What the Methods Account Must Make Reproducible
A clinical study report (CSR) methods account should let a reviewer reconstruct the analysis from the protocol, analysis plan, and data. For a Bayesian trial, naming a prior distribution or reporting a posterior probability is not enough. The account must connect the clinical question to the estimand, model, prior, computation, and decision rule—and make clear which choices were prespecified.
The focus here is the methods narrative, not the presentation of all posterior results. Those results belong in the report’s results sections. The methods account should nevertheless define the quantities and rules used there precisely enough that a reader can tell what was calculated and why.
A useful starting point is to write the analysis as a chain: the clinical question defines the estimand; the observed data are related to it through a likelihood; prior distributions encode information about unknown parameters before the trial data are analyzed; and Bayes’ rule combines prior and likelihood to produce a posterior distribution. A decision rule, if used, maps the posterior evidence to a specified action.
Here, \(y\) denotes the observed trial data, \(\theta\) the model parameters, \(L(y\mid\theta)\) the likelihood, \(\pi(\theta)\) the prior distribution, and \(\pi(\theta\mid y)\) the posterior distribution. The proportionality omits the normalizing constant, which makes the posterior integrate or sum to one. A methods account should specify each component, not merely state that a Bayesian analysis was performed.
State the Plan Before the Implementation
Begin with the estimand in clinical terms: the population, treatment conditions, endpoint, time point, and approach to events after randomization that affect interpretation. Identify the analysis population and the rule for including participants and observations. Then name the model and its assumptions. For example, for a binary endpoint, a binomial likelihood assumes that each participant contributes a binary outcome and, conditional on the modeled response probability, responses are independent. If the model has treatment, site, or covariate terms, define their coding and role.
Describe the prior on every parameter relevant to the primary estimand or decision. Include its distribution, parameter values, scale, and rationale. If the prior is derived from external evidence, state the source and how that evidence was translated into the distribution. If it is weakly informative or chosen to regularize estimates, explain what values it permits and how that choice is checked. Do not use “noninformative” as a substitute for specifying a prior: most priors have consequences, and a prior that is diffuse on one scale need not be diffuse on another.
The methods account should also state the posterior quantity used for inference and the precise decision criterion. If success requires a posterior probability to exceed a threshold, define the clinical event in that probability, the threshold, and whether equality meets the criterion. The threshold should match the prespecified decision rule; its justification belongs in the rationale for the design and analysis, not in an undocumented change made after observing results.
Finally, state how the posterior was computed. For conjugate calculations, provide the updating rule. For simulation-based methods, identify the software and version, algorithm, chain and iteration settings, warm-up or adaptation, convergence checks, and any decisions about Monte Carlo error. The goal is not to reproduce every software setting in prose, but to give enough detail to reproduce and assess the computation.
Worked Example: Writing a Prespecified Binary-Endpoint Analysis
Hypothetical design. A single-arm trial plans to enroll 20 participants and measures response by a fixed assessment date. Let \(p\) be the response probability in the target population. The success rule is that the posterior probability that \(p\) exceeds 0.50 must be at least 0.85. The prespecified prior is \(p\sim\mathrm{Beta}(1,1)\), and the response count \(X\) is modeled as \(X\mid p\sim\mathrm{Binomial}(20,p)\). These assumptions mean the binary responses are conditionally independent with a common response probability and that the prior is uniform on \(p\) from 0 to 1.
A report-ready methods sentence could read: “The primary analysis modeled the number of responders among 20 participants as binomial with response probability \(p\). The prior distribution for \(p\) was \(\mathrm{Beta}(1,1)\). Trial success was defined as \(P(p>0.50\mid X)\geq0.85\), where \(X\) is the number of responders by the prespecified assessment date. The posterior was calculated using the beta-binomial conjugate update.”
