Measures of Association, Impact and Causal Inference

Introduction

Epidemiology does more than describe how frequently disease occurs. It asks whether an exposure is associated with an outcome, how large that association is, how much disease could be prevented by changing the exposure, and whether the observed relationship is plausibly causal. These questions are central to health-care management because decisions about screening, prevention, occupational safety, resource allocation and service design depend on interpreting evidence correctly.

An association is a statistical relationship between an exposure and an outcome. Causation is a stronger claim: it means that changing the exposure would change the outcome, under specified conditions. Confusing association with causation can lead to ineffective policies, inappropriate clinical recommendations and misallocation of limited health resources.

This chapter develops measures of association, measures of public-health impact and practical approaches to causal inference. It integrates epidemiological designs, counterfactual reasoning, causal criteria and directed acyclic graphs (DAGs).

Learning objectives

By the end of this chapter, the reader should be able to:

1. Construct and interpret two-by-two epidemiological tables.
2. Calculate risk ratios, odds ratios, rate ratios and risk differences.
3. Calculate attributable fractions and population-level impact measures.
4. Distinguish measures appropriate for cohort, case-control and person-time studies.
5. Assess temporality, dose-response patterns and alternative explanations.
6. Use causal criteria, DAGs and counterfactual reasoning to evaluate causality.
7. Translate epidemiological results into health-care and managerial decisions while recognizing uncertainty and bias.

1. The basic epidemiological table

Consider an exposure such as hypertension treatment status and an outcome such as stroke. A two-by-two table is written as follows:

The risk among exposed individuals is:

The risk among unexposed individuals is:

Here, risk is a cumulative probability over a specified follow-up period. It has no physical unit, although it may be expressed as a proportion or percentage. The notation R₁ and R₀ refers to exposed and unexposed groups, respectively.

These measures require a defined population, a clearly specified exposure, a valid outcome definition and an appropriate observation period. If follow-up times differ substantially, incidence rates rather than risks may be preferable.

2. Measures of association

2.1 Risk ratio

The risk ratio (RR), also called the relative risk, compares cumulative risk in the exposed group with cumulative risk in the unexposed group:

Interpretation:

• RR = 1: no relative difference in risk.
• RR > 1: exposure is associated with higher risk.
• RR < 1: exposure is associated with lower risk.

An RR of 2.0 means that risk in the exposed group is twice the risk in the unexposed group. It does not mean that two times as many exposed people necessarily develop disease, nor does it reveal the absolute number of excess cases.

The risk ratio is most naturally estimated in cohort studies and randomized trials when participants can be followed over a common period. It is generally not directly estimable from a conventional case-control study because the investigator fixes the number of cases and controls rather than observing disease risk in the source population.

2.2 Odds ratio

Odds are defined as the probability of an event divided by the probability of no event. The odds among exposed individuals are a/b, and those among unexposed individuals are c/d. The odds ratio (OR) is therefore:

An OR of 1 indicates equal odds. An OR above 1 indicates higher odds among the exposed, and an OR below 1 indicates lower odds.

The odds ratio is the natural measure of association in case-control studies. It can also be estimated using logistic regression. When the outcome is rare, the OR approximates the RR because, for small p, odds p/(1−p) are close to p. When the outcome is common, the OR can substantially exaggerate the apparent magnitude of a positive association compared with the risk ratio.

2.3 Rate ratio

When individuals contribute different amounts of observation time, incidence rates are appropriate. Let D₁ and D₀ denote the numbers of new events in exposed and unexposed groups, and PT₁ and PT₀ their respective person-time denominators:

The incidence rate ratio is:

Incidence rates have units such as cases per 1,000 person-years. The rate ratio is dimensionless because the units cancel. It is especially useful when follow-up varies, populations are dynamic, or events can recur, provided the statistical model and event definition are appropriate.

2.4 Risk difference

The risk difference (RD), also called the attributable risk among the exposed, measures the absolute excess risk associated with exposure:

RD is expressed in risk units, such as 0.04 or 4 percentage points. A positive value indicates excess risk; a negative value indicates a potentially protective association.

Relative and absolute measures answer different questions. A small baseline risk can produce a large RR but a small RD. Conversely, a modest RR may correspond to a substantial number of preventable cases when baseline risk is high. Health-care managers should generally report both.

