Likelihood ratios

Sensitivity & Specificity - Sickness vs Health

• Sensitivity: The ability of a test to correctly identify individuals with a disease (True Positive Rate).

  • Formula: $TP / (TP + FN)$
  • Use when the cost of a false negative is high (e.g., missing a serious, treatable disease).
  • 📌 SNOUT: a highly SeNsitive test, when Negative, rules OUT disease.

• Specificity: The ability of a test to correctly identify individuals without a disease (True Negative Rate).

  • Formula: $TN / (TN + FP)$
  • Use when the cost of a false positive is high (e.g., unnecessary, invasive procedures).
  • 📌 SPIN: a highly SPecific test, when Positive, rules IN disease.

⭐ Screening tests require high sensitivity to catch all potential cases (minimize false negatives), while confirmatory tests need high specificity to avoid misdiagnosing healthy individuals (minimize false positives).

Predictive Values - Patient-Centric Probabilities

• Positive Predictive Value (PPV): The probability that a patient with a positive test result truly has the disease. It answers the clinical question: “I tested positive, what is the chance I actually have the disease?”

  • Formula: $PPV = \frac{TP}{TP + FP}$

• Negative Predictive Value (NPV): The probability that a patient with a negative test result truly does not have the disease. It answers: “I tested negative, what is the chance I am disease-free?”

  • Formula: $NPV = \frac{TN}{TN + FN}$

• Dependence on Prevalence:
  • If disease prevalence ↑, PPV ↑ and NPV ↓.
  • If disease prevalence ↓, PPV ↓ and NPV ↑.
  • 📌 PPV follows Prevalence.

⭐ Predictive values are not fixed characteristics of a test. They are heavily influenced by the pre-test probability (i.e., prevalence) of the disease in the specific population being tested.

Likelihood Ratios - Odds Of Being Right

• Quantifies how much a test result changes the likelihood of disease, independent of prevalence.
• Positive Likelihood Ratio (LR+): How much to ↑ odds of disease given a positive test.
  • $LR+ = \frac{Sensitivity}{(1 - Specificity)}$
• Negative Likelihood Ratio (LR-): How much to ↓ odds of disease given a negative test.
  • $LR- = \frac{(1 - Sensitivity)}{Specificity}$

Interpreting LRs:

• LR+ > 10 or LR- < 0.1: Large, often conclusive change.
• LR+ 5-10 or LR- 0.1-0.2: Moderate change.
• LR+ 2-5 or LR- 0.2-0.5: Small change.
• LR = 1: No change in likelihood.

⭐ To calculate post-test probability, convert pre-test probability to odds, multiply by the LR, and then convert post-test odds back to probability. Post-test odds = Pre-test odds × LR.

High‑Yield Points - ⚡ Biggest Takeaways

• Likelihood Ratios (LRs) quantify the diagnostic power of a test, independent of prevalence.
• Positive LR (LR+) = sensitivity / (1 − specificity). A high LR+ (ideally >10) strongly rules in a disease.
• Negative LR (LR−) = (1 − sensitivity) / specificity. A low LR− (ideally <0.1) strongly rules out a disease.
• LRs are used to calculate post-test probability from pre-test probability.
• Unlike predictive values, LRs are not affected by disease prevalence.