A 38-year-old woman from Delhi undergoes screening mammography for breast cancer as part of a routine health check-up. She has no palpable breast mass and no family history of breast cancer. The mammogram is reported as suspicious (BI-RADS 4). In the screening population, the prevalence of breast cancer is approximately 1%. The mammography test has a sensitivity of 90% and specificity of 95% for detecting breast cancer. What is the negative likelihood ratio (LR−) of this test, and what does it tell us about the probability of disease if the mammogram were negative?

Explanation

## Negative Likelihood Ratio and Its Clinical Interpretation ### Calculating LR− **Key Point:** The negative likelihood ratio (LR−) quantifies how much a negative test result decreases the odds of disease. $$LR^- = \frac{1 - Sensitivity}{Specificity}$$ Given: - Sensitivity = 90% = 0.90 - Specificity = 95% = 0.95 $$LR^- = \frac{1 - 0.90}{0.95} = \frac{0.10}{0.95} = 0.105$$ ### Interpreting LR− | LR− Value | Clinical Meaning | |-----------|------------------| | < 0.1 | Strong evidence against disease (negative test is very reassuring) | | 0.1–0.5 | Moderate evidence against disease | | 0.5–1.0 | Weak evidence against disease | | 1.0 | Test is useless (no discriminatory power) | Our LR− = 0.105 falls in the **strong evidence** category. ### Calculating Post-Test Probability After a Negative Test Using the nomogram approach or Bayes' theorem: **Pretest odds** = $\frac{Prevalence}{1 - Prevalence} = \frac{0.01}{0.99} = 0.0101$ **Post-test odds** (after negative test) = Pretest odds × LR− $$Post\text{-}test\ odds = 0.0101 \times 0.105 = 0.00106$$ **Post-test probability** = $\frac{Post\text{-}test\ odds}{1 + Post\text{-}test\ odds}$ $$P(Disease | \text{Negative test}) = \frac{0.00106}{1.00106} \approx 0.001 \text{ or } 0.1\%$$ **High-Yield:** A negative mammogram in a low-prevalence screening population reduces the disease probability from 1% to approximately **0.1%** — a 10-fold reduction. This is why LR− is so powerful in ruling out disease. ### Clinical Pearl **Mnemonic: SnNOut, SpPIn** - **Sn**Nout: High **Sensitivity** rules **OUT** disease (negative test is reassuring). - **Sp**PIn: High **Specificity** rules **IN** disease (positive test is confirmatory). Mammography has high sensitivity (90%), so a negative result effectively excludes breast cancer in this low-risk woman. The post-test probability drops to 0.1%, making further imaging unnecessary unless clinical suspicion changes. ### Why LR− Matters More Than NPV in Screening | Metric | Value | Interpretation | |--------|-------|----------------| | Sensitivity | 90% | If cancer present, test detects it 90% of the time | | Specificity | 95% | If no cancer, test is negative 95% of the time | | LR− | 0.105 | Negative test is 10× more likely in disease-free than disease-present individuals | | NPV (at 1% prevalence) | ~99.9% | If test is negative, probability of no cancer is 99.9% | [cite:Park 26e Ch 11; Harrison 21e Ch 3]