Question at position 16 In a confusion matrix, which of the following is False?True Positive Rate (TPR) is also called SensitivityROC curve plots Sensitivity as y-axis against (1-Specificity) as x-axisTrue Negative Rate (TNR) is also called SpecificityThe values TP, FN, FP, TN in a confusion matrix do not depend on your choice of probability threshold

Question

BlackTom AI · Accepted Answer

TP, FN, FP and TN are counts computed after converting predicted probabilities into binary class labels using a chosen decision threshold; therefore they generally change when the threshold changes.

1) Relevant concepts
- Confusion matrix entries: TP, FN, FP, TN are counts of predicted vs actual class membership.
- True Positive Rate (TPR, a.k.a. Sensitivity) = TP / (TP + FN).
- True Negative Rate (TNR, a.k.a. Specificity) = TN / (TN + FP).
- ROC curve plots TPR (sensitivity) on the y-axis versus False Positive Rate (FPR) = 1 − Specificity on the x-axis.
- Converting model probabilities to class labels requires a threshold; changing the threshold changes predictions and thus the confusion matrix.

2) Step-by-step reasoning with concrete example
Consider six examples with actual labels and predicted probabilities:
1: actual +, p=0.90
2: actual +, p=0.60
3: actual +, p=0.40
4: actual −, p=0.80
5: actual −, p=0.30
6: actual −, p=0.20

Threshold = 0.5:
- Predicted positives: 1, 2, 4. Predicted negatives: 3, 5, 6.
- TP = {1,2} = 2; FN = {3} = 1; FP = {4} = 1; TN = {5,6} = 2.
- TPR = 2/(2+1) = 2/3 ≈ 0.667; TNR = 2/(2+1) = 2/3 ≈ 0.667.

Threshold = 0.7:
- Predicted positives: 1, 4. Predicted negatives: 2,3,5,6.
- TP = {1} = 1; FN = {2,3} = 2; FP = {4} = 1; TN = {5,6} = 2.
- TPR = 1/(1+2) = 1/3 ≈ 0.333; TNR = 2/(2+1) = 2/3 ≈ 0.667.

3) Intermediate calculation steps shown above demonstrate TP and FN changed when threshold moved from 0.5 to 0.7 (TP: 2 → 1, FN: 1 → 2), confirming dependence on threshold.

4) Why the correct answer is correct
Statement "The values TP, FN, FP, TN in a confusion matrix do not depend on your choice of probability threshold" is false because those counts are derived from binary predictions formed by applying a threshold to continuous probability outputs; altering the threshold changes which instances are predicted positive or negative, changing TP/FN/FP/TN.

5) Why other options are incorrect (i.e., why they are true)
- "TPR is also called Sensitivity": true by definition (TP/(TP+FN)).
- "ROC curve plots Sensitivity as y-axis against (1-Specificity) as x-axis": true—ROC uses TPR vs FPR where FPR = 1 − Specificity.
- "TNR is also called Specificity": true by definition (TN/(TN+FP)).

6) Conclusion
The false statement is exactly "The values TP, FN, FP, TN in a confusion matrix do not depend on your choice of probability threshold," which matches the provided answer.

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