Why the Scale Matters
For a binary clinical endpoint, each participant either experiences the event or does not. Yet the treatment effect is not a single number until its scale is specified. A difference in event probabilities answers an absolute question; a ratio compares risks or odds relatively. These contrasts can describe the same pair of risks while conveying different information about clinical impact.
The target population, endpoint, time point, and treatment strategies must be fixed before comparing scales. In the notation below, \(p_T\) and \(p_C\) are the marginal probabilities of the event under the treatment and control strategies, respectively, for that same estimand. The risk is a probability, not a count or an odds. If the endpoint is an adverse event, a negative treatment-minus-control difference indicates fewer events under treatment.
The signs and reference groups are part of the definition. Here, every contrast is treatment relative to control. Reversing the subtraction or ratio changes the numerical result and its interpretation. The risk difference is measured in probability units; it is often reported in percentage points. The risk ratio and odds ratio are dimensionless.
Three Scales, Three Questions
The risk difference asks how much the event probability changes in absolute terms. For example, \(RD=-0.06\) means a six-percentage-point lower event risk under treatment. It does not mean a six-percent relative reduction. The risk ratio asks how large the treatment risk is as a proportion of the control risk: \(RR=0.75\) means the treatment risk is 75% of the control risk, or 25% lower relative to control.
The odds of an event at risk \(p\) are \(p/(1-p)\). The odds ratio compares treatment odds with control odds. An \(OR\) of 0.75 means treatment odds are 75% of control odds; it does not in general mean treatment risk is 25% lower. Risk and odds are close only when the event is uncommon, and the approximation can become poor as risks increase.
Worked Example: One Trial, Three Contrasts
Suppose a hypothetical randomized trial observes the endpoint in 18 of 100 participants assigned to treatment and 24 of 100 assigned to control. Treat these observed proportions as plug-in estimates of the two risks for this calculation. The endpoint is an adverse event, and all quantities refer to the same population, time point, and treatment strategies.
The estimated risks are \(\hat p_T=18/100=0.18\) and \(\hat p_C=24/100=0.24\). The risk difference is
Thus the observed treatment risk is six percentage points lower than the observed control risk. The estimated risk ratio is
The observed treatment risk is 0.75 times the control risk, or 25% lower relative to control. The estimated odds ratio is
The treatment odds are about 0.695 times the control odds, or about 30.5% lower. That is an odds comparison, not a 30.5% reduction in risk. These are plug-in contrasts from hypothetical data, not posterior summaries or proof that treatment caused the observed difference. Causal interpretation also depends on the trial design and on the stated estimand.
How the Scales Relate
The odds ratio and risk ratio are linked by a simple identity. Starting from their definitions and assuming \(0<p_T,p_C<1\),
This identity explains why an odds ratio is not generally a risk ratio. If treatment lowers an adverse-event risk, so \(p_T<p_C\), then \(1-p_C<1-p_T\), and the multiplier \((1-p_C)/(1-p_T)\) is less than one. Consequently \(OR<RR\): when both ratios are below one, the odds ratio is farther below one. If the event is rare in both groups, both complements are near one, so the multiplier is near one and the ratios are more similar. Rarity is an approximation condition, not an identity.
A risk difference is bounded: because both risks lie in \([0,1]\), \(RD\) lies in \([-1,1]\). Its scale directly expresses absolute change, but its possible magnitude depends on the starting risk. By contrast, relative measures can remain the same across different baseline risks while absolute changes differ. Neither scale is universally superior; the relevant question is what comparison is needed for the clinical decision.
Worked Example: Same Relative Risk, Different Absolute Effects
Consider two hypothetical settings in which treatment halves the risk of an adverse event. In the first, \(p_C=0.02\) and \(p_T=0.01\). In the second, \(p_C=0.20\) and \(p_T=0.10\). The assumptions for these arithmetic comparisons are simply that each pair refers to the same endpoint and target population within its setting.
In both settings, \(RR=p_T/p_C=0.5\). In the first, \(RD=0.01-0.02=-0.01\), a one-percentage-point reduction. In the second, \(RD=0.10-0.20=-0.10\), a ten-percentage-point reduction. Thus a 50% relative risk reduction does not determine the absolute reduction without a baseline risk.
The odds ratios also differ. In the first setting,
In the second,
The same risk ratio therefore corresponds to different risk differences and odds ratios. The arithmetic does not establish which setting is clinically more important; that judgment also requires the clinical consequences of the event and the decision context.
Bayesian Estimands as Posterior Distributions
In Bayesian analysis, uncertainty about the risks is represented by their joint posterior distribution given the observed data and the specified model. A Bayesian risk-difference estimand is the posterior distribution of \(p_T-p_C\); corresponding distributions for \(p_T/p_C\) and the odds ratio are obtained by applying those transformations to posterior draws of the two risks. Each draw must use the treatment and control risks from the same draw of the joint posterior.
