Premed · Premed · Statistics Biostatistics
Lecture 25: Relative Risk, Odds Ratios, and Epidemiologic Measures
Statistics / Biostatistics
Learning Objectives
By the end of this lecture, students will be able to:
- Calculate and interpret relative risk, absolute risk reduction, and number needed to treat
- Calculate and interpret the odds ratio
- Explain when to use RR vs. OR based on study design
- Calculate and interpret incidence, prevalence, and attributable risk
- Recognize common pitfalls in interpreting epidemiologic measures
Lecture Content
I. Measures of Disease Frequency
Prevalence is the proportion of a population that has a disease at a specific point in time, calculated as the number of existing cases divided by the total population. It is a snapshot measure, typical of cross-sectional studies. Incidence (also called cumulative incidence or risk) is the proportion of an at-risk population that develops the disease over a specified time period, calculated as the number of new cases divided by the population at risk during that period. The incidence rate (person-time rate) is the number of new cases divided by the total person-time at risk, which accounts for varying follow-up times and is expressed in units such as events per person-year.
Prevalence depends on both incidence and disease duration. When prevalence is low, the approximation Prevalence is roughly equal to Incidence multiplied by Average duration holds reasonably well.
II. The 2x2 Table for Epidemiologic Studies
| Disease (D+) | No Disease (D-) | Total | |
|---|---|---|---|
| Exposed (E+) | a | b | a + b |
| Unexposed (E-) | c | d | c + d |
| Total | a + c | b + d | N |
From this table, the risk in the exposed group is R_E = a / (a + b), the risk in the unexposed group is R_U = c / (c + d), the odds of disease in the exposed is a / b, and the odds of disease in the unexposed is c / d.
III. Relative Risk (Risk Ratio)
The relative risk is calculated as RR = R_E / R_U = [a/(a+b)] / [c/(c+d)]. It tells how many times more (or less) likely the exposed group is to develop the disease compared to the unexposed group. An RR of 1 indicates no association, an RR greater than 1 indicates that exposure increases risk, and an RR less than 1 indicates that exposure decreases risk (is protective).
For example, in a cohort study of smoking and lung cancer, if the risk in smokers is 0.15 and the risk in non-smokers is 0.01, then RR = 0.15 / 0.01 = 15.0, meaning smokers are 15 times more likely to develop lung cancer. The relative risk can be calculated from cohort studies and RCTs, where risks can be directly measured, but it cannot be calculated from case-control studies because disease frequency in those designs is determined by the researcher's sampling scheme.
<image>A 2x2 table worked example for a cohort study of smoking and lung cancer. The table is filled with hypothetical numbers: a = 150, b = 850, c = 10, d = 990. Calculations shown below: Risk in exposed = 150/1000 = 0.15, Risk in unexposed = 10/1000 = 0.01, RR = 15.0, AR = 0.14, ARR or RD = 0.14, NNH = 1/0.14 = 7.1. Each measure is labeled and interpreted in one sentence.</image>
IV. Odds Ratio
The odds ratio is calculated as OR = (ad) / (bc) = (a/b) / (c/d). It represents the ratio of the odds of disease in the exposed to the odds of disease in the unexposed. An OR of 1 indicates no association, an OR greater than 1 indicates exposure is associated with higher odds of disease, and an OR less than 1 indicates exposure is associated with lower odds.
The odds ratio can be calculated from any study design -- case-control, cohort, cross-sectional, or RCT. In case-control studies, the OR is the primary measure of association because the relative risk cannot be computed. The OR approximates the RR when the disease is rare (prevalence less than 10%), which is known as the "rare disease assumption." As disease prevalence increases, the OR increasingly overestimates the RR. The OR is also the natural output of logistic regression.
V. Absolute Risk Measures
The Absolute Risk Reduction (ARR), also called the Risk Difference (RD), equals the risk in the control group minus the risk in the treatment group: ARR = R_C - R_T. It measures the absolute difference in event rates. For example, if mortality with placebo is 10% and mortality with the drug is 6%, the ARR is 4 percentage points.
The Relative Risk Reduction (RRR) expresses the reduction as a fraction of the baseline risk: RRR = ARR / R_C = (R_C - R_T) / R_C = 1 - RR. In the example above, RRR = 0.04 / 0.10 = 40%.
The Number Needed to Treat (NNT) equals 1 / ARR and represents the number of patients who need to be treated to prevent one additional adverse event. In the example, NNT = 1/0.04 = 25, meaning 25 patients must be treated to prevent one death. A lower NNT indicates a more effective treatment. The Number Needed to Harm (NNH) = 1 / ARI (absolute risk increase) gives the number of patients treated for one additional adverse outcome to occur.
<image>A comparison of relative vs. absolute risk reduction. Scenario A: Baseline risk = 40%, treatment risk = 20%. RRR = 50%, ARR = 20%, NNT = 5. Scenario B: Baseline risk = 2%, treatment risk = 1%. RRR = 50%, ARR = 1%, NNT = 100. Both scenarios have the same RRR (50%) but vastly different ARR and NNT. A bar chart shows the two scenarios side by side, with annotations emphasizing that RRR alone can be misleading and ARR/NNT provide the clinically meaningful picture.</image>
VI. Attributable Risk and Population Attributable Risk
The attributable risk (AR) equals R_E - R_U (the risk difference), representing the amount of risk in the exposed group that is attributable to the exposure, assuming a causal relationship. The attributable risk percent (AR%) = (R_E - R_U) / R_E 100 = (RR - 1) / RR 100 gives the percentage of disease in the exposed that is due to the exposure.
The population attributable risk (PAR) is the proportion of disease in the total population attributable to the exposure: PAR = R_total - R_U. The PAR% = PAR / R_total * 100 depends on both the strength of association (RR) and the prevalence of exposure, making it important for public health policy because it identifies which exposures cause the most disease burden. For example, if smoking causes an AR% of 93% for lung cancer among smokers and 25% of the population smokes, the PAR% tells us what fraction of all lung cancer in the population is attributable to smoking.
VII. Confidence Intervals for RR and OR
Both the relative risk and odds ratio are typically reported with 95% confidence intervals. These CIs are computed on the log scale because ratios have skewed distributions: ln(RR) +/- 1.96 * SE(ln(RR)), then exponentiate, and similarly for the OR. If the 95% CI includes 1.0, the association is not statistically significant at alpha = 0.05. If the CI excludes 1.0, the association is statistically significant.
VIII. Interpreting and Reporting Epidemiologic Measures
Both relative measures (RR or OR) and absolute measures (ARR, NNT) should always be reported together. Relative measures indicate the strength of association, while absolute measures indicate the public health or clinical impact. The base rate must always be considered: a relative risk reduction of 50% sounds impressive but may correspond to a very small absolute risk reduction. Transparency about the study design is important when choosing between RR and OR. Forest plots are commonly used to display RR or OR with confidence intervals from multiple studies in systematic reviews and meta-analyses.
<image>A forest plot from a hypothetical meta-analysis of five studies examining the effect of statin therapy on cardiovascular events. Each row shows a study name, the number of events/total in treatment and control groups, and a point estimate (OR) with 95% CI displayed as a square and horizontal line. A vertical line at OR = 1.0 marks no effect. The diamond at the bottom represents the pooled estimate. Studies favoring treatment are to the left of 1.0. The pooled OR and its CI are annotated. Heterogeneity statistics (I-squared, Q test) are displayed below.</image>