3. Worked example: interpreting association and impact

Illustrative cohort study

Suppose 1,000 workers exposed to a respiratory irritant and 2,000 unexposed workers are followed for one year. The results are:

Risk among exposed:

Risk among unexposed:

Risk ratio:

Risk difference:

Thus, exposed workers experienced three times the risk, or an excess of 8 percentage points over one year. The excess corresponds to 80 additional cases per 1,000 exposed workers, assuming the comparison is causal and the unexposed risk represents the relevant counterfactual risk.

Odds ratio:

The OR is larger than the RR because disease is not rare. Reporting the OR as if it were a risk ratio would overstate the relative risk.

4. Measures of attributable and population impact

4.1 Attributable fraction among the exposed

The attributable fraction among the exposed (AFₑ) is the proportion of disease among exposed individuals that is attributable to the exposure, under a causal interpretation:

For the illustrative study:

Approximately 66.7% of disease among exposed workers is attributable to the exposure, assuming no important bias or uncontrolled confounding and that the unexposed risk represents the risk that exposed workers would have experienced without exposure.

AFₑ is useful for estimating the preventive potential of eliminating exposure among exposed people. It is not the percentage of all cases in the population attributable to exposure.

4.2 Population attributable risk

Let Rₚ be the risk in the total population. The population attributable risk (PAR) is:

It estimates the excess population risk associated with exposure, under causal assumptions. The population attributable fraction (PAF) is:

If Pₑ is the prevalence of exposure in the population and the causal risk ratio is applicable across exposure groups, then:

and:

This formula assumes a binary exposure, a correctly specified RR and a population in which the exposure distribution and effect estimate are relevant.

Population impact depends on both effect size and exposure prevalence. A moderately harmful, highly prevalent exposure may create more preventable disease than a strongly harmful but rare exposure.

4.3 Prevented fraction and number needed to treat

For a protective exposure or intervention, the prevented fraction among the exposed may be expressed as:

when RR is below 1. For interventions, the absolute risk reduction is:

and, when ARR is positive, the number needed to treat or implement (NNT) is:

ARR must be expressed as a proportion rather than a percentage. NNT depends on follow-up duration and baseline risk; it should not be presented without those conditions.

5. Choosing the appropriate measure

Confidence intervals should accompany point estimates. A 95% confidence interval describes sampling uncertainty under the model and study assumptions; it does not guarantee that the interval contains the true causal effect. Statistical significance does not establish practical importance or causality.

6. From association to causation

Causal inference asks a counterfactual question: what would have happened to the same individuals, at the same time, had they experienced a different exposure? Because one person cannot simultaneously be observed exposed and unexposed, the causal effect is inferred from comparisons designed to approximate this impossible contrast.

For person i, let Yᵢ(1) be the outcome that would occur under exposure and Yᵢ(0) the outcome under no exposure. The individual causal effect is:

Only one of these potential outcomes is observed for each person. A population average causal effect can be expressed as:

where E denotes the population mean or expected outcome. For binary outcomes, analogous causal contrasts include a risk ratio, risk difference or odds ratio comparing the two potential-outcome distributions.

Randomization makes exposure independent of potential outcomes, in expectation, so that observed group differences can estimate causal effects. In observational studies, exchangeability must instead be supported through design, measurement and adjustment for confounding.

6.1 The causal pathway

A useful causal model distinguishes:

• Exposure: the factor being evaluated.
• Outcome: the health event or state of interest.
• Confounder: a common cause of exposure and outcome that is not on the causal pathway.
• Mediator: an intermediate variable through which exposure affects outcome.
• Effect modifier: a variable at which the effect differs across subgroups.
• Collider: a variable caused by both exposure and another variable; conditioning on it can create bias.

6.2 Causal criteria

The Bradford Hill considerations remain useful as structured prompts, although they are not a mechanical checklist or proof of causation:

1. Temporality: exposure must precede outcome.
2. Strength: large associations may be less readily explained by bias, although small effects can be causal.
3. Consistency: similar findings across studies, settings and methods strengthen confidence.
4. Specificity: a particular exposure-outcome pairing may support causality, but lack of specificity does not disprove it because most causes have multiple effects.
5. Biological gradient: increasing exposure associated with increasing risk supports a dose-response pattern, but non-linear relationships are possible.
6. Plausibility and coherence: the finding should fit biological and wider scientific knowledge.
7. Experiment: intervention evidence is particularly informative.
8. Analogy: comparable causal relationships may increase plausibility, but analogy is weak evidence alone.

