Risk Ratios, Odds Ratios, and Hazard Ratios: Definition, Calculation, Examples

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Contents

• Foundational Concepts
• The Three Core Ratio Measures
• Step-by-Step Calculations: A Worked Example
• Absolute Effect Measures: Risk Difference, ARR, RRR, NNT, and NNH
• Comprehensive Comparison of Effect Measures
• Which Measure to Use: Study Design Matters
• The OR-as-RR Error: A Common Pitfall
• Controlling for Confounders: Regression Models
• Best Practices for Reporting Statistical Measures
• Real-World Example: Smoking and Lung Cancer Mortality
• Glossary of Key Terms
• Frequently Asked Questions

In biomedical research and clinical literature, the terms risk, odds, and rate appear constantly  yet they are frequently misunderstood or used interchangeably. In reality, each describes a distinct mathematical relationship, and choosing the wrong measure can lead to misleading conclusions. This guide explains the foundational concepts, walks through step-by-step calculations with numerical examples, compares all major effect measures, and provides practical guidance on which measure to use for different study designs.

Foundational Concepts

Before diving into ratios, it is essential to understand the building blocks.

Probability, Risk, and Odds

These three terms are related but mathematically distinct:

• Probability is the fraction of times an outcome is expected to occur across many trials. It ranges from 0 to 1 (0% to 100%).
• Risk is the probability of an adverse outcome in a defined population over a specified time period. Risk is dimensionless and confined to values between 0 and 1. Example: if 30 out of 100 patients develop an infection, the risk is 0.30 or 30%.
• Odds express the ratio of the probability of an event occurring to the probability of it not occurring. If the probability of an event is p, then: Odds = p / (1 − p). Using the example above: Odds = 0.30 / 0.70 = 0.43. Odds have no upper bound and can take any non-negative value.

Key Distinction: Risk vs. Odds

When disease prevalence is low (under approximately 10%), risk and odds are numerically similar and odds ratios closely approximate risk ratios. As prevalence rises, the two diverge — and the odds ratio will always exaggerate the effect size compared to the risk ratio. This is one of the most important practical points in biostatistics.

Feature	Risk (Probability)	Odds
Definition	Events / Total people at risk	Events / Non-events
Range	0 to 1 (0% to 100%)	0 to infinity
Example (30 of 100 infected)	30/100 = 0.30	30/70 = 0.43
When they are similar	—	When disease prevalence < 10%

Rate vs. Risk

A rate expresses change over time. For example, the incidence rate of a disease equals the number of new cases divided by the total person-time at risk. Unlike risk, a rate has a time dimension and can exceed 1.0. A risk is a cumulative probability over a fixed follow-up period, whereas a rate is an instantaneous measure expressed per unit time (e.g., per 100 person-years).

Risk Factors

A risk factor is any variable that increases the probability of developing a disease or experiencing an adverse outcome. Examples include smoking (lung cancer), obesity (type 2 diabetes), and radiation exposure (certain cancers). Identifying and quantifying risk factors is the cornerstone of epidemiology and preventive medicine.

The Three Core Ratio Measures

Risk Ratio (Relative Risk)

The risk ratio (RR), also called relative risk, compares the probability of an outcome in an exposed group to the probability in an unexposed (control) group.

Formula: 

RR = Risk in exposed group / Risk in unexposed group

Interpretation:

• RR = 1: No association; both groups have equal risk.
• RR > 1: The exposure is associated with higher risk (harmful).
• RR < 1: The exposure is associated with lower risk (protective).

Odds Ratio

The odds ratio (OR) compares the odds of an outcome in an exposed group to the odds in an unexposed group.

Formula: 

OR = Odds in exposed group / Odds in unexposed group  =  (a/b) / (c/d)  =  ad / bc

where a = exposed with outcome, b = exposed without outcome, c = unexposed with outcome, d = unexposed without outcome.

Interpretation:

• OR = 1: No association.
• OR > 1: Positive association between exposure and outcome.
• OR < 1: Negative (protective) association.

Hazard Ratio

The hazard ratio (HR) is used in time-to-event (survival) analyses. Unlike RR or OR — which are cumulative over a study period — the HR compares the instantaneous rate of events between groups at any given moment in time. It is derived from survival analysis techniques such as the Cox proportional hazards model.

Formula: 

HR = Hazard rate in intervention group / Hazard rate in control group

Interpretation:

• HR = 1: Both groups experience events at the same rate at any given time.
• HR > 1: The intervention group experiences events at a higher rate at any point in time.
• HR < 1: The intervention group experiences events at a lower rate at any point in time (protective effect).

