Worked Example: Reporting a Binary Endpoint

What a Binary-Endpoint Report Needs to Say

A useful results report for a binary endpoint lets a reader answer four questions: how often did the event occur in each arm, what was the absolute difference between arms, how uncertain is that difference, and how probable is a benefit large enough to matter clinically? These quantities are connected, but they are not interchangeable. In particular, a posterior probability that a treatment benefit exceeds a threshold says more than whether the posterior interval includes zero, and neither statement replaces the event rates themselves.

The example below concerns a hypothetical randomized trial of a treatment intended to reduce an undesirable event by a fixed follow-up time. The event and trial are fictional. Let \(X_T\) and \(X_C\) be the event counts among \(n_T\) treatment-arm and \(n_C\) control-arm participants, and let \(p_T\) and \(p_C\) be their respective event probabilities by that follow-up time. Define the absolute reduction as \(D=p_C-p_T\). With this direction, positive \(D\) favours treatment.

Model and assumptions: Conditional on the arm-specific event probabilities, the observed counts are independent binomial outcomes: \(X_T\sim\operatorname{Binomial}(n_T,p_T)\) and \(X_C\sim\operatorname{Binomial}(n_C,p_C)\). The analysis assumes a clearly defined binary endpoint and follow-up time, appropriate allocation to the two arms, and complete or appropriately handled endpoint data. The binomial model also treats participants’ outcomes as conditionally independent within each arm and assumes a common event probability within each arm. These assumptions should be reviewed against the trial design and data; they are not guaranteed merely by using this model.

From Event Counts to Posterior Event Rates

For a compact conjugate analysis, suppose the prior distributions are independent and uniform: \(p_T\sim\operatorname{Beta}(1,1)\) and \(p_C\sim\operatorname{Beta}(1,1)\). This is a modelling choice, not a consequence of randomization. For an arm with \(x\) events among \(n\) participants and a \(\operatorname{Beta}(a,b)\) prior, the likelihood is proportional to \(p^x(1-p)^{n-x}\). Multiplying by the prior density gives a posterior density proportional to

p^{a-1}(1-p)^{b-1}p^x(1-p)^{n-x} = p^{a+x-1}(1-p)^{b+n-x-1},

which is the kernel of a \(\operatorname{Beta}(a+x,b+n-x)\) distribution. Thus the prior parameters are updated by adding the observed event and non-event counts. For a \(\operatorname{Beta}(a,b)\) distribution, the posterior mean is \(a/(a+b)\); this is a mean of the uncertain event probability, not the observed proportion.

Worked Example: Updating the Two Event Rates

Suppose 30 of 200 participants in the treatment arm and 45 of 200 in the control arm experience the event. Under the stated independent \(\operatorname{Beta}(1,1)\) priors, the treatment posterior is

p_T\mid \text{data}\sim\operatorname{Beta}(1+30,\,1+200-30) =\operatorname{Beta}(31,171).

Its posterior mean is \(31/(31+171)=31/202=0.1535\), or 15.3%. For control,

p_C\mid \text{data}\sim\operatorname{Beta}(1+45,\,1+200-45) =\operatorname{Beta}(46,156),

with posterior mean \(46/202=0.2277\), or 22.8%. The observed proportions are \(30/200=15.0\%\) and \(45/200=22.5\%\); the posterior means differ slightly because the prior contributes one unit to each beta parameter. These are estimated event probabilities by the specified follow-up time, not rates per unit of person-time.

Summarizing the Absolute Contrast and Clinical Threshold

For this example, the target contrast is \(D=p_C-p_T\), measured in probability units. A value of \(D=0.05\) means an absolute reduction of five percentage points, not a five-percent relative reduction. The difference of posterior means is a convenient summary of the posterior mean of \(D\), because expectation is linear:

E(D\mid\text{data}) =E(p_C\mid\text{data})-E(p_T\mid\text{data}) =\frac{46}{202}-\frac{31}{202} =\frac{15}{202} =0.0743.

The posterior mean absolute reduction is therefore 7.43 percentage points. Uncertainty in \(D\), however, must be calculated from the joint posterior distribution of \(p_C\) and \(p_T\); subtracting interval endpoints for the two event rates is not a valid general way to obtain an interval for their difference. Under the independent priors and separate-arm likelihoods here, the posterior distributions are independent. One may draw paired posterior values from the two beta distributions, calculate \(D\) for each pair, and summarize the resulting draws.

Worked Example: Reporting Uncertainty and the Threshold Probability

Continue with the posterior distributions \(\operatorname{Beta}(31,171)\) for \(p_T\) and \(\operatorname{Beta}(46,156)\) for \(p_C\). Suppose numerical posterior calculation gives an equal-tailed 95% credible interval for \(D=p_C-p_T\) of approximately \(-0.002\) to \(0.151\). In percentage-point units, that is an interval from about a 0.2-point increase in the event probability to a 15.1-point reduction. It includes zero, so the posterior remains compatible with a small difference in either direction, as well as with a larger reduction.

Suppose the prespecified clinically important reduction is five percentage points. The relevant posterior probability is

P(D>0.05\mid\text{data}) = P(p_C-p_T>0.05\mid\text{data}) \approx 0.73.

For example, in a simulation with one million paired posterior draws, this estimate corresponds to about 730,000 draws with \(p_C-p_T>0.05\), divided by 1,000,000. The result supports the statement that the posterior probability of a reduction exceeding five percentage points is about 73%, under the specified model and prior. It does not mean there is a 73% probability that the treatment works in some unconditional sense, and it does not establish that the threshold is clinically appropriate; that threshold should be justified on clinical grounds.

Define the endpoint and contrast.
Name the event, follow-up time, arms, and direction of the contrast. Here, a positive \(D=p_C-p_T\) indicates fewer events with treatment.

