Bayes theorem application

Diagnostic Accuracy - The 2x2 Tango

-   $Sensitivity = TP / (TP + FN)$

• Specificity: Probability of testing negative if you don't have the disease. Rules IN.
  • $Specificity = TN / (TN + FP)$
• Positive Predictive Value (PPV): Probability of having the disease if you test positive.
  • $PPV = TP / (TP + FP)$
• Negative Predictive Value (NPV): Probability of not having the disease if you test negative.
  • $NPV = TN / (TN + FN)$

📌 SPIN & SNOUT: SPecific test, when Positive, rules IN. SNensitive test, when Negative, rules OUT.

⭐ Increasing disease prevalence increases PPV and decreases NPV. Sensitivity and specificity are intrinsic test characteristics and are unaffected by prevalence.

Prevalence's Power - The PPV/NPV Pivot

• Prevalence = Pre-test Probability: The baseline chance of having a disease in a specific population before testing.

• This directly influences the post-test probabilities (PPV and NPV).

• High Prevalence Setting (e.g., specialist clinic):

  • • PPV ↑: A positive test is more trustworthy.
  • • NPV ↓: A negative test is less reliable.

• Low Prevalence Setting (e.g., general screening):

  • • PPV ↓: More false positives are expected.
  • • NPV ↑: A negative test is very reassuring.

⭐ A positive screening test for a rare disease in the general population has a low PPV. Always confirm with a more specific test before diagnosing.

Bayesian Logic - The Probability Update

Bayes' theorem updates pre-test probability to post-test probability based on a test result. This is done by converting probabilities to odds, applying a likelihood ratio, and then converting back.

• Pre-test Odds: Odds of disease before testing.
  • $Pre-test odds = Prevalence / (1 - Prevalence)$
• Likelihood Ratio (LR): The power of a test to change our certainty.
  • For a positive test: $LR+ = Sensitivity / (1 - Specificity)$
  • For a negative test: $LR- = (1 - Sensitivity) / Specificity$
• Bayesian Update: The core calculation.
  • $Post-test odds = Pre-test odds × Likelihood Ratio$

⭐ A truly useful diagnostic test has an LR+ > 10 or an LR- < 0.1. These values cause large shifts in post-test probability, often confirming or ruling out a diagnosis.

• Pre-test probability (often prevalence) is the crucial starting point before applying a diagnostic test.
• Positive Predictive Value (PPV) is directly proportional to prevalence; as disease prevalence ↑, PPV ↑.
• Negative Predictive Value (NPV) is inversely proportional to prevalence; as disease prevalence ↑, NPV ↓.
• Sensitivity and Specificity are intrinsic test characteristics and are not affected by disease prevalence.
• A high-Sensitivity test, when negative, effectively rules out disease (SNOUT).
• A high-Specificity test, when positive, helps rule in disease (SPIN).