Why Report More Than One Scale?
A binary endpoint records whether an event occurred or did not occur for each participant during a specified period. Examples include response by week 12, hospitalization within 30 days, or a serious adverse event during follow-up. Reporting only that treatment “reduced the event rate” leaves important questions unanswered: how frequent was the event in each arm, how large was the change in absolute terms, and how does the result relate to the untreated risk?
Absolute and relative scales answer different questions. An absolute difference expresses the change in event probability in percentage points, which is often directly useful for discussing how many events might be prevented or caused. A relative measure compares risks or odds proportionally. Neither scale replaces the arm-specific event rates: presenting those rates alongside the contrast lets readers understand both the baseline frequency and the comparative effect.
The event and its time window must be defined before interpreting any measure. In what follows, \(p_T\) is the event probability in the treatment arm and \(p_C\) is the event probability in the control arm for the stated endpoint and follow-up period. For an undesirable event, a lower treatment probability is generally favorable; for a desirable event, the direction of benefit is reversed.
Model and Posterior Quantities
A basic model for a randomized trial treats the event count in each arm as binomial, conditional on that arm’s event probability. Let \(Y_T\) and \(Y_C\) be the numbers of participants with the event among \(n_T\) and \(n_C\) participants, respectively. The model is \(Y_T\mid p_T\sim\operatorname{Binomial}(n_T,p_T)\) and \(Y_C\mid p_C\sim\operatorname{Binomial}(n_C,p_C)\). This assumes a binary endpoint measured consistently for every participant, a fixed analysis population and follow-up definition, and conditionally independent outcomes within each arm with a common event probability. The simple model does not itself account for clustering, varying follow-up, covariates, or missing outcomes; those features require an appropriate analysis model and estimand.
For illustration, suppose the arm-specific priors are independent, with \(p_T\sim\operatorname{Beta}(a_T,b_T)\) and \(p_C\sim\operatorname{Beta}(a_C,b_C)\). Combining the binomial likelihood with the beta prior gives posterior distributions \(p_T\mid Y_T\sim\operatorname{Beta}(a_T+Y_T,b_T+n_T-Y_T)\) and \(p_C\mid Y_C\sim\operatorname{Beta}(a_C+Y_C,b_C+n_C-Y_C)\). Other models and prior structures are possible; the chosen analysis should match the trial design and be described in the report.
The posterior distribution of each contrast follows from the joint posterior of \(p_T\) and \(p_C\). In practice, draw paired values \(p_T^{(m)},p_C^{(m)}\) from that joint posterior and calculate \(p_T^{(m)}-p_C^{(m)}\), \(p_T^{(m)}/p_C^{(m)}\), and the corresponding odds ratio for each draw. Posterior summaries and credible intervals are then calculated from the resulting draws. This preserves uncertainty and dependence in the contrast rather than treating estimated arm rates as fixed.
Worked Example: From Event Counts to Posterior Arm Rates
Suppose, hypothetically, 12 of 100 participants assigned to treatment and 20 of 100 assigned to control experience an undesirable event by week 12. Use independent \(\operatorname{Beta}(1,1)\) priors for the two event probabilities. The binomial model is appropriate here if the endpoint is recorded for all 200 participants over the same defined period and participants’ outcomes are conditionally independent within each arm.
The treatment posterior is \(\operatorname{Beta}(1+12,1+100-12)=\operatorname{Beta}(13,89)\). The control posterior is \(\operatorname{Beta}(1+20,1+100-20)=\operatorname{Beta}(21,81)\). A \(\operatorname{Beta}(a,b)\) distribution has mean \(a/(a+b)\), so the posterior mean event rates are:
The posterior mean risk difference is the difference of the posterior means:
This is a posterior mean reduction of about 7.84 percentage points in the event probability by week 12. It is an absolute-scale summary; it does not state the probability that treatment is beneficial or give the uncertainty interval for the difference. Those require summaries of the posterior contrast itself.
Deriving and Interpreting the Scales
The risk difference is obtained by subtracting event probabilities, so its unit is an absolute probability difference. For example, a risk difference of \(-0.05\) means that the event probability in the treatment arm is five percentage points lower than in control. It does not mean a five percent relative reduction. The risk ratio instead divides probabilities. A risk ratio below 1 indicates a lower treatment event probability for an undesirable endpoint; a value of 0.70 means the treatment risk is 70% of the control risk, or 30% lower relative to control.