A key feature of the HR: two trials with the same final RR can have very different HRs if one group experienced earlier events. Survival analysis captures this temporal nuance.

Step-by-Step Calculations: A Worked Example

The following example uses a hypothetical influenza vaccine trial to show how RR and OR are calculated from a 2×2 contingency table.

The 2×2 Contingency Table

Suppose 48 vaccinated and 40 unvaccinated individuals are followed to see who becomes infected:

	Infected	Not Infected	Total
Vaccinated (exposed)	a = 16	b = 32	48
Not vaccinated (unexposed)	c = 30	d = 10	40
Total	46	42	88

Calculating Risk Ratio

• Risk in vaccinated group = a / (a + b) = 16 / 48 = 0.33 (33%)
• Risk in unvaccinated group = c / (c + d) = 30 / 40 = 0.75 (75%)
• RR = 0.33 / 0.75 = 0.44

Interpretation:

Vaccinated individuals are 0.44 times as likely to become infected compared to unvaccinated individuals — the vaccine reduces the risk of infection by about 56%.

Calculating Odds Ratio

• Odds of infection (vaccinated) = a / b = 16 / 32 = 0.50
• Odds of infection (unvaccinated) = c / d = 30 / 10 = 3.0
• OR = 0.50 / 3.0 = 0.17  (equivalently: ad / bc = 16×10 / 32×30 = 160/960 = 0.17)

Interpretation:

The odds of infection among vaccinated individuals are 0.17 times the odds among unvaccinated individuals.

Why RR and OR Differ in This Example

The infection rate among unvaccinated individuals is 75%, far above the 10% threshold. This is why the OR (0.17) is considerably smaller than the RR (0.44). When the outcome is common, OR exaggerates the effect compared to RR. The relationship between OR and RR is:

OR = RR × [(1 − risk in unexposed) / (1 − risk in exposed)]

When both risks are small (< 10%), this multiplier approaches 1 and OR ≈ RR.

Absolute Effect Measures: Risk Difference, ARR, RRR, NNT, and NNH

Ratio measures (RR, OR, HR) describe relative effects. Absolute measures describe the actual magnitude of benefit or harm — and are often more relevant for clinical decision-making.

Risk Difference (Absolute Risk Reduction)

Formula:  ARR = Risk in control group − Risk in treatment group

Using the vaccine example: ARR = 0.75 − 0.33 = 0.42 (42 percentage points). This means that for every 100 people vaccinated, 42 infections are prevented.

A large RR can accompany a trivially small ARR. For example, an RR of 2.0 (doubled risk) sounds alarming, but if the baseline risk is 0.1%, doubling it to 0.2% is clinically insignificant. This is why absolute measures should always accompany relative ones.

Relative Risk Reduction

Formula:  RRR = 1 − RR  (or equivalently, ARR / Risk in control group)

Using the vaccine example: RRR = 1 − 0.44 = 0.56 (56%). The vaccine reduces relative risk by 56%.

Caution: RRR reported without ARR can be misleading. A drug that reduces risk from 0.2% to 0.1% has an RRR of 50% — impressive-sounding — but an ARR of only 0.1%, meaning 1,000 people must be treated to prevent one event.

Number Needed to Treat (NNT)

Formula:  NNT = 1 / ARR

NNT answers: how many patients must receive the treatment for one additional patient to benefit? Lower NNT = more effective treatment.

Using the vaccine example: NNT = 1 / 0.42 = 2.4 (approximately 2–3 people must be vaccinated to prevent one infection).

Number Needed to Harm (NNH)

Formula:  NNH = 1 / Absolute Risk Increase

NNH answers: how many patients must be exposed to a risk factor or treatment for one additional patient to be harmed? Higher NNH = safer intervention.

NNT and NNH together provide a balanced picture of benefit versus risk, and are widely used in evidence-based medicine to communicate clinical significance.

Survival Ratio

The survival ratio is the complement of the risk ratio. It compares the probability of not experiencing the outcome between groups:

Survival Ratio = (1 − Risk in exposed) / (1 − Risk in unexposed)

In the vaccine example: (1 − 0.33) / (1 − 0.75) = 0.67 / 0.25 = 2.68. Unvaccinated individuals are 2.68 times as likely to survive infection-free, compared to vaccinated individuals but wait, the direction flips: vaccinated individuals are 2.68 times as likely to remain infection-free. This framing can be intuitive when communicating protective effects.