For a simple illustration, suppose the endpoint is fully observed for every participant, outcomes within each arm are modeled as independent Bernoulli observations with a common arm-specific risk, and the two arm risks have independent \(\operatorname{Beta}(1,1)\) priors. These assumptions yield binomial likelihoods and beta posteriors. This is a deliberately simple model: if the data structure or target estimand requires other features, the model must reflect them. The scale definitions themselves do not require this particular likelihood or prior.
Worked Example: Transforming Posterior Risk Draws
Use the hypothetical counts from the first example: 18 events among 100 treatment participants and 24 among 100 control participants. Under the independent binomial likelihood and independent \(\operatorname{Beta}(1,1)\) priors just stated, conjugate updating gives
The posterior means are \(E(p_T\mid\text{data})=19/102\approx0.1863\) and \(E(p_C\mid\text{data})=25/102\approx0.2451\). Linearity of expectation gives the posterior mean of the risk difference:
This posterior mean difference is about a 5.88-percentage-point lower risk under treatment. For a posterior draw with \(p_T=0.16\) and \(p_C=0.20\), the transformed draw is \(RD=-0.04\), \(RR=0.16/0.20=0.80\), and
Repeating these transformations for every joint posterior draw produces a posterior distribution on each scale. In general, the posterior mean of a ratio is not the ratio of posterior means: \(E(p_T/p_C\mid\text{data})\) need not equal \(E(p_T\mid\text{data})/E(p_C\mid\text{data})\). Therefore, a ratio summary should be computed from transformed posterior draws, not inferred by dividing two marginal posterior summaries.
Conditions, Diagnostics, and Limitations
The mathematical contrasts require risks that refer to the same binary endpoint, time point, target population, and treatment strategies. For a causal interpretation, the design and analysis must support identification of those risks; in a randomized trial, this includes conducting and analyzing the trial consistently with its design and handling outcome ascertainment in a way appropriate to the estimand. A conditional model coefficient should not automatically be reported as a marginal risk contrast: the risks must be obtained for the target population if the intended estimand is marginal.
The risk difference is defined even at boundary risks. The risk ratio requires \(p_C>0\); if \(p_T=0\) and \(p_C>0\), it equals zero, while \(p_C=0\) makes the ratio undefined. The odds ratio as defined here requires both risks strictly between 0 and 1. Boundary cases lie outside that definition; limits depend on which risk approaches which boundary. For instance, with \(p_T=0\) and \(0<p_C<1\), the odds-ratio limit is 0. If both risks approach a boundary, the limit can depend on how they approach it.
In Bayesian computation, check that the fitted model actually represents the intended risks and that each posterior draw is transformed consistently. Sparse events deserve particular attention: a risk ratio or odds ratio may be highly sensitive to the model and prior when one arm has few or no events. Ratios can also have long-tailed posterior distributions when the control risk is close to zero, making a posterior mean unstable or even nonexistent under some models. In such settings, report the posterior distribution and relevant uncertainty carefully rather than relying on a single ratio summary.
The main alternatives are not competing calculations of one universal number. The risk difference is often clearest for absolute event reduction; the risk ratio communicates proportional change in risk; the odds ratio is a comparison of odds and may arise naturally in a model. Reporting more than one scale can aid interpretation, provided each is labelled and the risks or reference level are clear. The limitation is that emphasizing one scale can obscure information on another: relative effects do not reveal absolute impact without baseline risk, while a risk difference alone does not state the proportional change.
Check Your Understanding
Answer each question using the treatment-minus-control direction and the definitions above. Short solutions follow the questions.
- If \(p_T=0.12\) and \(p_C=0.15\), calculate the risk difference and risk ratio. Interpret each for an adverse event.
- For \(p_T=0.10\) and \(p_C=0.20\), calculate the odds ratio and explain why it is not a 50% reduction in risk.
- Two hypothetical settings have \(RR=0.5\), with control risks 0.04 and 0.30. What are the corresponding treatment risks and risk differences?
- With independent beta posteriors whose means are 0.25 for treatment risk and 0.20 for control risk, can the posterior mean risk ratio be obtained by dividing those means? Explain.
- Is the odds ratio defined by the formula here if \(p_T=0\) and \(p_C=0.10\)? What is the relevant boundary limit?
Solutions. (1) \(RD=0.12-0.15=-0.03\), a three-percentage-point lower treatment risk; \(RR=0.12/0.15=0.80\), so treatment risk is 20% lower relative to control. (2) \(OR=(0.10/0.90)/(0.20/0.80)=0.444\). This compares odds; the risk ratio is \(0.10/0.20=0.50\), a 50% lower risk. (3) The treatment risks are \(0.02\) and \(0.15\); the risk differences are \(0.02-0.04=-0.02\) and \(0.15-0.30=-0.15\). (4) Not in general: the posterior mean of a ratio is generally not the ratio of posterior means, so transform joint posterior draws and summarize those values. (5) The odds-ratio definition requires both risks strictly between 0 and 1, so it is outside its domain at \(p_T=0\). For \(0<p_C<1\), its boundary limit is 0.