Measures of Association & Public Health Impact: Risk Ratios, Odds Ratios & Attributable Fractions

The standard analytic format for evaluating dichotomous exposures and binary disease outcomes is the classical $2 \times 2$ contingency table. In a prospective cohort study of $N$ participants initially free of disease, participants are classified by exposure status ($E^+$ vs $E^-$) and subsequently monitored for disease occurrence ($D^+$ vs $D^-$): ### Mathematical Definition of the Risk Ratio The **Risk Ratio** ($RR$), also universally designated as the **Relative Risk**, is the ratio of the cumulative incidence of disease in the exposed group to that in the unexposed group over the designated time interval: $$RR = \frac{R_1}{R_0} = \frac{a / n_1}{c / n_0} = \frac{a(c + d)}{c(a + b)}$$ The risk ratio is a non-negative dimensionless scalar spanning $[0, \infty)$: - $RR = 1.0$: Null hypothesis of no association; the exposure neither increases nor decreases disease risk. - $RR > 1.0$: Positive association; the exposure increases disease risk (harmful exposure / risk factor). For example, $RR = 2.5$ indicates that exposed individuals experience $2.5$ times the risk of unexposed individuals. - $RR < 1.0$: Negative (inverse) association; the exposure reduces disease risk (protective factor / intervention efficacy). When disease occurrence is measured using incidence density across person-time denominators ($PT_1$ and $PT_0$), the corresponding relative measure is the **Rate Ratio** ($IRR$): $$IRR = \frac{IR_1}{IR_0} = \frac{a / PT_1}{c / PT_0}$$ ### Asymptotic Sampling Distribution of ln(RR) via the Delta Method Because the empirical ratio $\widehat{RR} = \widehat{R}_1 / \widehat{R}_0$ is constrained to be non-negative, its finite-sample sampling distribution is severely skewed right. To perform valid statistical inference, we apply a logarithmic transformation $\theta = \ln(RR) = \ln(R_1) - \ln(R_0)$, which maps the parameter onto the entire real line $(-\infty, \infty)$ and accelerates convergence to Gaussian normality. Assuming independent binomial sampling in the two exposed cohorts: $$a \sim \text{Binomial}(n_1, R_1), \quad c \sim \text{Binomial}(n_0, R_0)$$ The sample estimates $\widehat{R}_1 = a / n_1$ and $\widehat{R}_0 = c / n_0$ have variances: $$\text{Var}(\widehat{R}_1) = \frac{R_1(1 - R_1)}{n_1}, \quad \text{Var}(\widehat{R}_0) = \frac{R_0(1 - R_0)}{n_0}$$ Applying the first-order **Delta Method** to the transformation $g(p) = \ln(p)$, where $g'(p) = 1/p$: $$\text{Var}(\ln \widehat{R}_1) \approx [g'(R_1)]^2 \text{Var}(\widehat{R}_1) = \frac{1}{R_1^2} \frac{R_1(1 - R_1)}{n_1} = \frac{1 - R_1}{n_1 R_1} = \frac{1 - a/n_1}{a} = \frac{1}{a} - \frac{1}{n_1}$$ Because the two cohorts are statistically independent: $$\text{Var}(\ln \widehat{RR}) = \text{Var}(\ln \widehat{R}_1) + \text{Var}(\ln \widehat{R}_0) \approx \left(\frac{1}{a} - \frac{1}{n_1}\right) + \left(\frac{1}{c} - \frac{1}{n_0}\right)$$ The standard error on the natural logarithmic scale is: $$\text{SE}(\ln \widehat{RR}) = \sqrt{\frac{1}{a} - \frac{1}{n_1} + \frac{1}{c} - \frac{1}{n_0}} = \sqrt{\frac{b}{a n_1} + \frac{d}{c n_0}}$$ A $100(1 - \alpha)\%$ confidence interval for $\ln(RR)$ is $\ln \widehat{RR} \pm z_{1 - \alpha/2} \text{SE}(\ln \widehat{RR})$. Exponentiating both bounds yields the exact confidence limits for the Risk Ratio: $$CI_{1 - \alpha}(RR) = \left[ \widehat{RR} \cdot \exp\left(-z_{1 - \alpha/2} \text{SE}(\ln \widehat{RR})\right), \, \widehat{RR} \cdot \exp\left(z_{1 - \alpha/2} \text{SE}(\ln \widehat{RR})\right) \right]$$

The odds of an event is defined as the probability that the event occurs divided by the probability that it does not occur: $\text{Odds} = p / (1 - p)$. In the $2 \times 2$ table, the odds of disease among exposed individuals is:

Similarly, the odds of disease among unexposed individuals is:

The Disease Odds Ratio ($OR$) is defined as the ratio of these two odds:

Because this algebraic formula multiplies the diagonal cells and divides by the off-diagonal cells, the odds ratio is universally termed the cross-product ratio.