Comprehensive Comparison of Effect Measures

	Risk Ratio (RR)	Odds Ratio (OR)	Hazard Ratio (HR)	Risk Difference (ARR)	NNT / NNH
Also called	Relative risk	—	—	Absolute risk reduction	—
Calculation	Risk_exp / Risk_ctrl	(a/b) / (c/d)	Survival analysis (Cox model)	Risk_ctrl − Risk_exp	1 / ARR
Null value	1	1	1	0	∞ (no effect)
Range	0 to ∞	0 to ∞	0 to ∞	−1 to +1	Any positive number
Type of effect	Relative	Relative	Relative (time-varying)	Absolute	Absolute
Best study design	Cohort / RCT	Case-control, logistic regression	RCT, cohort (time-to-event)	Cohort / RCT	Cohort / RCT
Time dimension	No	No	Yes	No	No
Accounts for censoring	No	No	Yes	No	No
When OR ≈ RR	—	When outcome prevalence < 10%	—	—	—

Which Measure to Use: Study Design Matters

The choice of effect measure is not arbitrary — it is constrained by study design.

Cohort Studies and Randomised Controlled Trials

In cohort studies and clinical trials, participants are not selected based on outcome status. Both RR and OR can be calculated. RR is generally preferred because it is more directly interpretable.

• Use RR when the outcome is binary and follow-up time is equal or short.
• Use HR when follow-up times vary across participants, or when censoring occurs (i.e., survival analysis is needed).
• OR from logistic regression is valid but should be clearly labelled, especially when outcome prevalence is above 10%.

Case-Control Studies

In case-control studies, participants are selected based on their outcome status (cases vs. controls). Because the sampling is outcome-based, true population prevalence cannot be estimated. So RR cannot be directly calculated.

• OR is the appropriate and only valid measure of association in standard case-control studies.
• The OR from a case-control study can approximate the RR only when the disease is rare in the source population (< 10%).

Cross-sectional Studies

Cross-sectional studies measure exposure and outcome at the same point in time. The prevalence ratio (PR), which is computed identically to RR but using prevalence rather than incidence, is the preferred measure. OR can be reported but is less intuitive.

Quick Reference: Study Design and Appropriate Measures

Study design	RR valid?	OR valid?	HR valid?
Randomised controlled trial	Yes	Yes (with caution if outcome common)	Yes (if time-to-event)
Cohort study	Yes	Yes (with caution if outcome common)	Yes (if time-to-event)
Case-control study	No (usually)	Yes — preferred	No
Cross-sectional study	Prevalence ratio	Yes (less preferred)	No

The OR-as-RR Error: A Common Pitfall

One of the most documented errors in clinical literature is treating the odds ratio as if it were a risk ratio. Published research has found this misinterpretation in approximately 23–26% of studies in fields such as obstetrics and gynaecology and obesity research.

The consequences:

• When OR > 1 and the outcome is common, interpreting OR as RR overestimates the strength of the association.
• When OR < 1, interpreting OR as RR exaggerates the protective effect.
• The error grows as the event rate increases and as the OR moves further from the null (further from 1.0).

A useful rule of thumb: if the outcome prevalence in unexposed individuals is under 10%, OR and RR will differ by less than 20%, and the error is unlikely to change clinical conclusions. Above 10%, always report both or use the appropriate measure.

Alternatives when OR is not appropriate but logistic regression has been used: researchers can use Poisson regression with robust variance, log-binomial regression, or modified Poisson regression to estimate RR directly from multivariable models.

Controlling for Confounders: Regression Models

In observational studies, simple 2×2 table calculations do not account for confounding variables. Multivariable models are needed to estimate adjusted effect measures.

Logistic Regression and Adjusted Odds Ratios

Logistic regression is the standard method for obtaining adjusted odds ratios while controlling for multiple covariates simultaneously. The exponentiated coefficient (exp(β)) from logistic regression equals the adjusted OR for a one-unit change in the predictor, holding all other variables constant.

Logistic regression is appropriate when:

• The outcome is binary.
• The study is a case-control design.
• The outcome is rare (< 10%) in a cohort study and OR is an adequate approximation of RR.

Cox Proportional Hazards Model and Adjusted Hazard Ratios

The Cox model estimates the adjusted HR while accounting for time-to-event data and censoring. It assumes proportional hazards, meaning the ratio of hazards between groups remains constant over time. When this assumption is violated, alternative models (e.g., time-varying covariates, restricted mean survival time) are needed.