Temporality is indispensable. The other considerations increase or decrease confidence but cannot substitute for correct temporal ordering.

7. Directed acyclic graphs

A DAG represents causal assumptions using nodes and directed arrows. It helps identify variables that should be adjusted for and variables that should not.

Figure: A DAG showing confounding through C and mediation through M.

The path X ← C → Y is a backdoor path. If C is measured adequately and is not itself a mediator or collider, adjustment for C can block this path. Adjusting for M estimates a direct effect rather than the total effect and may remove part of the effect of X. Therefore, the correct adjustment set depends on the causal question.

A collider should generally not be adjusted for. For example, if both exposure and an unmeasured factor cause hospital admission, conditioning on admission can create an artificial association between exposure and the factor.

8. Confounding, bias and effect modification

Confounding occurs when the observed exposure-outcome association differs from the association that would be observed after removing the influence of a common cause. Important controls include restriction, randomization, matching, stratification, standardization and multivariable regression. Statistical adjustment cannot repair poor measurement, missing essential variables or a fundamentally incorrect causal model.

Selection bias arises when inclusion, retention or analysis depends on variables related to exposure and outcome. Information bias arises from inaccurate measurement, including recall bias, interviewer bias and outcome misclassification. Differential misclassification can bias results in either direction.

Effect modification is not a bias. It occurs when the effect differs across levels of another variable, such as age, sex, genetic susceptibility or baseline disease severity. It should be reported rather than automatically controlled away. Researchers should distinguish interaction on additive scales, such as risk differences, from interaction on multiplicative scales, such as risk ratios; conclusions can differ by scale.

9. A practical causal-inference workflow

A defensible analysis usually follows these steps:

1. Define the causal question, target population, exposure, intervention, outcome, time zero and estimand.
2. Specify a substantive causal model before examining results.
3. Establish temporality and prevent immortal-time bias by defining eligibility and follow-up correctly.
4. Identify confounders using subject knowledge and DAGs, not only statistical significance tests.
5. Measure exposure, outcome and covariates with valid, consistent methods.
6. Select a design and analysis appropriate to the estimand, such as standardization, regression, propensity scores, inverse-probability weighting or g-methods.
7. Examine missing data, selection mechanisms, measurement error and positivity.
8. Conduct sensitivity analyses for unmeasured confounding and alternative outcome definitions.
9. Report absolute and relative effects with uncertainty and explain the target of inference.
10. Assess transportability: determine whether the findings apply to the population and health system in which action is contemplated.

Positivity requires that each relevant subgroup has a non-zero probability of receiving each exposure level. If no older adults receive an intervention in the data, the effect in older adults cannot be learned reliably from that dataset without strong extrapolation.

10. Applied case: reducing hospital-acquired venous thromboembolism

Background

This illustrative case concerns a hospital network evaluating whether a computerized venous-thromboembolism (VTE) alert reduces hospital-acquired VTE among eligible inpatients.

Problem

A before-and-after comparison shows VTE risk falling from 8 per 1,000 admissions to 5 per 1,000 after implementation. Leaders must decide whether the alert caused the reduction and whether expansion is justified.

Analysis

The crude risk difference is:

Thus, there were 3 fewer VTE events per 1,000 admissions. The risk ratio is:

The relative risk reduction is 1 − 0.625 = 0.375, or 37.5%. The apparent number needed to implement is approximately:

This means approximately 333 admissions would need to occur under the intervention, under the observed conditions, to prevent one event. It is not a universal value: it depends on baseline risk, follow-up and implementation fidelity.

However, the before-and-after design is vulnerable to confounding by secular trends. During the same period, the hospital introduced earlier mobilization, changed coding procedures and experienced a reduction in average length of stay. A DAG would include calendar time as a cause of both alert exposure and VTE risk, and perhaps length of stay as a mediator or competing process. A stronger design would compare intervention hospitals with similar hospitals that adopted the alert later, using a difference-in-differences analysis if the parallel-trends assumption is plausible. Patient-level adjustment and interrupted time-series analysis could provide additional evidence.

