From Analysis Output to a Results Section
A trial results section has to do more than list posterior means and intervals. Readers need to know which participants and endpoint the analysis concerns, what contrast is being estimated, what model and prior produced the posterior, and how the estimated effect relates to clinical importance. The numerical summary, table, figure, and conclusion must all tell the same story.
The earlier material on credible intervals, posterior probabilities, endpoint scales, tables, figures, calibrated conclusions, and computation diagnostics covers the component reporting skills. Here the task is to assemble them into a coherent account. The example below is hypothetical; its numbers are provided solely to demonstrate reporting.
Set the Analysis Context Before Giving Results
A reader should be able to tell what was compared before encountering the estimate. State the randomized groups, endpoint definition and assessment time, analysis population, effect scale and direction, and the main model assumptions. If participants with missing outcomes are handled by an imputation or model, name that approach rather than silently describing the analysis as if every outcome had been observed.
For the main example, suppose a hypothetical randomized trial assigned 160 adults equally to a new treatment or control. The primary endpoint is clinical improvement by week 12, recorded as a binary outcome. All 160 randomized participants have an observed week-12 outcome. Let \(p_T\) and \(p_C\) denote the probabilities of improvement under assignment to treatment and control, respectively, in the trial’s target population. Define the absolute treatment contrast as \(\Delta=p_T-p_C\), so a positive value favors treatment.
Assume outcomes are independent between participants conditional on their group response probabilities, and that each group follows a binomial model. Use independent Beta(1,1) priors for \(p_T\) and \(p_C\). This simple analysis presumes that the binary endpoint is measured consistently, randomization and follow-up are handled as specified, and the prior adequately represents the intended baseline information. It does not adjust for covariates or account for clustering.
Worked Example: Constructing the Primary Endpoint Results
Worked Example: Constructing the Primary Endpoint Results
Suppose 40 of 80 participants assigned to treatment improved by week 12, compared with 30 of 80 assigned to control. Under the stated binomial model and independent Beta(1,1) priors, show the posterior response rates and construct a concise report of the treatment contrast.
Update each group’s response probability. If \(Y\) improvements are observed among \(n\) participants and \(p\sim\operatorname{Beta}(a,b)\), the binomial likelihood is proportional to \(p^Y(1-p)^{n-Y}\). Multiplying by the prior density gives a posterior proportional to \(p^{a+Y-1}(1-p)^{b+n-Y-1}\), which is a Beta distribution. Thus the posterior is \(\operatorname{Beta}(a+Y,b+n-Y)\).
For treatment, the posterior is \(\operatorname{Beta}(1+40,1+80-40)=\operatorname{Beta}(41,41)\). Its posterior mean is \(41/(41+41)=0.500\). For control, the posterior is \(\operatorname{Beta}(1+30,1+80-30)=\operatorname{Beta}(31,51)\), with posterior mean \(31/82=0.378\). The difference between these posterior means is \(0.500-0.378=0.122\), or 12.2 percentage points.
The contrast’s posterior distribution is the distribution of \(p_T-p_C\), not a Beta distribution. For this example, suppose one million independent draws were generated from the two stated posterior distributions and the contrasts calculated draw by draw. An illustrative Monte Carlo summary is a posterior mean difference of 0.122, a 95% equal-tailed credible interval of about \(-0.03\) to \(0.27\), and \(P(\Delta>0\mid\text{data})\) of about 0.94. The interval is expressed as a difference in response probabilities: from about 3 percentage points lower to 27 points higher for treatment. It includes no difference.
Suppose clinical collaborators had specified an absolute improvement of 10 percentage points as an important threshold before examining these results. If the same posterior draws give \(P(\Delta>0.10\mid\text{data})\approx0.62\), the report should distinguish that probability from the probability of any benefit. It is not strong evidence that the effect exceeds the threshold; nor does it mean the effect is exactly 10 percentage points with probability 0.62.
A concise report could read: “Among participants with observed week-12 outcomes, improvement occurred in 40/80 (50.0%) assigned to treatment and 30/80 (37.5%) assigned to control. Under the independent binomial model with independent Beta(1,1) priors, the posterior mean absolute difference was 12.2 percentage points favoring treatment (95% credible interval, approximately −3 to 27 percentage points). The posterior probability of any benefit was approximately 0.94, while the probability that benefit exceeded 10 percentage points was approximately 0.62.”