The odds of an event at probability \(p\) are \(p/(1-p)\). Thus the odds ratio compares treatment and control odds. An odds ratio of 0.70 means treatment odds are 30% lower than control odds; it does not generally mean the event risk is 30% lower. Odds and risks are close when events are rare but can differ substantially when events are common.
A useful algebraic check makes the distinction explicit. If the control risk is \(p_C\) and the risk ratio is \(r\), then \(p_T=rp_C\), and the odds ratio is:
Except in special cases, this is not equal to \(r\). The odds ratio approaches the risk ratio when both event probabilities are small, but the approximation becomes less reliable as the risks increase.
Worked Example: Comparing the Contrasts for the Trial Above
Using the posterior mean risks from the hypothetical trial, the risk ratio formed from those means is:
This plug-in ratio says the treatment posterior mean risk is about 61.9% of the control posterior mean risk, or about 38.1% lower relative to that control risk. It is not the posterior mean of the risk ratio, which must be calculated by taking the ratio for each posterior draw and then summarizing those ratios.
The odds calculated from the posterior mean risks are \(0.12745098/(1-0.12745098)=13/89\approx0.1461\) for treatment and \(0.20588235/(1-0.20588235)=21/81\approx0.2593\) for control. Their ratio is:
Thus the odds based on these posterior mean risks are about 43.7% lower in treatment than control, whereas the risk ratio based on the same means is 0.6190. The odds ratio calculation is a ratio of odds evaluated at posterior mean risks; it is not the posterior mean odds ratio. For a Bayesian report, calculate the posterior distribution of the odds ratio by transforming each posterior draw, just as for the risk ratio.
Worked Example: Transforming a Posterior Draw
Suppose one hypothetical joint posterior draw for an undesirable event gives \(p_T=0.12\) and \(p_C=0.20\). This single draw is not an estimate of the whole posterior distribution; it demonstrates how to calculate each contrast consistently for a draw.
The risk difference is \(0.12-0.20=-0.08\), an 8-percentage-point lower treatment risk. The risk ratio is \(0.12/0.20=0.60\), meaning treatment risk is 60% of control risk in this draw. The odds ratio is:
Repeating these calculations for every posterior draw produces a posterior distribution for each contrast. Its median or mean and an appropriate credible interval can then be reported. The single draw above cannot be used to claim that the posterior probability of benefit is any particular value.
Choose Scales for Clinical Meaning
The absolute scale is especially useful when readers need to understand the change in event frequency. In the first example, the difference in posterior mean risks is approximately 7.84 fewer events per 100 participants over 12 weeks. That wording is a direct rescaling of the risk difference, not a statement about a number needed to treat. Any number-needed-to-treat summary requires careful definition and treatment of uncertainty, especially when the risk difference is near zero.
A relative scale can help compare proportional effects across settings with different baseline risks, but it may obscure the absolute number of events involved. For instance, a similar relative reduction can correspond to a small absolute reduction when control risk is low and a larger absolute reduction when control risk is high. For this reason, a clear results report usually gives both arm-specific risks and at least one contrast that matches the clinical question. The risk ratio is often more directly interpretable as a proportional change in risk than the odds ratio.
| Quantity | Question it answers | Interpretation guide |
|---|---|---|
| Arm-specific risks | How often did the event occur in each arm? | Report probabilities or percentages with the endpoint and time window. |
| Risk difference | How far apart are the event probabilities? | Absolute difference; report in percentage points or as a probability. |
| Risk ratio | How many times the control risk is the treatment risk? | Relative comparison of risks; 1 indicates equal risks. |
| Odds ratio | How do event odds compare between arms? | Relative comparison of odds; 1 indicates equal odds, but it is not generally a risk ratio. |
For each posterior contrast, report a point summary and uncertainty interval on that same scale, stating whether the interval is a credible interval and the probability content used. Pair the contrast with the event rates by arm and specify which direction is favorable. For example: “By week 12, the posterior mean event risks were 12.7% with treatment and 20.6% with control; the posterior mean risk difference was \(-7.84\) percentage points.” If available, add a credible interval for the risk difference calculated from posterior draws. Avoid supplying a contrast interval from one scale beside a point estimate from another without labeling the difference.