Poisson Regression for Risk Ratios

When the goal is to estimate adjusted RR from a cohort study with a common outcome, Poisson regression with robust (sandwich) standard errors is a well-established alternative to logistic regression. This approach avoids the OR-as-RR error and produces directly interpretable relative risk estimates.

Best Practices for Reporting Statistical Measures

Always Accompany Relative Measures with Absolute Measures

Reporting a relative measure alone (RR, OR, HR) is insufficient for clinical interpretation. Always include:

• The baseline risk (or control group event rate).
• The absolute risk reduction or increase (ARR / ARI).
• NNT or NNH where relevant.
• 95% confidence intervals for all estimates.
• The p-value (though confidence intervals are more informative).

Match the Measure to the Study Design

Do not report RR from a case-control study. Do not interpret OR as RR when the outcome is common. Use HR when follow-up time is variable or censoring has occurred.

Transparency About Biases

Clearly report:

• Inclusion and exclusion criteria to define the population.
• Data sources and their limitations.
• Potential biases: selection bias, measurement bias, confounding, and loss to follow-up.
• Whether estimates are crude (unadjusted) or adjusted, and for which covariates.

Interpreting Values Near the Null

A ratio of 1.0 (or a risk difference of 0) means no association. Always consider whether a confidence interval crossing 1.0 (for ratios) or 0 (for differences) renders the finding statistically non-significant. Statistical significance does not equal clinical significance, and vice versa.

Real-World Example: Smoking and Lung Cancer Mortality

Smoking is among the most studied risk factors in epidemiology. Data from large US prospective cohort studies illustrate how different effect measures tell different parts of the story:

Outcome	Risk in smokers	Risk in non-smokers	Risk Ratio	Absolute Risk Increase
Lung cancer mortality (men)	~1.4%	~0.07%	~21×	~1.3 percentage points
Heart disease mortality	~6.5%	~3.5%	~2×	~3 percentage points

This illustrates that the risk ratio for lung cancer (approximately 21) is far higher than for heart disease (approximately 2). Yet in absolute terms, smoking causes far more deaths from heart disease simply because heart disease is more common. Both relative and absolute measures are essential for a complete picture.

Glossary of Key Terms

Term	Definition
Absolute Risk Reduction (ARR)	The arithmetic difference in risk between two groups (control risk minus treatment risk). Indicates how much a treatment reduces the absolute probability of an outcome.
Case-control study	A study design in which participants are selected based on their outcome status (cases vs. controls). Exposure history is then compared between groups. Only OR — not RR — can be directly calculated.
Censoring	In survival analysis, censoring occurs when a participant leaves the study before experiencing the outcome, or the study ends. Censored individuals contribute follow-up time but not an event to the analysis.
Cohort study	A study design in which participants are enrolled based on exposure status (exposed vs. unexposed) and followed over time to observe outcomes. Both RR and OR can be calculated.
Confounding	A distortion of the true association between exposure and outcome, caused by a third variable related to both. Adjusted analyses (e.g., logistic regression, Cox model) are used to control for confounders.
Cox proportional hazards model	A regression model used in survival analysis to estimate adjusted hazard ratios while accounting for time-to-event data and censoring.
Hazard rate	The instantaneous probability of an event occurring at a specific point in time, given that the individual has survived until that time.
Hazard Ratio (HR)	The ratio of the hazard rate in the intervention group to the hazard rate in the control group at any given time. Used in survival analyses with time-to-event outcomes.
Incidence rate	The number of new cases of a disease per unit of person-time at risk. Has a time dimension (e.g., per 100 person-years).
Logistic regression	A multivariable statistical model for binary outcomes that yields adjusted odds ratios. The standard method for estimating ORs while controlling for covariates.
Null value	The value of an effect measure that indicates no association. For ratio measures (RR, OR, HR), the null value is 1. For difference measures (ARR, risk difference), it is 0.
Number Needed to Harm (NNH)	The number of individuals who must be exposed to a risk factor or intervention for one additional person to experience harm. Calculated as 1 / Absolute Risk Increase.
Number Needed to Treat (NNT)	The number of patients who must receive a treatment for one additional patient to benefit. Calculated as 1 / ARR. A smaller NNT indicates greater treatment efficacy.
Odds	The ratio of the probability of an event occurring to the probability of it not occurring: p / (1 − p).
Odds Ratio (OR)	The ratio of the odds of an outcome in an exposed group to the odds in an unexposed group. The preferred measure in case-control studies.
Person-time at risk	The accumulated time that all individuals in a study contribute while being at risk of the outcome. Used to calculate incidence rates.
Poisson regression	A regression model that can estimate risk ratios directly from cohort data, particularly useful when the outcome is common and logistic regression would yield ORs that overestimate RR.
Prevalence ratio (PR)	The ratio of the prevalence of an outcome in exposed vs. unexposed groups. Conceptually equivalent to RR but used in cross-sectional studies.
Probability	The expected frequency of an event across many trials. Ranges from 0 to 1.
Proportional hazards assumption	The assumption in the Cox model that the ratio of hazards between groups remains constant over time.
Relative Risk Reduction (RRR)	The proportional reduction in risk in the treatment group compared to the control group: 1 − RR, or ARR / control risk.
Risk	The probability of an adverse outcome in a defined population over a specific time period. Ranges from 0 to 1.
Risk Difference	Synonymous with Absolute Risk Reduction. The arithmetic difference in risk between two groups.
Risk factor	Any variable associated with an increased probability of developing a disease or adverse outcome.
Risk Ratio (RR)	The ratio of the risk of an outcome in the exposed group to the risk in the unexposed group. Also called relative risk. Preferred in cohort studies and RCTs.
Survival analysis	A set of statistical methods for analysing time-to-event data. Accounts for censoring and variable follow-up times. Produces hazard ratios.
Survival ratio	The ratio of the probability of not experiencing the outcome in one group versus another: (1 − Risk in exposed) / (1 − Risk in unexposed).

