odds ratio

**Definition:** Odds = probability of event / (1 − probability of event). The odds ratio is the ratio of two odds.

**2x2 table:** - a = exposed, diseased (cases) - b = exposed, no disease (controls) - c = unexposed, diseased - d = unexposed, no disease

OR = (a/c) / (b/d) = **ad / bc** (the cross-product ratio). In a case-control study, this is the odds of exposure in cases divided by the odds of exposure in controls. It equals the odds of disease in exposed divided by the odds of disease in unexposed.

**Interpretation:** - OR = 1: no association between exposure and outcome (null value) - OR > 1: exposure is associated with higher odds of disease (risk factor) - OR < 1: exposure is associated with lower odds of disease (protective) - A 95% confidence interval that includes 1 means the result is not statistically significant at p < 0.05.

**Why it is used in case-control studies:** Investigators select subjects by disease status, so they set the proportion of cases. Incidence cannot be calculated, and neither can relative risk (RR). The OR is the only valid measure of association in this design.

**Also used in:** - Logistic regression: exponentiating the coefficient gives the adjusted OR. - Cross-sectional studies and cohort studies (OR can be calculated, but RR is preferred in cohorts). - Meta-analyses.

**Rare disease assumption:** When disease prevalence is low (<10%), a and c are small relative to b and d. Then OR ≈ RR. When the disease is common, the OR overestimates the RR (it moves further from 1).

**Comparison with RR:** - RR = [a/(a+b)] / [c/(c+d)], the ratio of incidences. It is used in cohort studies and RCTs. - OR is a ratio of odds, not probabilities.

**Worked example:** In a case-control study of smoking and lung cancer: 80 cases smoke and 20 cases do not; 40 controls smoke and 60 controls do not. OR = (80 × 60) / (20 × 40) = 4800/800 = 6. Cases had 6 times higher odds of smoking exposure than controls.

**High-yield board pearls:** - "Case-control → odds ratio; cohort → relative risk" is the classic pairing. - Case-control studies are good for rare diseases and for diseases with long latency. They are prone to recall bias and selection bias. - OR cannot give incidence or prevalence. - Hazard ratio is used in survival analysis (time-to-event), and is a different measure. - Number needed to treat (NNT) is derived from absolute risk reduction, not directly from the OR. - Do not confuse OR with attributable risk or relative risk reduction.