2x2 contingency tables

2x2 Contingency Table - The Grid Game

• Organizes diagnostic test results against the true disease status (gold standard).
• Rows: Test Result (Positive / Negative)
• Columns: Disease Status (Present / Absent)
  • TP (True Positive): Sick people correctly identified as sick.
  • FP (False Positive): Healthy people incorrectly identified as sick.
  • FN (False Negative): Sick people incorrectly identified as healthy.
  • TN (True Negative): Healthy people correctly identified as healthy.

⭐ Columns represent the truth (actual disease status), while rows represent the test's conclusion.

Sensitivity & Specificity - Trusting Your Test

• Sensitivity: Measures a test's ability to correctly identify patients with a disease. A highly sensitive test will capture most true cases.
  • Formula: $Sensitivity = TP / (TP + FN)$
  • 📌 SNOUT: High Sensitivity, when Negative, rules OUT disease. Ideal for screening.
• Specificity: Measures a test's ability to correctly identify people without a disease. A highly specific test will have few false positives.
  • Formula: $Specificity = TN / (TN + FP)$
  • 📌 SPIN: High Specificity, when Positive, rules IN disease. Ideal for confirmation.

⭐ These are fixed, intrinsic properties of a test, calculated vertically from the 2x2 table. They do not change with disease prevalence.

PPV & NPV - Patient Prediction Power

• Positive Predictive Value (PPV): Probability that a + test result means the patient truly has the disease.

  • Calculated from rows: $PPV = \frac{TP}{TP + FP}$
  • Directly varies with prevalence: ↑ prevalence → ↑ PPV.

• Negative Predictive Value (NPV): Probability that a - test result means the patient is truly disease-free.

  • Calculated from rows: $NPV = \frac{TN}{TN + FN}$
  • Inversely varies with prevalence: ↑ prevalence → ↓ NPV.

⭐ Unlike sensitivity/specificity, PPV & NPV are not intrinsic to the test. They are heavily influenced by the pre-test probability (prevalence) of the disease in the specific population being tested.

Prevalence & Predictive Values - The Prevalence Effect

• Prevalence: Proportion of a population with a disease at a given time.
• While Sensitivity & Specificity are fixed test characteristics, PPV & NPV are heavily influenced by prevalence.
• The Prevalence Effect:
  • Prevalence ↑ → PPV ↑, NPV ↓
  • Prevalence ↓ → PPV ↓, NPV ↑
• A test has a higher PPV in a high-prevalence population vs. a low-prevalence one.

⭐ In low-prevalence (e.g., general screening) settings, positive results have a lower PPV. In high-prevalence (e.g., symptomatic patient) settings, positive results have a much higher PPV.

Likelihood Ratios - Odds & Ends

• A measure of a test's diagnostic power, combining sensitivity and specificity. They are independent of prevalence.
• Positive Likelihood Ratio (LR+): How much the odds of disease increase with a positive test.
  • $LR+ = sensitivity / (1 - specificity)$
• Negative Likelihood Ratio (LR-): How much the odds of disease decrease with a negative test.
  • $LR- = (1 - sensitivity) / specificity$
• Interpretation:
  • LR+ > 10 is strong evidence to rule IN.
  • LR- < 0.1 is strong evidence to rule OUT.

⭐ Post-test odds can be calculated from pre-test odds and the likelihood ratio: Post-test odds = Pre-test odds × LR.

High‑Yield Points - ⚡ Biggest Takeaways

• Sensitivity is the True Positive Rate (TP / [TP+FN]); a high sensitivity test, when negative, rules out disease (SNOUT).
• Specificity is the True Negative Rate (TN / [TN+FP]); a high specificity test, when positive, rules in disease (SPIN).
• Screening tests for dangerous diseases require high sensitivity to avoid missing cases.
• Confirmatory tests need high specificity to ensure a positive result is truly a positive.
• Prevalence impacts PPV and NPV but not sensitivity or specificity.