Suppose, hypothetically, 7 of the 20 participants respond. The prior density is proportional to \(p^{1-1}(1-p)^{1-1}\), and the likelihood is proportional to \(p^7(1-p)^{13}\). Multiplying them gives a posterior proportional to \(p^7(1-p)^{13}\), hence \(p\mid X=7\sim\mathrm{Beta}(8,14)\). In general, with a \(\mathrm{Beta}(a,b)\) prior and \(x\) responses among \(n\) participants, the posterior is \(\mathrm{Beta}(a+x,b+n-x)\).
For this example, the posterior probability of exceeding 0.50 can be evaluated using the integer-parameter beta-binomial identity:
The posterior probability is approximately 0.09462, below the prespecified 0.85 threshold, so this hypothetical analysis does not meet the success criterion. This conclusion applies to the stated model, prior, endpoint, and rule; it is not a probability that a null hypothesis is true.
Prior Choices and Sensitivity Analyses
The primary prior should be identified separately from sensitivity priors. For each, state its distribution and parameters, the reason for considering it, and whether it was prespecified. A sensitivity analysis asks whether conclusions depend materially on a defensible alternative assumption. It does not replace the primary analysis after results are known.
For a conjugate illustration, suppose a hypothetical study observes 12 responses among 30 participants. Under a \(\mathrm{Beta}(1,1)\) prior, the posterior is \(\mathrm{Beta}(13,19)\), with posterior mean \(13/(13+19)=13/32=0.4063\). Under a more skeptical \(\mathrm{Beta}(2,8)\) prior, the posterior is \(\mathrm{Beta}(14,26)\), with posterior mean \(14/(14+26)=14/40=0.3500\). Both updates use the same binomial likelihood; the difference comes from the prior.
A report should not present these two means as a complete sensitivity analysis if the actual decision depends on a different posterior quantity. It should apply the prespecified decision rule to each prior, or otherwise compare the posterior features relevant to the estimand, and explain whether the conclusion changes. If prior parameters were elicited from experts, describe the elicitation sufficiently to understand how judgments became a distribution. If borrowing from historical or external data, specify the borrowing model and assumptions about comparability. A brief rationale such as “the prior was chosen to be conservative” is inadequate unless the report explains what conservative means on the parameter scale.
Worked Example: Documenting a Prior Sensitivity Analysis
Suppose the primary analysis uses a \(\mathrm{Beta}(1,1)\) prior for a response probability, and the prespecified sensitivity analysis uses \(\mathrm{Beta}(2,8)\) to assess the impact of a lower prior expectation. In hypothetical data, there are 12 responses among 30 participants. The primary posterior is \(\mathrm{Beta}(1+12,1+30-12)=\mathrm{Beta}(13,19)\); its mean is \(13/32=0.4063\). The sensitivity posterior is \(\mathrm{Beta}(2+12,8+30-12)=\mathrm{Beta}(14,26)\); its mean is \(14/40=0.3500\).
A suitable methods account would state that these priors and the purpose of the sensitivity analysis were prespecified. The results section would then report the relevant posterior comparison under both priors. If the formal decision is \(P(p>0.50\mid X)\geq0.85\), that probability—not just the posterior mean—must be evaluated under each prior before describing whether the decision is robust. The prior means above alone do not establish whether either analysis crosses that threshold.
Deviations: Preserve the Planned Analysis and Describe What Happened
A CSR must distinguish the original plan from the analysis actually performed. For every material deviation, identify the planned method, the implemented method, when and why the change occurred, who authorized it, whether anyone making the change had access to unblinded data, and the likely consequence for interpretation. State whether the change was documented in an amendment, statistical analysis plan, or other controlled record. Do not silently rewrite the original plan to match the final analysis.
A change made before unblinding for an operational reason is different from a choice made after reviewing outcomes. Neither circumstance makes a departure automatically invalid, but the timing and access to data affect how readers assess it. Preserve the prespecified primary analysis where it remains implementable, and clearly label additional analyses as revised, supplementary, or exploratory as appropriate.
Worked Example: Reporting a Change to the Analysis Window
In a hypothetical randomized trial, the protocol defines response using an assessment scheduled at day 28. During the trial, a documented site-system outage delays assessments for some participants. Before the database is locked, the sponsor approves a sensitivity analysis that uses the first assessment recorded from day 28 through day 35 for participants with no day-28 assessment. The original day-28 endpoint definition remains the primary analysis.
