Measures of Association in Epidemiology

Why This Matters

When you're analyzing epidemiological data, you need more than just raw numbers—you need tools that reveal whether an exposure actually matters and how much it matters. Measures of association are the mathematical backbone of every study you'll encounter, from landmark cohort studies to clinical trials. You're being tested on your ability to select the right measure for a given study design, interpret what the values mean, and explain the public health implications of your findings.

These measures fall into two fundamental categories: relative measures (ratios that compare groups) and absolute measures (differences that quantify actual impact). Understanding this distinction is crucial because a huge relative risk might have minimal public health significance if the baseline risk is tiny, while a modest risk ratio could represent thousands of preventable cases. Don't just memorize formulas—know when each measure is appropriate, what study designs require which measures, and how to translate numbers into actionable public health insights.

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Relative Measures: Comparing Groups with Ratios

Relative measures express association as a ratio, telling you how many times more (or less) likely an outcome is in one group compared to another. These are intuitive for communicating strength of association but can overstate importance when baseline risks are low.

Risk Ratio (Relative Risk)

• The gold standard for cohort studies—directly compares the probability of disease in exposed versus unexposed groups
• Formula: $RR = \frac{\text{Incidence in exposed}}{\text{Incidence in unexposed}}$—requires knowing incidence in both groups, which means you need to follow people over time
• Interpretation thresholds: $RR > 1$ indicates increased risk, $RR < 1$ indicates protective effect, $RR = 1$ means no association

Odds Ratio

• Essential for case-control studies—the only valid ratio measure when you're sampling based on outcome rather than exposure
• Approximates the risk ratio when the outcome is rare (the rare disease assumption), making it interpretable in similar terms
• Formula: $OR = \frac{a \times d}{b \times c}$ from a 2×2 table—memorize this structure for quick calculations

Rate Ratio

• Accounts for variable follow-up time—compares incidence rates (events per person-time) rather than cumulative incidence
• Preferred in cohort studies with unequal observation periods—participants who drop out or die contribute only their actual time at risk
• Interpretation mirrors risk ratio: $\text{Rate Ratio} > 1$ means higher event rate in exposed group

Hazard Ratio

• The measure of choice for survival analysis—compares instantaneous risk of an event at any given time point
• Derived from Cox proportional hazards models—assumes the ratio of hazards remains constant over time (proportional hazards assumption)
• Common in clinical trials and time-to-event research—a $HR = 2.0$ means the exposed group experiences the event at twice the rate at any moment

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Absolute Measures: Quantifying Real-World Impact

Absolute measures express association as a difference, answering the question: how many additional cases (or prevented cases) result from the exposure? These are critical for public health planning because they reflect actual disease burden.

Risk Difference

• The simplest absolute measure—subtracts the risk in unexposed from risk in exposed: $RD = R_{exposed} - R_{unexposed}$
• Directly interpretable: a risk difference of 0.05 means 5 additional cases per 100 people exposed
• Sign indicates direction: positive values mean excess risk from exposure, negative values indicate protective effect

Incidence Rate Difference

• Absolute difference accounting for person-time—subtracts incidence rates rather than cumulative risks
• Quantifies excess events per unit of person-time—useful when comparing populations with different follow-up durations
• Complements the rate ratio by showing the actual magnitude of excess disease burden

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Attributable Measures: Assigning Responsibility to Exposures

Attributable measures answer a causal question: how much of the disease can we blame on this exposure? These measures are essential for prioritizing interventions and estimating the potential impact of prevention programs.

Attributable Risk

• Proportion of disease in exposed group due to exposure—also called attributable risk percent or etiologic fraction
• Formula: $AR\% = \frac{RR - 1}{RR} \times 100$—tells you what percentage of cases among the exposed would disappear if exposure were eliminated
• Assumes causal relationship—only meaningful if the association is truly causal, not just statistical

Population Attributable Risk

• Extends attributable risk to the entire population—accounts for how common the exposure is, not just how dangerous
• Formula incorporates prevalence: $PAR\% = \frac{P_e(RR - 1)}{1 + P_e(RR - 1)} \times 100$ where $P_e$ is exposure prevalence
• Guides public health priorities—a moderately risky but common exposure may cause more total disease than a highly risky but rare one

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Clinical Application Measures: Translating Research to Practice

These measures bridge the gap between statistical findings and clinical decision-making, expressing results in terms that directly inform treatment choices.

Number Needed to Treat (NNT)

• The clinical translation of risk difference—calculated as $NNT = \frac{1}{\text{Absolute Risk Reduction}}$
• Lower is better: an NNT of 10 means treating 10 patients prevents one adverse event; NNT of 100 means the intervention is less efficient
• Context-dependent interpretation—an NNT of 50 might be acceptable for preventing death but unacceptable for preventing mild side effects

Correlation Coefficient

• Measures linear association between continuous variables—ranges from $-1$ (perfect negative) to $+1$ (perfect positive)
• Denoted as $r$ for Pearson correlation—$r = 0$ indicates no linear relationship, though nonlinear associations may still exist
• Critical limitation: correlation does not imply causation—ecological studies using correlation are particularly vulnerable to confounding

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Quick Reference Table

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Self-Check Questions

1. A case-control study examines the association between smoking and lung cancer. Which measure of association is appropriate, and why can't you use risk ratio in this design?

2. Compare and contrast attributable risk and population attributable risk. If a rare exposure has a very high risk ratio, which measure would show greater public health significance—and why?

3. Two interventions both have a risk ratio of 0.5 for preventing heart attacks. Intervention A has an NNT of 20; Intervention B has an NNT of 200. Explain why these NNTs differ despite identical risk ratios.

4. When would you choose rate ratio over risk ratio? Identify the key study characteristic that makes person-time analysis necessary.

5. A correlation coefficient of $r = 0.85$ is found between ice cream sales and drowning deaths. Explain why this finding does not support a causal relationship and identify the likely explanation for the association.