Learning Objective
Interpret and apply positive and negative likelihood ratios (LR+ and LR–) to modify pretest probability into posttest probability for diagnostic reasoning in clinical scenarios.
Introduction
Likelihood ratios are statistical tools used to quantify how much a test result changes the probability that a patient has a disease.
They combine sensitivity and specificity into a single measure of diagnostic accuracy.
Key Definitions
| Concept | Formula | Interpretation |
|---|---|---|
| LR+ (Positive Likelihood Ratio) | Sensitivity / (1 − Specificity) | How much the odds of the disease increase when a test is positive |
| LR– (Negative Likelihood Ratio) | (1 − Sensitivity) / Specificity | How much do the odds of the disease decrease when a test is negative |
Key Points
- LR+ > 10 → Strongly rules in disease (high specificity).
- LR– < 0.1 → Strongly rules out disease (high sensitivity).
- Likelihood ratios are independent of disease prevalence, unlike predictive values.
- They allow transition from pretest odds → posttest odds using:
Posttest Odds=Pretest Odds×Likelihood Ratio\text{Posttest Odds} = \text{Pretest Odds} \times \text{Likelihood Ratio}
Then convert odds to probability:
Posttest Probability=Posttest Odds1+Posttest Odds\text{Posttest Probability} = \frac{\text{Posttest Odds}}{1 + \text{Posttest Odds}}
Conversion Table (Approximation Guide)
| LR | Effect on Disease Probability |
|---|---|
| >10 | Strong evidence to rule in |
| 5–10 | Moderate evidence to rule in |
| 2–5 | Small evidence to rule in |
| 1 | No change |
| 0.5–1 | Small evidence to rule out |
| 0.1–0.5 | Moderate evidence to rule out |
| <0.1 | Strong evidence to rule out |
Clinical Example
A test for Disease X has:
- Sensitivity = 90% (0.9)
- Specificity = 80% (0.8)
Then:
- LR+ = 0.9 / (1 − 0.8) = 0.9 / 0.2 = 4.5
- LR– = (1 − 0.9) / 0.8 = 0.1 / 0.8 = 0.125
Interpretation:
A positive test moderately increases the probability of Disease X;
A negative test substantially decreases it.
Summary Table
| Metric | Formula | Interpretation | Strength Indicator |
|---|---|---|---|
| LR+ | Sensitivity / (1 − Specificity) | The probability test is positive if the disease is present | >10 strong rule-in |
| LR– | (1 − Sensitivity) / Specificity | The probability test is negative if the disease is absent | <0.1 strong rule-out |
| Posttest Odds | Pretest Odds × LR | Adjusted probability | Used for decision-making |
| Posttest Probability | Odds / (Odds + 1) | Converts odds → probability | Clinically interpretable |
