Relationship with false positive/negative rates

The 2x2 Table - Truth vs. Test

• The 2x2 table is the foundation for calculating a test's accuracy against a gold standard "truth."

-   $Sensitivity = \frac{TP}{TP + FN}$
-   Related Error: False Negative Rate ($1 - Sensitivity$)

• Specificity: Ability to detect true absence of disease (True Negative Rate).
  • $Specificity = \frac{TN}{TN + FP}$
  • Related Error: False Positive Rate ($1 - Specificity$)

⭐ Increasing a test's cutoff threshold ↑ Specificity but ↓ Sensitivity. Lowering the cutoff has the opposite effect.

📌 Mnemonic: SPIN & SNOUT

• A SPecific test, when Positive, rules IN disease.
• A SNensitive test, when Negative, rules OUT disease.

Sensitivity & Specificity - SNOUT & SPIN

• Sensitivity: Probability of a positive test in patients with the disease. Ability to correctly identify true positives (TP).

  • Formula: $Sensitivity = TP / (TP + FN)$
  • Related to False Negative Rate (FNR): $FNR = 1 - Sensitivity$.
  • 📌 SNOUT: A highly SeNsitive test, when Negative, rules OUT the disease.

• Specificity: Probability of a negative test in patients without the disease. Ability to correctly identify true negatives (TN).

  • Formula: $Specificity = TN / (TN + FP)$
  • Related to False Positive Rate (FPR): $FPR = 1 - Specificity$.
  • 📌 SPIN: A highly SPecific test, when Positive, rules IN the disease.

⭐ Increasing a test's cut-off value (making it harder to test positive) will increase specificity but decrease sensitivity. This is a fundamental trade-off visualized by the ROC curve.

The Inverse Relationship - Errors & ROC Curves

• Sensitivity & False Negatives (FN): Inversely related.

  • Sensitivity = $1 - \text{False Negative Rate (FNR)}$.
  • A highly sensitive test, if negative, helps rule out a disease. 📌 SNOUT (SeNsitive test, Negative, rules OUT).

• Specificity & False Positives (FP): Inversely related.

  • Specificity = $1 - \text{False Positive Rate (FPR)}$.
  • A highly specific test, if positive, helps rule in a disease. 📌 SPIN (Specific test, Positive, rules IN).

• Receiver Operating Characteristic (ROC) Curve:

  • Plots Sensitivity (TPR) vs. 1 - Specificity (FPR) for various cut-off points.
  • Area Under the Curve (AUC) reflects test accuracy:
    • AUC = 1.0: Perfect test.
    • AUC = 0.5: No better than chance.

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⭐ Adjusting a test's cut-off point creates a trade-off. Lowering the threshold to catch more cases (↑ True Positives) also increases false alarms (↑ False Positives). This results in ↑ Sensitivity but ↓ Specificity.

High-Yield Points - ⚡ Biggest Takeaways

• Sensitivity is inversely related to the false-negative rate (FNR); a high sensitivity means a low FNR (Sensitivity = 1 − FNR).
• Specificity is inversely related to the false-positive rate (FPR); a high specificity means a low FPR (Specificity = 1 − FPR).
• The SNOUT mnemonic: a highly Sensitive test, when Negative, helps to rule OUT a disease.
• The SPIN mnemonic: a highly Specific test, when Positive, helps to rule IN a disease.
• Both are intrinsic properties of a diagnostic test and are not affected by disease prevalence.