Approach and solution

The management team should define the outcome and time zero consistently, audit alert delivery and clinician response, monitor false alerts and examine effects by baseline risk. They should report both absolute and relative effects, confidence intervals, unintended consequences and implementation costs. If the alert reduces VTE but substantially increases unnecessary anticoagulation, the net clinical benefit must be reassessed.

Outcome and learning points

The observed reduction is promising but is not automatically causal. Temporality is satisfied, but alternative explanations remain. A robust decision combines effect estimates, uncertainty, patient safety, cost, feasibility and equity. The case demonstrates why causal inference is a management discipline as well as a statistical exercise.

11. Ethical, governance and contemporary issues

Causal analyses in health care use sensitive information and may affect access to services. Governance requires lawful data use, confidentiality, secure linkage, transparent algorithms and proportionality. Automated risk models can reproduce inequities when historical data reflect unequal diagnosis, treatment or access. Analysts should examine performance and treatment effects across demographic and socioeconomic groups rather than assume that a single average effect is equitable.

Machine-learning systems may improve prediction but prediction is not causation. A model can accurately identify high-risk patients without identifying which intervention will reduce their risk. Causal machine learning, target-trial emulation and routinely collected electronic health-record analyses are developing rapidly, but they remain dependent on valid temporal structure, measurement, exchangeability, positivity and well-defined interventions.

Best practice includes preregistering major analyses where feasible, preserving an auditable analysis trail, reporting deviations from the protocol, sharing code when permitted, distinguishing exploratory from confirmatory findings and involving clinicians, patients and affected communities in defining meaningful outcomes.

Challenges and limitations

No single measure establishes causality. Relative measures may appear impressive when baseline risk is low, while absolute measures may obscure important proportional changes. Rare outcomes can make estimates unstable. Odds ratios are easily misinterpreted for common outcomes. Rate ratios depend on the validity of person-time denominators and assumptions about the event process. Attributable fractions require a causal interpretation and may be misleading when exposure cannot realistically be removed.

Observational causal inference is limited by residual confounding, exposure changes over time, treatment switching, missing data, selection, measurement error and violations of positivity. Even randomized trials may have non-adherence, contamination, loss to follow-up and limited generalizability. Consequently, conclusions should be calibrated to the design and assumptions rather than stated with unwarranted certainty.

Chapter Summary

Measures of association quantify relationships between exposure and outcome. The risk ratio compares cumulative risks, the odds ratio compares odds and is particularly natural for case-control studies, the rate ratio compares incidence rates, and the risk difference quantifies absolute excess or reduced risk. Relative and absolute measures should be reported together because they support different clinical and managerial decisions.

Attributable fractions translate associations into potential preventive impact. The attributable fraction among the exposed concerns disease among exposed people, whereas the population attributable fraction also depends on exposure prevalence. These measures are causal interpretations, not merely arithmetic transformations.

Causal inference requires more than statistical association. Temporality, confounding, selection, measurement, dose-response, consistency, biological knowledge and intervention evidence must be considered. Counterfactual reasoning defines causal effects as contrasts between potential outcomes. DAGs clarify backdoor paths, mediators, colliders and appropriate adjustment sets. Sound health-care decisions combine quantitative estimates with uncertainty, feasibility, safety, equity and applicability to the target population.

Key Terms

Risk: Probability of an outcome over a specified period.

Incidence rate: New events divided by accumulated person-time at risk.

Risk ratio: Ratio of cumulative risk in exposed and unexposed groups.

Odds ratio: Ratio of odds of outcome in exposed and unexposed groups.

Rate ratio: Ratio of incidence rates.

Risk difference: Difference between exposed and unexposed risks.

Attributable fraction among the exposed: Proportion of exposed-group disease attributable to exposure under causal assumptions.

Population attributable fraction: Proportion of population disease attributable to exposure under causal assumptions.

Confounding: Distortion caused by a common cause of exposure and outcome.

Mediator: Intermediate variable on a causal pathway.

Effect modification: Variation in an effect across levels of another variable.

Collider: Variable caused by two or more variables, whose conditioning may induce bias.

Temporality: Requirement that the cause precede the effect.

Dose-response: Change in outcome frequency across exposure levels.

