rate (TPR) and the false positive rate (FPR) by the true positive rate.

In formula form:

$$\text{Eberhardt's statistic} = \frac{\text{TPR} - \text{FPR}}{\text{TPR}}$$

Explanation

• True positive rate (TPR) is the proportion of positive cases correctly identified by the model.
• False positive rate (FPR) is the proportion of negative cases incorrectly identified as positive by the model.
• Eberhardt’s statistic expresses how much of the model’s true positive performance remains after accounting for false positives, by normalizing the difference (TPR − FPR) by TPR.

Examples

Medical diagnosis example

Suppose a binary classification model is designed to identify individuals who have a certain disease. If the model correctly identifies 80 out of 100 individuals with the disease, then the TPR would be 80%. If the model incorrectly identifies 20 out of 100 individuals without the disease as having the disease, then the FPR would be 20%. Eberhardt’s statistic is:

$$(80\% - 20\%) / 80\% = 75\%$$

Fraud detection example

For a model designed to identify fraudulent credit card transactions, the TPR is the proportion of fraudulent transactions correctly identified, and the FPR is the proportion of legitimate transactions incorrectly flagged as fraudulent. Dividing the difference between these rates by the TPR yields Eberhardt’s statistic for the model.

Use cases

• Medical diagnosis (where minimizing false positives is important)
• Fraud detection (where minimizing false positives is important)

Related terms

• True positive rate (TPR)
• False positive rate (FPR)
• Binary classification