Recall, also known as sensitivity or true positive rate, is a statistical metric used to evaluate the performance of a binary classification model. It measures the fraction of actual positive cases that are correctly identified by the model, calculated as True Positives divided by the sum of True Positives and False Negatives. Recall is particularly important in scenarios where missing a positive instance carries a high cost, such as medical diagnostics or fraud detection.
1 Definition and Formula
1.1 Mathematical expression
Recall is mathematically expressed as:
\[ \text{Recall} = \frac{\text{True Positives (TP)}}{\text{True Positives (TP)} + \text{False Negatives (FN)}} \]
In this formula, True Positives are correctly identified positive instances, and False Negatives are positive instances incorrectly classified as negative. The ratio indicates how many of all actual positives the model successfully detects.
1.2 Relationship to confusion matrix
In a confusion matrix for binary classification, the cells are arranged as:
| Predicted Positive | Predicted Negative | ||
|---|---|---|---|
| Actual Positive | TP | FN | |
| Actual Negative | FP | TN |
Recall corresponds to the TP cell divided by the sum of the TP and FN cells (the entire actual positive row). It is sometimes called the row-wise proportion for the positive class.
1.3 Range and interpretation
Recall ranges from 0 to 1 (or 0% to 100%). A recall of 0 means no positive cases were identified; a recall of 1 means all positive cases were correctly detected. Values closer to 1 indicate the model rarely misses a positive instance, while lower values indicate frequent missed detections.
2 Relationship with Other Metrics
2.1 Recall vs. Precision
Precision measures the fraction of predicted positives that are truly positive: \(\text{TP} / (\text{TP} + \text{FP})\). While recall focuses on capturing all positives, precision focuses on the correctness of positive predictions. Both are essential for understanding classifier behavior.
2.1.1 Precision–Recall trade-off
There is often an inverse relationship between precision and recall. Adjusting a classifier’s decision threshold typically improves one metric at the expense of the other. For example, lowering the threshold may increase recall (catching more positives) but also increase false positives, decreasing precision. This trade-off is intrinsic to most classification systems.
2.1.2 Precision–Recall curve
The precision–recall curve plots precision (y-axis) against recall (x-axis) for different threshold values. It provides a comprehensive view of a model’s performance, especially in imbalanced datasets. The curve’s shape indicates how well the model balances the two metrics.
2.2 Recall vs. Specificity
Specificity (or true negative rate) measures the proportion of actual negatives correctly identified: \(\text{TN} / (\text{TN} + \text{FP})\). While recall addresses the positive class, specificity addresses the negative class. Both are complementary; a high recall does not guarantee high specificity, and vice versa.
2.3 F1 Score
The F1 score combines precision and recall into a single metric. It is the harmonic mean of the two, providing a balanced measure when both are important.
2.3.1 Harmonic mean of precision and recall
The F1 score is calculated as:
\[ F1 = 2 \times \frac{\text{Precision} \times \text{Recall}}{\text{Precision} + \text{Recall}} \]
The harmonic mean penalizes large disparities between precision and recall more strongly than the arithmetic mean.
2.3.2 Balanced vs. weighted F1
A balanced F1 (the standard form) gives equal importance to precision and recall. Weighted variants, such as \(F_\beta\), allow adjusting the emphasis using a parameter \(\beta\). When \(\beta > 1\), recall is weighted more; when \(\beta < 1\), precision is weighted more.
3 Use Cases
3.1 Imbalanced datasets
In datasets where the positive class is rare (e.g., 1% of all instances), accuracy can be misleadingly high. Recall becomes critical to evaluate how well the model captures those few positive cases.
3.1.1 Rare event detection
Examples include detecting fraudulent credit card transactions, diagnosing rare diseases, or identifying security intrusions. High recall is desired because failing to detect a rare event often has severe consequences.
3.1.2 Cost-sensitive classification
In cost-sensitive scenarios, costs of false negatives differ from false positives. Recall guides threshold selection to minimize the overall cost when false negatives are more costly than false positives.
3.2 Information retrieval
In information retrieval systems, such as search engines and document retrieval, recall measures the fraction of relevant documents retrieved out of all relevant documents in the collection.
3.2.1 Recall in search engines
For a given query, recall indicates how many of the total relevant web pages are returned. A high recall means the user is likely to see all relevant results, though it may include irrelevant ones.
3.2.2 Relevance assessment
Recall is often used alongside precision to evaluate relevance ranking. An ideal system achieves both high recall (finding all relevant items) and high precision (showing only relevant items).
3.3 Medical diagnostics
In medical testing, recall (sensitivity) measures how well a test identifies patients with a condition. A high recall reduces the risk of false negatives, which could lead to missed diagnoses.
3.3.1 Screening tests
Screening tests (e.g., mammograms for breast cancer) prioritize high recall to avoid missing cases. False positives are tolerated because they are later resolved by more specific tests.
3.3.2 Threshold selection
The threshold for a positive test result can be adjusted to balance recall and specificity. For example, lowering the threshold increases recall but may raise false positives, which must be weighed against the cost of missed diagnoses.
4 Limitations and Considerations
4.1 Ignoring true negatives
Recall does not account for true negatives (correctly identified negative cases). A model could achieve perfect recall by classifying all instances as positive, but this would produce many false positives and be useless in practice. Therefore, recall is rarely used alone.
4.2 Sensitivity to class imbalance
While recall is useful for imbalanced datasets, it is itself sensitive to the class distribution if interpreted without context. A high recall in a dataset with very few positive instances may be less impressive than the same recall in a balanced dataset. Comparing recall across different populations requires caution.
4.3 Dependence on threshold
Recall varies with the decision threshold used to convert a model’s probability or score into a binary prediction. A single recall value provides only a snapshot, not a complete picture of model performance.
4.3.1 Receiver operating characteristic (ROC) curve
The ROC curve plots the true positive rate (recall) on the y-axis against the false positive rate on the x-axis for various thresholds. It shows how recall changes as the trade-off with false positives is adjusted.
4.3.2 Area under the curve (AUC)
The area under the ROC curve (AUC) summarizes overall model performance across all thresholds. A higher AUC indicates better discriminative ability, independent of a specific threshold. However, AUC does not directly report recall at any particular threshold.