The posterior means and contrast are calculated directly from the stated model; the contrast interval and probabilities are illustrative numerical summaries of the specified posterior computation. A report based on an actual analysis must use its verified computation output, not these example values.
Make the Table and Narrative Agree
A table should let readers compare observed outcomes with posterior summaries while preserving the meaning and units of each quantity. The table below keeps arm-level observed proportions separate from the posterior treatment contrast. The safety row is also hypothetical: suppose 9 of 80 treatment participants and 12 of 80 control participants experienced a serious adverse event. The negative contrast is treatment minus control, so its direction favors treatment for this harm endpoint, but its interval is wide.
| Endpoint and time point | Treatment, observed | Control, observed | Posterior contrast: treatment minus control | Posterior interpretation |
|---|---|---|---|---|
| Clinical improvement by week 12 | 40/80 (50.0%) | 30/80 (37.5%) | 12.2 percentage points (95% CrI approximately −3 to 27) | \(P(\Delta>0\mid\text{data})\approx0.94\); \(P(\Delta>0.10\mid\text{data})\approx0.62\) |
| Serious adverse event during follow-up | 9/80 (11.3%) | 12/80 (15.0%) | Approximately −3.7 percentage points (95% CrI approximately −14 to 7) | Posterior uncertainty includes both lower and higher event rates with treatment |
The safety contrast’s approximate posterior mean can be checked using the same Beta(1,1) updating: treatment gives \(\operatorname{Beta}(10,72)\), with mean \(10/82=0.122\); control gives \(\operatorname{Beta}(13,69)\), with mean \(13/82=0.159\). Their difference is \(0.122-0.159=-0.037\), or about −3.7 percentage points. The table’s safety interval is an illustrative posterior contrast summary; it should not be inferred from the difference in the two arm-level interval endpoints.
The narrative should explain the primary endpoint, then briefly address other prespecified outcomes without implying that a favorable point estimate establishes benefit or safety. A figure can show posterior distributions for \(p_T-p_C\), with a vertical reference at zero and, when clinically justified, another at the 10-point threshold. Its caption should name the contrast, units, time point, and any displayed credible interval. A plot is a complement to—not a replacement for—the endpoint definition and numerical results.
Worked Example: Reporting an Uncertain Safety Contrast
Worked Example: Reporting an Uncertain Safety Contrast
Using the serious adverse event counts above, write a two-sentence interpretation that reports the observed data, gives the posterior contrast in the correct direction, and avoids claiming that treatment is safer.
The observed event proportions are \(9/80=0.1125\), or 11.3%, with treatment and \(12/80=0.150\), or 15.0%, with control. The posterior mean contrast is \(0.122-0.159=-0.037\), or approximately 3.7 percentage points fewer events with treatment. The illustrative 95% credible interval, about −14 to 7 percentage points, includes no difference and values in both directions.
A suitable report is: “Serious adverse events occurred in 9/80 (11.3%) participants assigned to treatment and 12/80 (15.0%) assigned to control. The posterior mean risk difference was approximately −3.7 percentage points (95% credible interval, approximately −14 to 7), leaving substantial uncertainty about whether treatment decreases or increases the event rate.”
This wording describes the observed counts and posterior uncertainty. It does not convert a negative posterior mean into a claim of proven safety, and it avoids the opposite mistake of concluding that the groups are equivalent because the interval includes zero.
Justify the Summary and Check the Reporting
The posterior updating in the primary example follows directly from conjugacy. For a binomial observation \(Y=y\) among \(n\) participants, the likelihood is proportional to \(p^y(1-p)^{n-y}\). A Beta(\(a,b\)) prior has density proportional to \(p^{a-1}(1-p)^{b-1}\). Their product is proportional to \(p^{a+y-1}(1-p)^{b+n-y-1}\), the kernel of a Beta(\(a+y,b+n-y\)) density. This justifies the arm-specific posteriors, but it does not give a closed-form Beta distribution for their difference. Posterior draws or numerical integration are needed for the contrast interval and threshold probabilities.
Before finalizing a results section, check the data, analysis, and presentation together. The denominators in the table must match the named analysis population; the endpoint and time point must match the protocol; the contrast sign must match the stated favorable direction; and all intervals must be labeled as credible intervals with units. Verify that posterior probabilities use the same definition of benefit or clinical importance stated in the text.
