Bayesian approach to diagnosis

Bayes' Theorem - Odds On Favourite

• An intuitive way to update diagnostic probability. It uses odds, not direct probabilities.
• Core Formula: $Pre-test Odds × Likelihood Ratio (LR) = Post-test Odds$

• From Probability to Odds:

  • Odds = Probability / (1 - Probability)

• From Odds to Probability:

  • Probability = Odds / (1 + Odds)

• Likelihood Ratios (LR): The factor by which the odds of disease change.

  • LR+ for a positive test: $Sensitivity / (1 - Specificity)$
  • LR- for a negative test: $(1 - Sensitivity) / Specificity$

⭐ A powerful diagnostic test has an LR+ > 10 or an LR- < 0.1, causing large shifts in post-test probability.

Pre & Post-Test Probability - Before & After Story

• Pre-test Probability (PTP): The probability of a patient having a disease before a diagnostic test is performed. It's often based on prevalence, clinical history, and physical exam findings.

• Likelihood Ratios (LRs): Quantify the diagnostic power of a test. They modify the pre-test odds to give you post-test odds.

  • Positive LR (LR+): For a positive test result. $LR+ = Sensitivity / (1 - Specificity)$
  • Negative LR (LR-): For a negative test result. $LR- = (1 - Sensitivity) / Specificity$

• Post-test Probability (Post-TP): The revised probability of disease after considering the test result.

⭐ A test with an LR+ > 10 or an LR- < 0.1 is considered very strong evidence to rule in or rule out a disease, respectively.

Likelihood Ratios - Test Power-Up

• Likelihood Ratios (LRs) quantify the diagnostic power of a test, indicating how much a test result will shift the pre-test probability to the post-test probability. They are independent of disease prevalence.

• Positive Likelihood Ratio (LR+): How much to increase the probability of disease with a positive test.

  • $LR+ = \frac{Sensitivity}{1 - Specificity}$

• Negative Likelihood Ratio (LR-): How much to decrease the probability of disease with a negative test.

  • $LR- = \frac{1 - Sensitivity}{Specificity}$

• Interpreting LRs:

• Bayes' Theorem formally updates the probability of a disease based on new test results.
• Start with pre-test probability, which is often the disease prevalence in the relevant population.
• Use Likelihood Ratios (LRs) to quantify a test's power to change probability.
• Post-test odds are calculated by multiplying pre-test odds by the appropriate LR.
• A high LR+ (>10) significantly rules in disease; a low LR- (<0.1) strongly rules it out.
• This avoids common cognitive biases by systematically integrating new data.