Mathematical Invariance Under Retrospective Case-Control Sampling

The odds ratio possesses a profound mathematical property that makes it the cornerstone of case-control studies: directional symmetry. Suppose that instead of sampling prospectively forward from exposure to disease, we sample retrospectively based on disease status, recruiting a fixed sample of $m_1 = a + c$ cases and $m_0 = b + d$ controls.

In this retrospective case-control design, the Exposure Odds among diseased individuals is:

The exposure odds among healthy controls is:

Evaluating the ratio of exposure odds:

Thus, the odds ratio calculated from a retrospective case-control study is mathematically identical to the odds ratio that would have been obtained had the entire underlying population been followed prospectively in a cohort study! Neither cumulative risk nor the risk ratio can be computed directly from a case-control study because the investigator artificially fixes the ratio of cases to controls ($m_1 / m_0$), completely distorting the marginal disease probability $R$. The odds ratio circumvent this limitation entirely.

The Rare Disease Assumption

How does the Odds Ratio relate to the Risk Ratio? Expanding the ratio:

When the disease is rare in both the exposed and unexposed populations—specifically, when $R_1 < 0.05$ and $R_0 < 0.05$—the proportions of non-cases approach unity: $1 - R_1 \approx 1$ and $1 - R_0 \approx 1$. Consequently:

Under the rare disease assumption, the odds ratio calculated from a case-control study serves as an extraordinarily accurate numerical proxy for the true population relative risk. However, if the outcome is common (e.g., childhood obesity with $R_1 = 0.40$ and $R_0 = 0.20$), the odds ratio severely exaggerates the relative risk:

Woolf's Logit Method for Odds Ratio Confidence Intervals

By the multivariate delta method applied to the natural logarithm of the cross-product ratio $\ln(\widehat{OR}) = \ln(a) - \ln(b) - \ln(c) + \ln(d)$, the asymptotic variance under a multinomial or product-binomial model is:

This elegant formulation, derived by Barnet Woolf (1955), yields the standard error:

A $95\%$ asymptotic confidence interval for the odds ratio is:

If any cell count is small, Haldane and Gart's correction adds $0.5$ to each cell to reduce finite-sample bias.

While relative measures ($RR, OR$) evaluate the strength of an etiologic association and are essential for biological causal inference, public health policymakers require measures on the additive scale to assess the actual human burden of disease that could be eliminated by eradicating an exposure.

Risk Difference (Attributable Risk)

The Risk Difference ($RD$), historically termed the Attributable Risk ($AR$) or Excess Risk, is the arithmetic difference between the cumulative incidence in the exposed group and the cumulative incidence in the unexposed group:

The risk difference represents the absolute excess probability of disease attributable specifically to the exposure among exposed individuals. Its units are the same as risk (a proportion per time interval), and it spans the interval $[-1.0, +1.0]$:

• $RD = 0$: Null hypothesis of no excess risk.
• $RD > 0$: The exposure inflicts an absolute excess disease burden.
• $RD < 0$: The exposure is protective, representing an absolute risk reduction ($ARR = R_0 - R_1$).

Because $\widehat{R}_1$ and $\widehat{R}_0$ are independent binomial proportions, the variance of the risk difference is simply the sum of their independent sampling variances:

The standard Wald 95% confidence interval is $\widehat{RD} \pm 1.96 \sqrt{\text{Var}(\widehat{RD})}$.

Attributable Fraction Among the Exposed (Etiologic Fraction)

The Attributable Fraction in the Exposed ($AF_e$), also termed the Attributable Proportion in the Exposed or Etiologic Fraction, measures the proportion of disease occurrences among exposed individuals that would be prevented if the exposure were entirely removed:

For example, if cigarette smoking produces a lung cancer relative risk of $RR = 10.0$ among heavy smokers, then $AF_e = (10 - 1) / 10 = 0.90$ ($90\%$). This means that ninety percent of lung cancer cases diagnosed in heavy smokers are directly attributable to their tobacco exposure, and would not have occurred had they remained lifelong non-smokers (assuming causal exchangeability).