Assumptions, Diagnostics, and Common Pitfalls
The binomial model assumes that the analysis uses a defined denominator and that each participant contributes an outcome classified consistently as event or no event. If follow-up differs materially, censoring or competing events affect ascertainment, or missing outcomes are present, the simple two-binomial model may not represent the intended clinical estimand. The assumptions and handling of these features should be stated rather than hidden by reporting a percentage alone. A randomized comparison also does not automatically fix errors in outcome definition or missing-data handling.
Check that reported event counts do not exceed denominators, that the rates agree with the counts, and that the endpoint and time point are the same across arms. When posterior contrasts are computed from simulation, check computation diagnostics and confirm that every draw uses the intended arm pairing and the stated transformation. For risk ratios, draws with a control probability near zero can produce very large ratios; for odds ratios, probabilities close to 0 or 1 can produce extreme odds. Such behavior can make posterior summaries sensitive to priors or model details, particularly with sparse data. Examine the posterior distribution and relevant prior sensitivity rather than relying only on a point estimate.
A common mistake is to report “a 38% reduction” after calculating \(1-\mathrm{OR}\) and then describe it as a reduction in risk. The correction is to call it a 38% reduction in odds, if that is the quantity calculated, or to calculate and report the risk ratio for a relative comparison of risks. Another error is to divide posterior mean risks and label the result the posterior mean risk ratio. In general, the mean of a ratio is not the ratio of means; calculate the ratio for every posterior draw before summarizing its posterior distribution.
The main alternatives are complementary rather than competing summaries. A risk difference emphasizes absolute impact; a risk ratio emphasizes proportional risk change; an odds ratio emphasizes proportional odds change and can be useful in models that naturally parameterize odds. Their numerical values are not interchangeable, and there is no universally best scale. The appropriate presentation depends on the clinical decision, baseline risk, and model. The limitations of each should be visible: absolute differences depend on baseline risk, while relative measures can understate or overstate intuitive clinical impact if readers confuse odds with risks.
State what counts as an event, the population, and the follow-up window.
Give event counts and denominators or posterior event-rate summaries, with units.
Use posterior draws to obtain the contrast’s posterior distribution and interval.
State whether the scale is absolute risk, relative risk, or odds, and identify the favorable direction.
Worked Example: Sparse Events and an Undefined Empirical Ratio
Suppose, hypothetically, 0 of 40 treatment participants and 4 of 40 control participants experience an undesirable event by week 12. With independent \(\operatorname{Beta}(1,1)\) priors and the binomial assumptions stated above, the posteriors are \(p_T\mid Y_T\sim\operatorname{Beta}(1,41)\) and \(p_C\mid Y_C\sim\operatorname{Beta}(5,37)\). Their posterior mean risks are:
The difference of these means is \(0.02381-0.1190=-0.09524\), or about 9.52 percentage points lower for treatment. The ratio of the posterior mean risks is \(0.02381/0.1190=0.2000\). The observed event-rate ratio, however, is undefined because the observed treatment event rate is zero. A Bayesian prior yields a proper posterior in this example, but the resulting relative contrast may depend strongly on the prior when events are sparse. Report the prior and assess sensitivity; do not imply that the data alone established a precise relative effect.
Check Your Understanding
Answer the questions, then compare with the solutions.
- If treatment risk is 0.10 and control risk is 0.25, what are the risk difference and risk ratio, and how would you interpret them for an undesirable event?
- Why is an odds ratio of 0.70 not generally interpreted as a 30% reduction in event risk?
- For posterior draws \(p_T=0.15\) and \(p_C=0.30\), calculate the risk difference, risk ratio, and odds ratio.
- Why is the ratio of posterior mean risks not necessarily the posterior mean risk ratio?
- What should be checked before interpreting a posterior risk contrast from a two-arm binomial model?
Solutions. (1) The risk difference is \(0.10-0.25=-0.15\), a 15-percentage-point lower treatment risk; the risk ratio is \(0.10/0.25=0.40\), so treatment risk is 40% of control risk, or 60% lower relative to control. (2) An odds ratio compares \(p/(1-p)\), not \(p\); the difference between odds and risks can be substantial when events are common. (3) The risk difference is \(-0.15\), the risk ratio is \(0.15/0.30=0.50\), and the odds ratio is \((0.15/0.85)/(0.30/0.70)\approx0.4118\). (4) A ratio is nonlinear, so averaging ratios across posterior draws generally does not equal the ratio of the averages; compute each draw’s ratio and summarize those values. (5) Check event definition, follow-up window, denominators, outcome completeness, model assumptions, and computation of the posterior contrast; sparse events also warrant attention to prior sensitivity.