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

Question

BlackTom AI · Accepted Answer

TP, FN, FP and TN are counts computed after converting model scores/probabilities into discrete class labels using a chosen decision threshold. Relevant concepts: confusion matrix entries; how thresholds convert probabilities to predicted classes; definitions of sensitivity/TPR, specificity/TNR; ROC curve axes (TPR vs FPR).

1) Relevant knowledge points and definitions
- Given predicted probability p and threshold t, predicted class = 1 if p ≥ t, else 0.  
- TP = count(true label = 1 and predicted = 1). FN = count(true = 1 and predicted = 0). FP = count(true = 0 and predicted = 1). TN = count(true = 0 and predicted = 0).  
- TPR (Sensitivity) = TP / (TP + FN). TNR (Specificity) = TN / (FP + TN). FPR = 1 − Specificity = FP / (FP + TN). ROC plots TPR (y-axis) vs FPR (x-axis).

2) Step-by-step reasoning showing dependence on threshold
- Because predicted class depends on threshold t, the membership of each instance in TP/FN/FP/TN changes when t changes.  
- Example: four instances with probabilities and true labels:
  - p = [0.8, 0.6, 0.4, 0.3], true = [1, 0, 1, 0].
  - If t = 0.5, predicted = [1, 1, 0, 0] → TP=1 (first), FP=1 (second), FN=1 (third), TN=1 (fourth).
  - If t = 0.7, predicted = [1, 0, 0, 0] → TP=1, FP=0, FN=1, TN=2.
  - The counts TP, FP, FN, TN clearly changed when t changed.

3) Intermediate formulas
- TP(t) = |{i: yi=1 and pi ≥ t}|, FN(t) = |{i: yi=1 and pi < t}|, etc. These are explicit functions of t.

4) Why the correct answer is correct
- The statement "The values TP, FN, FP, TN in a confusion matrix do not depend on your choice of probability threshold" is false because the counts are direct results of the thresholded predictions; changing the threshold alters which instances are predicted positive versus negative and therefore alters TP/FN/FP/TN.

5) Why the other options are correct
- "ROC curve plots Sensitivity as y-axis against (1-Specificity) as x-axis": True — ROC plots TPR vs FPR (FPR = 1 − Specificity).  
- "True Negative Rate (TNR) is also called Specificity": True — TNR = TN/(FP+TN) is the definition of specificity.  
- "True Positive Rate (TPR) is also called Sensitivity": True — TPR = TP/(TP+FN) is sensitivity (recall).

6) Conclusion
- Therefore the false statement is the first option. This matches the provided correct answer: "The values TP, FN, FP, TN in a confusion matrix do not depend on your choice of probability threshold."

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