Population Attributable Risk (PAR) and Population Attributable Fraction (PAF)

From the standpoint of municipal or national healthcare planning, exposed individuals represent only a subset of the total population. The Population Attributable Risk ($PAR$) measures the excess disease incidence observed in the total combined population ($R_{\text{total}}$) compared to the unexposed baseline:

Let $p_e = n_1 / N$ denote the prevalence of the exposure in the total population. By the Law of Total Probability, the total population incidence is a weighted average:

Subtracting $R_0$ yields:

The Population Attributable Fraction ($PAF$), originally derived by Morton L. Levin (1953), expresses this excess as a percentage of the total population disease burden:

Substituting $R_{\text{total}} = p_e R_1 + (1 - p_e) R_0$ and dividing numerator and denominator by $R_0$ yields Levin’s Classical Formula:

Alternatively, when working from case-control studies where exposure prevalence among diseased cases ($p_{e \mid D} = a / m_1$) is readily available, Miettinen’s Formula (1974) provides an identical calculation:

The PAF is arguably the most vital metric for setting public health priorities: an exposure with a modest relative risk ($RR = 1.3$) that is pervasive throughout the population ($p_e = 0.60$, such as physical inactivity) may produce a vastly larger PAF than an exposure with a gigantic relative risk ($RR = 20.0$) that is extraordinarily rare ($p_e = 0.001$, such as an industrial chemical).

The mathematical geometry of epidemiological effect measures reveals deep structural dualities between additive risk differences and multiplicative risk ratios. When risks are plotted in a Cartesian unit square with coordinates $(R_0, R_1) \in [0, 1]^2$, iso-relative risk contours represent straight rays emanating from the origin $(0, 0)$ with slope $RR$:

By contrast, iso-risk difference contours represent parallel lines with slope $+1$ and intercept $RD$:

Because human survival probabilities are bounded within $[0, 1]$, an exposure that produces a large relative risk (say, $RR = 5.0$) cannot mathematically maintain that constant multiplier across high baseline risks; if $R_0 = 0.30$, $RR = 5.0$ would imply an impossible risk $R_1 = 1.50$. Consequently, biological mechanisms frequently conform to additive limits, emphasizing why epidemiologists evaluate both additive and multiplicative scales when assessing public health interventions.

Score-Based Confidence Intervals

While Woolf's Wald-type interval for the odds ratio and the delta-method interval for the risk ratio are computationally simple, they rely on first-order Taylor series approximations that can exhibit poor coverage in small samples or when proportions approach $0$ or $1$. The Score Test Confidence Interval, originated by E. B. Wilson (1927) for a single proportion and extended by Miettinen and Nurminen (1985) for risk differences and risk ratios, test-inverts the efficient profile score statistic without relying on estimated standard errors in the denominator. For the Risk Difference, the score statistic enforces the null constraint $R_1 - R_0 = \Delta_0$ in the variance calculation:

where $\tilde{R}_1(\Delta_0)$ and $\tilde{R}_0(\Delta_0)$ are constrained maximum likelihood estimates under $H_0$. Inverting this quadratic score equation yields confidence intervals that never overshoot $[0, 1]$ and maintain exact nominal coverage across skewed contingency tables.

In clinical epidemiology, evidence-based medicine, and randomized controlled trial reporting, abstract proportions and logarithmic ratios can obscure direct clinical utility. To bridge this communication gap, Laupacis, Sackett, and Roberts (1988) introduced the Number Needed to Treat ($NNT$).

Number Needed to Treat (NNT)

When an experimental intervention reduces disease risk relative to a control or placebo ($R_1 < R_0$), the Absolute Risk Reduction is $ARR = R_0 - R_1 > 0$. The $NNT$ is defined as the mathematical reciprocal of the absolute risk reduction:

The $NNT$ represents the expected number of patients who must receive the medical therapy over a designated treatment duration in order to prevent exactly one additional adverse clinical outcome. For example, if a novel antihypertensive agent reduces 5-year stroke risk from $R_0 = 0.10$ ($10\%$) in the standard control group to $R_1 = 0.06$ ($6\%$) in the treatment group:

A clinician must treat 25 hypertensive patients for five years to prevent exactly one stroke.

Number Needed to Harm (NNH)

Conversely, when an exposure or pharmaceutical side effect increases the risk of an adverse event ($R_1 > R_0$), the absolute risk increase is $ARI = R_1 - R_0 > 0$. The Number Needed to Harm ($NNH$) is:

The $NNH$ represents the number of individuals exposed to the agent required for exactly one additional person to experience the harmful adverse outcome. Comparing $NNT$ for therapeutic benefit against $NNH$ for severe adverse drug reactions provides the fundamental quantitative basis for risk-benefit clinical decision analysis. When communicating with patients, presenting $NNT$ alongside baseline risks prevents framing biases that frequently arise when pharmaceuticals are marketed using relative risk reductions alone.

