Why the Endpoint Determines the Summary
A clinical trial may report a binary outcome such as whether a participant experienced an adverse event, a continuous outcome such as a change in a symptom score, or a time-to-event outcome such as time until disease progression. Bayesian analysis gives each target quantity a posterior distribution, but the target and its scale differ across these endpoints. A posterior mean event rate, a posterior mean difference in score points, and a posterior survival probability are not interchangeable summaries.
A useful comparison begins by naming the parameter, its units or scale, and the event to which it refers. Then summarize its posterior distribution with a measure of location and an uncertainty statement. A credible interval expresses posterior probability about the parameter, conditional on the specified model and observed data. It is not a repeated-sampling guarantee.
Three Targets, Three Scales
For a binary endpoint, the parameter is often an event probability \(p\) over a specified follow-up period. It lies between 0 and 1. If \(y\) events are observed among \(n\) independent participants with a common event probability \(p\), the likelihood is proportional to \(p^y(1-p)^{n-y}\). With a Beta prior, \(p\sim\operatorname{Beta}(a,b)\), the posterior is \(\operatorname{Beta}(a+y,b+n-y)\). Its posterior mean is \((a+y)/(a+b+n)\). This model assumes the observations are conditionally independent and the participants share the same event probability for the endpoint definition and follow-up period.
For a continuous endpoint, a common target is a treatment-minus-control difference in mean outcome, denoted here by \(\delta\). Its units are the outcome's units, such as score points or millimeters of mercury. Under a normal likelihood with known sampling variance and a normal prior, the posterior for \(\delta\) is normal. The difference can be positive, negative, or zero; unlike a rate, it is not restricted to \([0,1]\). A numerical value is meaningful only with the direction of subtraction and outcome units stated.
For survival analysis, the target may be the probability of remaining event-free beyond a particular time \(t\), \(S(t)\). It lies between 0 and 1 and refers to that fixed time, not to a rate or a median. Under an exponential event-time model with hazard \(\lambda\), \(S(t)=\exp(-\lambda t)\). If the posterior is available for \(\lambda\), the posterior for \(S(t)\) follows by transforming each possible hazard value. Since this transformation decreases as \(\lambda\) increases, interval endpoints reverse order.
| Endpoint target | Parameter scale | Example uncertainty statement |
|---|---|---|
| Binary event | Probability \(p\), from 0 to 1 | Posterior probability that \(p\) lies in a specified interval |
| Continuous outcome | Mean difference \(\delta\), in outcome units | Posterior probability that \(\delta\) lies in a range of score points |
| Time-to-event at fixed time | Survival probability \(S(t)\), from 0 to 1 | Posterior probability that the event-free proportion at time \(t\) lies in a range |
Worked Example: A Binary Event Rate
Worked Example: A Binary Event Rate
Suppose, hypothetically, 100 participants assigned to a treatment arm are followed for 12 weeks, and 18 experience a specified adverse event. Use a \(\operatorname{Beta}(2,8)\) prior for the 12-week event probability \(p\). Assume each participant has the same event definition and follow-up window, and event outcomes are conditionally independent given \(p\).
The observed likelihood contributes 18 event outcomes and 82 non-events. The posterior is therefore
The posterior mean event rate is
Thus the posterior mean is 18.18% for this 12-week adverse event probability. Numerical equal-tail 95% posterior quantiles are approximately 0.116 and 0.259. Conditional on this model and these observed data, the event probability has 95% posterior probability of lying between about 11.6% and 25.9%. This statement is about the probability \(p\), not about the chance that a future set of 100 participants will produce exactly that range of observed rates.
Worked Example: A Continuous Mean Difference
Worked Example: A Continuous Mean Difference
Suppose, hypothetically, a trial estimates a treatment-minus-control difference in a continuous symptom score of 2.4 points, with a known sampling standard error of 1.5 points. Assume the sampling distribution is normal, the reported standard error is known, and the prior for the true mean difference is \(\delta\sim N(0,4)\), where 4 is the prior variance in squared points.
The sampling model is \(\hat{\delta}\mid\delta\sim N(\delta,2.25)\). For a normal prior and normal likelihood, posterior variance \(V\) and mean \(m\) are
Therefore \(\delta\mid\text{data}\sim N(1.536,1.44)\), with posterior standard deviation \(\sqrt{1.44}=1.2\) points. The central 95% normal credible interval is
Conditional on the model and data, the treatment-minus-control mean difference has 95% posterior probability of lying between about \(-0.816\) and \(3.888\) score points. The interval includes zero, so the posterior assigns probability to both a negative and a positive difference. Under this posterior, \(P(\delta>0\mid\text{data})=\Phi(1.536/1.2)=\Phi(1.28)\approx0.900\), where \(\Phi\) is the standard normal distribution function. That probability does not by itself establish clinical importance; the clinical meaning depends on the score and a relevant threshold.
Worked Example: Survival at a Fixed Time
Worked Example: Survival at a Fixed Time
Suppose, hypothetically, a trial arm contributes 12 observed events and 180 person-months of total event-free follow-up. Assume an exponential event-time model with a constant hazard \(\lambda\), and non-informative right censoring conditional on the model. Use a Gamma prior with shape 2 and rate 20, so its density is proportional to \(\lambda^{2-1}\exp(-20\lambda)\). The shape-rate convention gives \(\lambda\) units of events per person-month.
