likelihood ratio

**Definition:** LR = P(test result | disease) / P(same result | no disease).

**Formulas:** - LR+ = sensitivity / (1 − specificity) = TPR / FPR - LR− = (1 − sensitivity) / specificity = FNR / TNR

**Interpretation:** - LR = 1: test result does not change the probability of disease (useless). - LR+ > 1: result raises the probability of disease. LR+ >10 is strong evidence to rule IN disease (think SpPIn: highly Specific test, Positive result rules IN). - LR− < 1: result lowers the probability of disease. LR− <0.1 is strong evidence to rule OUT disease (think SnNOut: highly Sensitive test, Negative result rules OUT). - Rough guide to change in probability: LR 2, 5, 10 raise the probability by about 15%, 30%, 45%. LR 0.5, 0.2, 0.1 lower it by about 15%, 30%, 45%.

**Use with Bayes' theorem:** - Pre-test odds × LR = post-test odds. - Pre-test odds = p / (1 − p); post-test probability = odds / (1 + odds). - Example: pre-test probability 20% (odds 0.25), LR+ = 10, so post-test odds = 2.5, which gives a post-test probability of about 71%.

**Key properties (high-yield):** - Independent of disease prevalence, because it is built from sensitivity and specificity. PPV and NPV do depend on prevalence. - It is a property of the test, but it can vary with the patient population (spectrum bias). - Can be calculated for multilevel or continuous results (e.g., ferritin or D-dimer ranges), not just dichotomous ones. - A high LR+ comes from high specificity. A low LR− comes from high sensitivity. - Fagan nomogram: a graphical tool that links pre-test probability, LR, and post-test probability.

**Exam tips:** - Be able to compute LR+ and LR− from a 2×2 table. - Know that LRs do not change with prevalence, while PPV and NPV do. - Know that a more specific test has a higher LR+, and a more sensitive test has a lower LR−. - Questions often ask which test is best for ruling in or ruling out a disease based on its LRs.

**Limitations:** LRs assume independence when multiple tests are combined in sequence. They also require a reasonable estimate of pre-test probability, which is often subjective.