In observational studies, investigators frequently encounter the skepticism that an observed relative risk ($RR$) could be entirely explained away by an unmeasured confounder $U$ (such as genetic susceptibility or subtle lifestyle differences). To address this challenge quantitatively without requiring complex parametric simulations, Tyler VanderWeele and Peng Ding (2017) formulated the E-Value.

Mathematical Definition of the E-Value

The E-value is defined as the minimum strength of association, on the risk ratio scale, that an unmeasured confounder $U$ would need to have with both the exposure and the outcome, conditional on the measured covariates, to fully explain away an observed exposure-outcome association:

for an observed $RR > 1.0$. (If $RR < 1.0$, one first takes the reciprocal $1 / RR$).

For example, if an observational study reports an observed risk ratio of $RR = 3.0$ between an occupational solvent and pancreatic cancer:

To explain away this observed association, an unmeasured confounder would have to be associated with both solvent exposure and pancreatic cancer by a risk ratio of at least $5.45$-fold, above and beyond all measured covariates. If no known plausible risk factor exhibits an association of that magnitude, the causal finding is remarkably robust.

To evaluate whether the association could be explained away by chance plus unmeasured confounding, the E-value is also evaluated for the confidence interval limit closest to the null ($RR_{\text{lower}}$):

If $RR_{\text{lower}} = 2.0$, the E-value for the lower limit is $2.0 + \sqrt{2(1)} \approx 3.41$. Reporting the E-value alongside relative risks transforms vague qualitative discussions of "residual confounding" into transparent, quantitative sensitivity assessments.

A central objective of modern epidemiological research is determining whether an observed statistical association between an exposure $A$ and a health outcome $Y$ represents a true causal relationship. Classical observational epidemiology relied on associative risk ratios and odds ratios, which are vulnerable to confounding and selection biases. Contemporary causal inference formalizes causality using the Counterfactual (Potential Outcomes) Framework, formulated by Donald Rubin and extended to epidemiology by James Robins and Miguel Hernán.

1. Potential Outcomes Definition

Let $A \in \{0, 1\}$ denote a binary exposure (e.g., $A = 1$ if treated with an antihypertensive medication, $A = 0$ if untreated). For each individual $i$ in the study population, we define two Potential Outcomes:

• $Y_i(1)$: The health outcome that would be observed if individual $i$ were exposed to $A = 1$.
• $Y_i(0)$: The health outcome that would be observed if individual $i$ were exposed to $A = 0$.

The Individual Causal Effect of exposure $A$ for person $i$ is defined as the difference between their potential outcomes:

The Fundamental Problem of Causal Inference: For any given individual, we can observe $Y_i(1)$ (if they received exposure) or $Y_i(0)$ (if they remained unexposed), but never both simultaneously. The unobserved potential outcome is the counterfactual.

2. Population Causal Contrasts

Because individual causal effects are unobservable, epidemiology focuses on Average Causal Effects across populations:

1. Average Treatment Effect (ATE): The expected difference in potential outcomes across the entire target population:

2. Average Treatment Effect on the Treated (ATT): The causal effect among those who actually received the exposure:

3. Causal Risk Ratio ($RR_{\text{causal}}$):

3. The Three Core Identifiability Conditions

To estimate causal effects from observational data (where treatment assignment is non-random), three rigorous mathematical identifiability conditions must hold:

1. Exchangeability (Conditional Unconfoundedness): Conditional on a vector of measured pre-exposure baseline confounders $L$, the potential outcomes are conditionally independent of actual exposure assignment:

Under exchangeability, the counterfactual outcome among the exposed had they been unexposed equals the observed outcome among the unexposed within strata of $L$: $\mathbb{E}[Y(0) \mid A = 1, L] = \mathbb{E}[Y \mid A = 0, L]$.

2. Positivity: Every individual in the population within covariate strata $L$ must have a non-zero probability of receiving either exposure level:

If positivity is violated (structural or random zeros), causal contrast is mathematically undefined because counterfactual trajectories cannot be observed.

3. Consistency: The observed outcome $Y$ for an individual receiving treatment $A = a$ equals their potential outcome under that specific treatment:

Consistency requires well-defined interventions with no hidden versions of treatment and no interference between individuals (Stable Unit Treatment Value Assumption - SUTVA).

When these three conditions are satisfied, the causal effect is non-parametrically identifiable from observational data via G-computation or propensity score weighting: