Measures of Epidemiological Association

Overview

Epidemiological association measures compare disease or outcome frequency across exposure groups.

They include:

• risk difference;
• risk ratio;
• prevalence difference;
• prevalence ratio;
• odds ratio;
• incidence-rate difference;
• incidence-rate ratio;
• attributable fraction.

No single measure is universally best; measure selection follows the epidemiological question and design [2].

Absolute measures describe additional outcome burden. Relative measures describe proportional contrast. Clinical and public-health decisions usually benefit from both.

The study design determines which frequencies can be estimated and how confounding, selection and measurement affect interpretation [3]. A case–control study generally cannot estimate population risks directly from sampled case and control counts.

Why this matters

An exposure that doubles risk from 1% to 2% has:

• risk ratio 2.0;
• risk difference 1 percentage point.

An exposure that doubles risk from 20% to 40% has the same risk ratio but a 20-percentage-point risk difference.

Relative effects alone can conceal the absolute burden.

Odds ratios can be especially misleading when interpreted as risk ratios for common outcomes [1].

When to use this method

Use risk measures when:

• a defined population is initially outcome-free or otherwise at risk;
• follow-up is sufficient to classify whether the event occurred;
• and group risks are estimable.

Use prevalence measures in cross-sectional studies of current status.

Use incidence rates when:

• person-time varies;
• events occur over follow-up;
• and the event-count model is appropriate.

Use odds ratios commonly in:

• case–control studies;
• logistic regression;
• contingency-table analyses.

Do not describe an odds ratio as a risk ratio.

Research question and variables

For a cohort two-by-two table:

The exposed and unexposed risks are:

$$R_E=\frac{a}{a+b}$$ $$R_U=\frac{c}{c+d}$$

Data structure

Simulated cohort table

simulated data

Table 1. Outcome by exposure

Exposure	Outcome	No outcome	Total
Exposed	60	240	300
Unexposed	30	270	300

Note: The table represents a simulated cohort with risks estimable from each exposure group.

Statistical assumptions

How the method works

Risk difference

Risk ratio

Odds ratio

Incidence-rate ratio

If $D_E$ and $D_U$ are event counts and $PT_E$ and $PT_U$ are person-time:

Attributable fraction among the exposed

Under a causal interpretation:

$$AF_E = \frac{R_E-R_U}{R_E} = \frac{RR-1}{RR}$$

This quantity describes the fraction of exposed-group risk attributable to exposure only if the comparison represents the counterfactual risk without exposure.

Worked example

Incidence-rate example

The exposed rate is:

$$45/1200=0.0375$$

or 3.75 events per 100 person-years.

The unexposed rate is:

$$25/1500=0.0167$$

or 1.67 events per 100 person-years.

The incidence-rate ratio is:

$$IRR=2.25$$

with approximate 95% CI 1.38 to 3.67.

Reproducible R analysis

a <- 60
b <- 240
c <- 30
d <- 270

risk_exposed <- a / (a + b)
risk_unexposed <- c / (c + d)

risk_difference <- risk_exposed - risk_unexposed
risk_ratio <- risk_exposed / risk_unexposed
odds_ratio <- (a * d) / (b * c)

rd_se <- sqrt(
  risk_exposed * (1 - risk_exposed) / (a + b) +
    risk_unexposed * (1 - risk_unexposed) / (c + d)
)
rd_ci <- risk_difference + c(-1, 1) * 1.96 * rd_se

rr_se <- sqrt(
  1 / a - 1 / (a + b) +
    1 / c - 1 / (c + d)
)
rr_ci <- exp(log(risk_ratio) + c(-1, 1) * 1.96 * rr_se)

or_se <- sqrt(1 / a + 1 / b + 1 / c + 1 / d)
or_ci <- exp(log(odds_ratio) + c(-1, 1) * 1.96 * or_se)

nnh <- 1 / risk_difference
af_exposed <- (risk_ratio - 1) / risk_ratio

events_exposed <- 45
pt_exposed <- 1200
events_unexposed <- 25
pt_unexposed <- 1500

rate_exposed <- events_exposed / pt_exposed
rate_unexposed <- events_unexposed / pt_unexposed
rate_ratio <- rate_exposed / rate_unexposed

irr_se <- sqrt(
  1 / events_exposed +
    1 / events_unexposed
)
irr_ci <- exp(log(rate_ratio) + c(-1, 1) * 1.96 * irr_se)

output <- data.frame(
  risk_exposed = risk_exposed,
  risk_unexposed = risk_unexposed,
  risk_difference = risk_difference,
  rd_lower = rd_ci[1],
  rd_upper = rd_ci[2],
  risk_ratio = risk_ratio,
  rr_lower = rr_ci[1],
  rr_upper = rr_ci[2],
  odds_ratio = odds_ratio,
  or_lower = or_ci[1],
  or_upper = or_ci[2],
  nnh = nnh,
  af_exposed = af_exposed,
  rate_ratio = rate_ratio,
  irr_lower = irr_ci[1],
  irr_upper = irr_ci[2]
)

round(output, 3)

Cleaned R output

 risk_exposed risk_unexposed risk_difference rd_lower rd_upper risk_ratio rr_lower rr_upper odds_ratio or_lower or_upper nnh af_exposed rate_ratio irr_lower irr_upper
          0.2            0.1             0.1    0.043    0.157        2.0    1.330    3.007       2.25    1.404    3.606  10        0.5       2.25     1.380     3.668

Interpreting the output

Presenting the results

Table 2. Epidemiological association measures

Note: Risk and rate examples use different simulated event counts and denominators.

Suggested reporting

Non-collapsibility and transportability MMed extension

Odds ratios can change after covariate adjustment even when the covariate is not a confounder because the odds ratio is non-collapsible.

Risk differences and risk ratios also vary across populations when baseline risk or effect modification differs.

No effect measure is automatically transportable. Report the target population and clinically relevant absolute risks.

Common mistakes

Knowledge check

What is the difference between risk difference and risk ratio?

Answer and reasoning

Risk difference measures absolute excess probability, while risk ratio compares probabilities multiplicatively.

Why is the odds ratio 2.25 when the risk ratio is 2.00?

Answer and reasoning

Odds and risks use different denominators. With a common outcome, the odds ratio often lies farther from one than the risk ratio.

Can a case–control study estimate the sampled case risk?

Answer and reasoning

No. The case and control numbers are selected by design, so sampled proportions do not represent population risks. The odds ratio remains estimable under suitable assumptions.

What does an incidence-rate ratio compare?

Answer and reasoning

It compares event counts per unit of person-time. It does not directly compare cumulative probabilities over a fixed period.

What is required before interpreting an attributable fraction causally?

Answer and reasoning

The exposed and comparison outcome frequencies must represent suitable counterfactual risks after accounting for confounding, selection, measurement and other biases.

Key take-home messages

References

1. Davies HTO, Crombie IK, Tavakoli M. When can odds ratios mislead? BMJ. 1998;316(7136):989–991. https://doi.org/10.1136/bmj.316.7136.989
2. Rothman KJ. Epidemiology: an introduction. 2nd ed. New York: Oxford University Press; 2012.
3. Lash TL, VanderWeele TJ, Haneuse S, Rothman KJ. Modern epidemiology. 4th ed. Philadelphia: Wolters Kluwer; 2021.
4. Agresti A. Categorical data analysis. 3rd ed. Hoboken (NJ): Wiley; 2013.