The posterior shape adds the number of events, and the posterior rate adds total exposure:
At \(t=12\) months, the target is \(S(12)=\exp(-12\lambda)\). The posterior mean survival probability can be calculated from the Gamma transform:
The posterior mean survival probability at 12 months is about 44.2%. Numerical equal-tail 95% posterior quantiles for \(\lambda\) are approximately 0.0383 and 0.111 events per person-month. Because \(\exp(-12\lambda)\) decreases with \(\lambda\), transform the upper hazard quantile for the lower survival endpoint and vice versa:
Thus, conditional on the exponential model and data, \(S(12)\) has 95% posterior probability of lying between about 26.4% and 63.2%. This is uncertainty about being event-free beyond 12 months, not an interval for the hazard and not a summary of the entire survival curve.
Why the Transform and the Assumptions Matter
The survival example illustrates a general point: a posterior summary does not always transform in the same way as the parameter itself. Since \(S(t)=\exp(-t\lambda)\) is nonlinear, \(E\{S(t)\mid\text{data}\}\) need not equal \(\exp[-tE(\lambda\mid\text{data})]\). For the example, the latter would be \(\exp[-12(14/200)]\approx0.432\), whereas the posterior mean survival probability is approximately 0.442. The posterior mean should be calculated on the target scale when that is the quantity to report. For an equal-tail interval under a decreasing transformation, transformed quantiles must be put in increasing order.
The assumptions behind each calculation also determine what its uncertainty means. A common event probability may be inappropriate if follow-up differs substantially or risks vary across participants. The normal continuous-outcome model may be inadequate when the sampling distribution is strongly skewed or the standard error is poorly estimated. The exponential survival model imposes a constant hazard over time; it can misrepresent data when hazards change. Informative censoring, in which censoring is related to event risk after accounting for the model, can also undermine the survival inference.
Useful diagnostics include comparing observed endpoint patterns with model implications, checking for unequal follow-up or influential participants, and examining whether estimates change materially under plausible alternatives. For survival data, assess whether the constant-hazard assumption is credible over the time range of interest. Such checks do not prove a model correct, but they can reveal a mismatch that makes a precise posterior interval misleading.
Comparing Approaches Without Confusing the Target
A posterior mean and a credible interval give a compact summary, but other summaries may answer a more relevant clinical question. For a binary outcome, a risk difference between trial arms answers a different question from a single-arm event probability. A continuous mean difference is expressed in the original outcome units, while a standardized difference changes the scale and may be less directly interpretable. Survival at a fixed time gives a point on a survival curve; a median survival time or hazard ratio is a different target, even when derived from the same survival model.
Likewise, a 95% credible interval and a posterior probability of crossing a clinical threshold are both valid posterior summaries, but they answer different questions. If \(\delta_{\min}\) is a prespecified clinically meaningful benefit on the chosen difference scale, then \(P(\delta>\delta_{\min}\mid\text{data})\) addresses the probability of exceeding that threshold. A probability of being greater than zero instead addresses direction alone. Neither summary should be substituted for the other without explaining the clinical question.
The central discipline is to report the parameter, scale, time horizon where relevant, and uncertainty statement together. For example, “the posterior mean 12-week event probability was 18.2%, with a 95% credible interval of 11.6% to 25.9%” identifies both the endpoint and its horizon. “The outcome improved” would not identify the parameter, direction, or amount of uncertainty.
Common Mistakes and Practical Checks
- Confusing posterior probability with repeated-sampling coverage. A 95% credible interval has 95% posterior probability under the specified model and observed data. It is not a claim that 95% of intervals from repeated trials contain a fixed parameter.
- Reporting a number without its scale. A value of 0.44 could be a rate or a survival probability. State the endpoint, units or probability scale, and time horizon.
- Transforming only a point estimate. Applying a nonlinear transformation to a posterior mean generally does not give the posterior mean of the transformed parameter. Transform the posterior distribution or use its appropriate quantiles.
- Reversing a transformed interval incorrectly. Since survival decreases as hazard increases, the upper hazard quantile gives the lower survival endpoint.
- Reading inclusion of zero as a decision by itself. A difference interval spanning zero describes uncertainty about direction; clinical importance additionally depends on the magnitude and relevant clinical threshold.
Check Your Understanding
Answer each question, then compare with the solutions.
- A binary endpoint has 10 events among 50 participants and a \(\operatorname{Beta}(1,1)\) prior. What is the posterior distribution and posterior mean event probability?
- A treatment-minus-control mean difference has posterior distribution \(N(3,1)\), with the outcome measured in points. Give its central 95% credible interval and describe the target in context.
- If the posterior interval for a constant hazard is \([0.02,0.08]\) per month, which hazard endpoint gives the lower endpoint for survival at 6 months? Write the transformed interval endpoints.
- Why does a posterior mean hazard generally not suffice to calculate the posterior mean survival probability at a fixed time?
Solutions. 1. The posterior is \(\operatorname{Beta}(11,41)\), because the prior parameters are updated by 10 events and 40 non-events. Its mean is \(11/(11+41)=0.2115\), or about 21.2%. 2. The interval is \(3\pm1.96(1)=[1.04,4.96]\) points. Conditional on the model and data, the treatment-minus-control mean difference has 95% posterior probability of lying in this range. 3. Survival decreases with hazard, so the upper hazard endpoint gives the lower survival endpoint. The transformed interval is \([\exp(-6(0.08)),\exp(-6(0.02))]\approx[0.619,0.887]\). 4. The exponential transformation is nonlinear, so transforming the posterior mean hazard does not generally equal the posterior mean of the transformed hazard. The posterior distribution must be transformed, or the expectation calculated on the survival-probability scale.