Counterfactual: Potential outcome under an exposure condition that may not have been observed.

Average causal effect: Population contrast between potential outcomes under two exposure conditions.

Directed acyclic graph: Graphical representation of assumed causal relationships without directed cycles.

Exchangeability: Condition that exposed and unexposed groups are comparable with respect to potential outcomes, conditional on relevant variables.

Positivity: Presence of each exposure option within relevant covariate strata.

Selection bias: Distortion caused by the mechanism determining inclusion, retention or analysis.

Review Questions

Short Answer Questions

1. Distinguish a risk ratio from a risk difference.
2. Why is the odds ratio commonly used in case-control studies?
3. What does temporality mean in causal inference?
4. Define a mediator and a confounder.
5. Why should absolute measures accompany relative measures?

Long Answer Questions

1. Explain how risk ratio, odds ratio and rate ratio are calculated, interpreted and selected according to study design.
2. Discuss attributable fractions and population impact, including the role of exposure prevalence.
3. Explain how a DAG can guide confounder adjustment and prevent inappropriate adjustment for mediators or colliders.
4. Critically evaluate the Bradford Hill considerations as a framework for causal inference.
5. Describe a complete workflow for conducting and reporting an observational causal analysis in health care.

Analytical / Application Questions

1. In a cohort, 60 of 600 exposed people and 30 of 900 unexposed people develop disease. Calculate the risk ratio, risk difference and attributable fraction among the exposed, and interpret each.
2. A screening intervention reduces five-year risk from 12% to 9%. Calculate the absolute risk reduction, relative risk and approximate NNT. Explain why all three measures matter to a health-care manager.
3. A hospital observes lower mortality after introducing an electronic alert, but the alert was adopted first in hospitals with better staffing and shorter delays. Identify likely confounding pathways and propose a stronger evaluation design.

Multiple Choice Questions with Answers

Question 1

In a cohort study, risk is 0.12 among exposed participants and 0.04 among unexposed participants. What is the risk ratio?

A. 0.33

B. 2.0

C. 3.0

D. 8.0

Correct Answer: C. 3.0

Question 2

Using a two-by-two table with cells a, b, c and d, which expression is the odds ratio?

A. (a+b)/(c+d)

B. ad/bc

C. a/(a+b) − c/(c+d)

D. (a+c)/(b+d)

Correct Answer: B. ad/bc

Question 3

If the risk ratio is 0.70 for a preventive intervention, what is the relative risk reduction?

A. 0.30

B. 0.70

C. 1.30

D. 70% increase

Correct Answer: A. 0.30

Question 4

Which measure is most directly appropriate for comparing disease occurrence when follow-up time differs substantially between groups?

A. Risk ratio

B. Odds ratio only

C. Rate ratio

D. Population attributable fraction

Correct Answer: C. Rate ratio

Question 5

An exposure has risk 0.20 among exposed people and 0.05 among unexposed people. What is the attributable fraction among the exposed?

A. 25%

B. 50%

C. 75%

D. 95%

Correct Answer: C. 75%

Question 6

Which condition is indispensable for a causal interpretation of an exposure-outcome relationship?

A. Specificity

B. Temporality

C. A large odds ratio

D. A linear dose-response relationship

Correct Answer: B. Temporality

Question 7

In a DAG, a variable caused by both the exposure and an independent cause of the outcome is called a:

A. Mediator

B. Confounder

C. Collider

D. Effect estimate

Correct Answer: C. Collider

Question 8

If an intervention reduces risk from 0.10 to 0.06, what is the approximate number needed to treat?

A. 4

B. 10

C. 25

D. 60

Correct Answer: C. 25

Question 9

Why can an odds ratio substantially overstate a risk ratio when the outcome is common?

A. Odds and risk are identical for common outcomes

B. Odds increase more rapidly than risk as probability becomes large

C. Risk ratios cannot be calculated in cohorts

D. Odds ratios always adjust for confounding

Correct Answer: B. Odds increase more rapidly than risk as probability becomes large

Question 10

Which statement about effect modification is correct?

A. It is always a form of selection bias

B. It should always be eliminated by regression adjustment

C. It indicates that the effect differs across levels of another variable

D. It occurs only in randomized trials

Correct Answer: C. It indicates that the effect differs across levels of another variable