Name the endpoint, time point, population, and treatment contrast, including its direction and units.
Summarize the likelihood, prior, and consequential assumptions, and identify how missing outcomes and computation were handled.
Give observed group data, posterior contrast summaries, and probabilities tied to defined clinical thresholds where relevant.
Describe what the posterior supports, what remains uncertain, and which limitations affect interpretation.
Diagnostics and failure modes matter because polished prose cannot repair an unreliable analysis. Check convergence and effective sample size when using Markov chain Monte Carlo; inspect trace plots and Monte Carlo error for the reported quantities. For direct independent draws from conjugate posteriors, verify that the sampled arm distributions match the analytic Beta posteriors and that increasing the number of draws stabilizes contrast summaries. Also check for coding errors in endpoint direction, inconsistent denominators, implausible prior implications, and unexplained departures from the planned analysis.
The Bayesian summaries answer questions about the posterior distribution under the specified model and prior. A frequentist confidence interval and p-value answer different questions and should not be inserted as if they were interchangeable with a credible interval and posterior probability. Nor does a posterior probability remove the need to explain assumptions, assess clinical relevance, or consider external validity.
Worked Example: Writing a Calibrated Conclusion
Worked Example: Writing a Calibrated Conclusion
Suppose the primary endpoint summaries are those in the first worked example, and the serious adverse event summaries are those in the table. Write a short conclusion that reflects both the evidence for improvement and the uncertainty about clinical importance and safety.
The posterior mean primary contrast is favorable, and the posterior probability of any benefit is about 0.94. However, its 95% credible interval includes zero, and the probability of exceeding the prespecified 10-point threshold is only about 0.62. The safety contrast is also uncertain, with its interval spanning lower and higher event rates under treatment.
A calibrated conclusion is: “In this hypothetical trial, the posterior distribution favored a higher week-12 improvement rate with treatment, but substantial uncertainty remained about whether the absolute benefit exceeded 10 percentage points. The observed serious adverse event data and posterior contrast were likewise inconclusive about the direction of a safety difference; these results do not establish that treatment is safer.”
This conclusion neither dismisses the posterior evidence nor turns it into certainty. It distinguishes the direction of the estimated effect from the probability of a clinically important effect and recognizes that the safety result is uncertain.
Limitations and Alternatives
A results section is only as informative as its estimand and assumptions. A simple independent binomial model may be inadequate if the trial uses repeated measurements, clustering, stratification that should be represented in the analysis, informative missingness, or a more complex endpoint definition. A uniform prior is transparent but is still a modeling choice; different reasonable priors can matter when data are sparse. These issues should be addressed by the analysis plan and reported when they materially affect interpretation, not hidden by a compact table.
Alternative presentations include reporting each arm’s posterior response rate alongside the contrast, showing a posterior density plot, or emphasizing a prespecified clinically meaningful threshold. These choices can complement one another, but none changes the estimand. A frequentist analysis may also be reported as a planned sensitivity or regulatory analysis, with its own interval and test clearly identified. The main report should not mix frameworks or select whichever summary sounds most favorable.
Check Your Understanding
Use the hypothetical trial and conventions in this tutorial. Each answer follows the question.
- Why is the primary contrast defined as \(p_T-p_C\), and what does a positive value mean here?
- With 40 improvements among 80 treatment participants and a Beta(1,1) prior, what is the posterior distribution and mean for \(p_T\)?
- Why does the interval for the posterior contrast need not be obtained by subtracting endpoints of the two arm-level credible intervals?
- How should \(P(\Delta>0.10\mid\text{data})\approx0.62\) be described?
- The safety contrast interval includes zero. What conclusion is justified, and what conclusion is not?
Solutions. (1) The definition fixes treatment minus control, so a positive difference means a higher improvement probability with treatment. (2) The posterior is \(\operatorname{Beta}(41,41)\), with mean \(41/82=0.500\). (3) The contrast is a random variable formed by subtracting paired posterior draws from two distributions; its quantiles are not generally the differences between corresponding marginal quantiles. (4) It is the posterior probability, under the stated model and prior, that the absolute benefit exceeds 10 percentage points; it does not establish that the benefit does exceed that threshold. (5) The data leave uncertainty about whether treatment reduces or increases the event rate. The interval does not establish equivalence, and a negative posterior mean does not prove treatment is safer.