Report the arm-level results.
Give posterior event-rate summaries with percentages and, where useful, credible intervals. Retain the denominators so readers can see the scale of the data.

Report the absolute contrast and uncertainty.
State the posterior summary and interval for \(D\) in percentage points, explaining what positive and negative values mean.

Address clinical importance and qualifications.
Give \(P(D>\delta\mid\text{data})\) for a justified threshold \(\delta\), then briefly identify material assumptions or limitations.

Worked Example: Putting the Results into a Concise Report

Use the hypothetical data and posterior summaries above. A compact report could read: “By the specified follow-up time, the posterior mean event probability was 15.3% in the treatment arm (30/200 observed events) and 22.8% in the control arm (45/200). The posterior mean absolute reduction, defined as control minus treatment, was 7.43 percentage points, with a 95% equal-tailed credible interval of approximately −0.2 to 15.1 percentage points. The posterior probability that the reduction exceeded the prespecified five-percentage-point clinical threshold was approximately 73%. Thus, the estimated reduction favoured treatment, while uncertainty remained about whether the event probability was reduced at all.”

This report gives the counts and posterior rates, declares the contrast direction, states the interval and threshold in the same units, and calibrates the conclusion to the results. The final sentence does not convert a 73% threshold probability into certainty or claim that treatment caused the observed difference beyond the assumptions of the randomized trial and analysis.

Checks, Alternatives, and Limitations

Before releasing a report, verify that the event definition, analysis population, denominators, and follow-up window match across the count, model, and written interpretation. A rate such as 15.3% is ambiguous if it is unclear whether it is an observed proportion or a posterior summary. State which it is. Also check that the reported difference uses the same arm order as its definition and that the threshold probability refers to the same contrast and units.

Posterior computation should be adequate for the reported precision. For simulation-based summaries, inspect the diagnostics appropriate to the sampling method and assess Monte Carlo uncertainty, particularly for a tail probability or interval endpoint. If posterior draws are autocorrelated or the effective sample is too small, additional computation may be needed before rounding a result to two decimal places. A reported numerical probability is conditional on the likelihood, prior, and endpoint assumptions; it is not a data-only quantity.

The main alternative summaries answer different questions. A relative measure, such as a risk ratio, expresses the event probability on a multiplicative scale; it may supplement the absolute reduction but should not replace it when the clinical threshold is defined in percentage points. A frequentist confidence interval has a different repeated-sampling interpretation from a Bayesian credible interval, and a frequentist p-value is not the posterior probability of exceeding a clinical threshold. Choose and explain the estimand and uncertainty summary that address the clinical question rather than presenting several measures without interpretation.

The simple model also has limits. A common probability within each arm may be unsuitable if important strata or clustering affect outcomes. Missing endpoint data, departures from the intended follow-up, or an endpoint definition that changes during the study can make the analyzed counts misleading. The uniform beta prior is not automatically noninformative for every clinical decision, and in smaller trials a different defensible prior can change the posterior. State material prior choices and investigate sensitivity when they could affect the conclusion.

Common Mistakes

Reversing the contrast. If the report defines \(D=p_C-p_T\), positive values favour treatment; defining \(p_T-p_C\) instead reverses that interpretation.
Confusing percentage points with percent change. A reduction from 22.8% to 15.3% is about 7.4 percentage points, not a 7.4% relative reduction.
Treating the interval as a threshold decision. An interval that includes zero does not itself give the probability of exceeding a specified clinical threshold. Calculate and report that posterior probability directly.
Overstating posterior probability. A 73% posterior probability of exceeding the threshold is not certainty, and it is conditional on the model and prior.
Reporting a number without its reference. State the event, follow-up, arm, denominator, contrast direction, and units so a percentage or interval is interpretable.

Key takeaway: A concise binary-endpoint report connects the observed counts to posterior arm-specific event probabilities, an explicitly defined absolute contrast, its uncertainty interval, and the posterior probability of exceeding a clinically justified threshold. Each quantity answers a distinct question.

Check Your Understanding

Answer the questions, then compare your reasoning with the solutions.

In a hypothetical trial, 18 of 120 treatment participants and 27 of 120 control participants have the event. With independent \(\operatorname{Beta}(1,1)\) priors, write the two posterior distributions.
For the counts in question 1, calculate the posterior means and the posterior mean absolute reduction \(D=p_C-p_T\).
In a report, what does a 90% credible interval for \(D\) describe, and how does that differ from \(P(D>0.05\mid\text{data})\)?
A report says “the treatment reduced risk by 5%,” but the estimated absolute reduction is 0.05. What clarification is needed?
Name two assumptions or data features that should be checked before interpreting a binomial posterior report.

Solutions. (1) The treatment posterior is \(\operatorname{Beta}(1+18,1+120-18)=\operatorname{Beta}(19,103)\); the control posterior is \(\operatorname{Beta}(28,94)\). (2) The posterior means are \(19/122=0.1557\) and \(28/122=0.2295\). Thus \(E(D\mid\text{data})=28/122-19/122=9/122=0.07377\), or a 7.38-percentage-point posterior mean reduction. (3) A 90% credible interval is a posterior interval containing 90% of the probability for \(D\), conditional on the model and data. \(P(D>0.05\mid\text{data})\) is the posterior probability that the reduction exceeds five percentage points; it is not determined simply by whether the interval includes zero. (4) Specify whether “5%” means five percentage points of absolute reduction or a 5% relative reduction, and state the direction and event probabilities. (5) For example, check that the endpoint and follow-up are consistently defined, that denominators and missing outcomes are handled appropriately, and that the common within-arm probability and conditional independence assumptions are credible.