Frequently Asked Questions

Can a hazard ratio and a risk ratio for the same study ever give opposite conclusions?

In theory, yes, though it is unusual. If events in the treatment group are concentrated early in the study (front-loaded), the HR can be greater than 1 (apparent harm) while the final RR is less than 1 (apparent benefit) because many treated patients who die early no longer appear in later risk windows. This is why examining Kaplan-Meier survival curves and testing the proportional hazards assumption is important. An HR alone, without the accompanying survival curves, can give an incomplete picture.

Is it possible to convert an odds ratio to a risk ratio after the fact?

Yes, if you know the baseline risk (the event rate in the control or unexposed group). The formula is: RR = OR / [(1 − baseline risk) + (baseline risk × OR)]. This conversion is increasingly used in meta-analyses and systematic reviews to standardise effect measures across studies with different designs. When the baseline risk is very low, OR and RR will already be close and conversion has little effect.

What does a confidence interval that crosses 1.0 mean for a ratio measure?

A 95% confidence interval (CI) that includes 1.0 for a ratio measure (RR, OR, HR) means the result is not statistically significant at the conventional α = 0.05 threshold; the data are compatible with there being no association. However, the width of the CI is equally important: a narrow CI crossing 1.0 suggests a small, precisely estimated null effect, while a very wide CI crossing 1.0 suggests the study was underpowered and cannot rule out a clinically meaningful effect in either direction.

When is it appropriate to pool odds ratios in a meta-analysis versus risk ratios?

OR is generally preferred for pooling in meta-analyses because it has better statistical properties (it is symmetric and less affected by variation in baseline risk across studies). However, if the outcome is common and the studies included mostly cohort designs, pooling RRs or converting ORs to RRs before pooling may give more clinically interpretable results. Some guidelines, such as those from the Cochrane Collaboration, recommend reporting both the pooled OR and an illustrative ARR at a representative baseline risk.

How does loss to follow-up affect the calculation of these measures?

Loss to follow-up (attrition) can bias estimates of RR, OR, and HR if the reasons for dropout are related to the outcome or the exposure, a form of informational censoring. For RR and OR in studies without censoring, lost participants are simply excluded from the denominator, which is acceptable only if missingness is completely at random (MCAR). In survival analyses, informative censoring violates the assumption that censoring is independent of the hazard, potentially biasing the HR. Sensitivity analyses, multiple imputation, and inverse probability weighting are common strategies for addressing attrition bias.

What is the difference between a rate ratio and a risk ratio?

A risk ratio compares cumulative risks (proportions) over a defined period; it is dimensionless and ranges from 0 to infinity. A rate ratio compares incidence rates, which are expressed per unit of person-time and can exceed 1.0. They are numerically similar when follow-up times are short and uniform, but diverge when follow-up is long or when person-time denominators are used instead of head counts. Rate ratios are appropriate when person-time data are available; risk ratios when only proportions are reported.

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