A complete account would state: “The prespecified primary endpoint was response at day 28. Because of a documented system outage, a supplementary sensitivity analysis used the first available response assessment through day 35 when no day-28 assessment was available. The day-28 analysis remained primary. The change was approved before database lock on the basis of the outage; the report records the approval date and confirms whether those approving the change had access to unblinded outcomes.”
For a concrete data reconciliation, suppose 52 participants were randomized to each group and, in the treatment group, 30 had a recorded day-28 response among 48 with an observed day-28 outcome. Then 4 treatment participants had no observed day-28 outcome, since \(52-48=4\). If 2 of those 4 had a day-35 assessment, the sensitivity analysis must say how those 2 outcomes were incorporated and how the other 2 were handled. This arithmetic check does not determine the missing-data model; the report must separately state the assumptions and method used for unavailable outcomes. It must not imply that a later observation is a day-28 response unless the endpoint definition explicitly supports that substitution.
Diagnostics, Alternatives, and Limits
Diagnostics should match the computation and model. For Markov chain Monte Carlo, report evidence that chains explored the relevant posterior adequately, such as trace-plot review, effective sample sizes, convergence statistics, and Monte Carlo error for decision-critical quantities. If sampling is poor, a nominal posterior probability can be numerically unreliable. For a closed-form calculation, verify the update, parameterization, and numerical evaluation. For any model, check whether the likelihood and prior imply plausible behavior, whether the model fits important features of the data, and whether influential observations or sparse data make the result fragile.
Common warning signs include divergent or poorly mixing chains, sensitivity to initial values, a posterior concentrated near parameter boundaries, implausible prior predictive outcomes, and conclusions that change under reasonable prior alternatives. State how such problems were investigated and whether they affected the primary analysis. A sensitivity analysis is not a guarantee of robustness: it assesses only the alternatives actually examined.
A Bayesian report should be compared with a frequentist analysis only when that alternative is part of the analysis plan or relevant to interpretation. The Bayesian account specifies a prior and reports posterior quantities conditional on the model and data. A frequentist account instead specifies its sampling model and procedure, and reports operating characteristics or confidence intervals according to that procedure. A posterior probability is not a frequentist p-value, and a credible interval is not a confidence interval. If both analyses are presented, identify which is primary and avoid describing one as a direct recalculation of the other.
The Bayesian analysis remains conditional on its model, prior, endpoint definition, and assumptions about missing or intercurrent outcomes. A well-written methods section makes these dependencies visible; it cannot remove uncertainty caused by weak data, model misspecification, or limited relevance of external information. It should not claim that a favorable posterior conclusion proves treatment effectiveness outside the studied population and conditions.
State the estimand, endpoint, time point, population, and analysis set before describing calculations.
Give the likelihood, parameterization, assumptions, priors, and any covariate or borrowing structure.
Define the posterior quantity, clinical threshold, and exact decision criterion.
Describe computation, diagnostics, sensitivity analyses, and each material deviation from the prespecified plan.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- What information must be supplied to make a prior reproducible rather than merely named?
- A trial uses a posterior probability threshold of 0.90. Why is it insufficient to report only the posterior mean?
- What should a report say when a sensitivity analysis was added after the protocol was finalized?
- A Bayesian primary analysis and a frequentist supplementary analysis are both reported. What distinction should the methods account preserve?
Solutions. (1) State the prior distribution, parameter values, parameter scale, and rationale; explain any external evidence or elicitation used to set it. (2) The mean is not the decision quantity: the account must define and calculate the specified posterior probability and compare it with 0.90. (3) Identify it as an additional analysis, give its timing and rationale, document approval and data access where relevant, and keep the prespecified primary analysis distinct. (4) Describe each analysis under its own framework: posterior quantities are conditional on the Bayesian model and prior, while frequentist quantities arise from a specified sampling procedure; neither should be mislabeled as the other.