Recording and Justifying Priors in a Trial Protocol

Why Prior Documentation Is Part of the Statistical Method

A prior is not fully specified by saying that it is “weakly informative,” “based on expert opinion,” or “centered on no effect.” Those phrases do not tell another analyst which parameter was modeled, what scale it was on, how the distribution was chosen, or what clinical judgments support it. In a clinical trial protocol, the prior specification is part of the analysis method: it should be clear enough that an independent statistician can implement the same model without guessing.

The focus here is not how to elicit a new judgment or choose among prior families. The earlier material on elicitation and distribution-parameter translation covers those tasks. The protocol-writing task is to preserve the decisions and reasoning in a reproducible record, including enough detail to reveal ambiguity or an unintended change in meaning.

Definition: A prior specification is a reproducible statement of the probability distribution assigned to each model parameter before observing the trial outcome data. A prior justification records why that specification is reasonable for the clinical question and identifies the evidence, judgments, and assumptions on which the choice depends.

For each parameter, define what it represents before giving its distribution. State the comparison direction, outcome coding, scale, and units. Then give the distribution and all its parameters, explain how those parameters were obtained, and record the source and clinical rationale. If parameters are related, state the joint structure rather than listing marginal priors alone. The specification should also identify the protocol version and the point at which it was finalized.

A Reproducible Specification

A useful protocol entry contains the fields below. Some details belong in an appendix or analysis plan, but the main protocol should point to the controlled document that contains them.

Parameter and estimand: define the quantity being modeled, including the population, outcome, time point, and treatment contrast where relevant.
Scale, direction, and units: state whether the parameter is a risk, risk difference, log odds ratio, mean difference, or another quantity. Identify which sign or ratio favors which treatment and give units for measured outcomes.
Distributional form and parameters: name the distribution and provide its parameters using an unambiguous convention. For example, state whether a Normal distribution is parameterized by mean and variance or mean and standard deviation.
Source and elicitation method: identify the evidence or expert group, the date or document, the questions asked, and the method used to translate responses into distribution parameters. If a published source is used, provide enough bibliographic detail to locate it.
Clinical rationale and assumptions: explain why the location, spread, support, and any dependence assumptions make sense for the trial. Separate clinical judgments from mathematical convenience.
Implementation and governance: record transformations, software-relevant conventions, who reviewed the specification, the date it was frozen, and how amendments will be handled.

The word “source” should not obscure the difference between data and judgment. A prior may draw on historical trial results, external registries, expert elicitation, or a stated combination. The protocol should say which information was used and how it informed the prior. If external data were discounted or adjusted for differences in population or care, document that choice rather than describing the prior as simply “historical.”

Key principle: A prior can be numerically precise yet poorly documented. Reproducibility requires both the mathematical distribution and the provenance and reasoning behind it.

Worked Specifications

Worked Example: A Beta Prior for the Control-Arm Event Risk

Suppose a hypothetical trial compares a new treatment with control for a binary adverse event by 30 days. Define \(p_C\) as the probability of the event by day 30 in the control arm. A clinician panel judges that a control risk near 0.20 is reasonable and asks the statistician to represent that judgment with a Beta distribution having an effective prior sample size of 10.

Distribution and translation. For \(p_C\sim\operatorname{Beta}(a,b)\), the mean is \(a/(a+b)\). Setting the mean to 0.20 and the total \(a+b\) to 10 gives

a=10(0.20)=2,\qquad b=10(1-0.20)=8, \qquad p_C\sim\operatorname{Beta}(2,8).

The prior mean is \(2/(2+8)=0.20\), or a 20% event risk. The Beta variance is \(ab/[(a+b)^2(a+b+1)]\), so here it is \(2(8)/[10^2(11)]=0.01455\). The effective sample size of 10 is a way to describe the chosen concentration; it is not a claim that ten actual control patients supplied these prior data.

Protocol record. The entry should name the parameter as the 30-day control-arm event probability, specify its support as 0 to 1, give \(\operatorname{Beta}(2,8)\), and state the mean and effective sample size used in the translation. It should identify the hypothetical panel and elicitation record as the source, describe how the panel was asked to judge the risk, and explain why a risk around 20% is clinically plausible for this example. The protocol should also state whether this prior is independent of the treatment-effect prior or linked to it through a joint model. Without that last statement, the full prior model may remain incomplete.

Worked Example: A Prior on a Risk Ratio Scale

Consider a hypothetical trial with an adverse event outcome. Let \(RR=p_T/p_C\), where \(p_T\) and \(p_C\) are the treatment- and control-arm event risks. Thus \(RR<1\) favors treatment. Suppose a hypothetical elicitation produced a central 95% range from 0.45 to 1.25 and a central value of 0.75, and the protocol represents this judgment as Normal on the log-risk-ratio scale.

Distribution and calculation. Write \(\log(RR)\sim N(\mu,\sigma^2)\), where \(\sigma\) is the standard deviation. The elicited central value gives \(\mu=\log(0.75)=-0.2877\). Using the Normal central 95% approximation, the log-scale endpoints are \(\mu\pm1.96\sigma\). The endpoints are symmetric around \(\log(0.75)\), since \(\log(0.45)+\log(1.25)=2\log(0.75)\). Therefore,

\sigma =\frac{\log(1.25)-\log(0.45)}{2(1.96)} =\frac{0.2231-(-0.7985)}{3.92} \approx 0.2606.

The resulting specification is \(\log(RR)\sim N(-0.2877,0.2606^2)\), or equivalently a lognormal prior on \(RR\). Its approximate central 95% interval on the log scale is \(-0.2877\pm1.96(0.2606)=[-0.7985,0.2231]\). Exponentiating gives approximately \([0.45,1.25]\). These are prior quantiles, not a statement that the parameter has a 95% probability of lying in a fixed interval after the parameter is treated as fixed.

Protocol record. The specification should state that the modeled parameter is the treatment-to-control risk ratio, define the event and 30-day time point, and say that ratios below 1 favor treatment. It should identify the log scale and use the Normal mean and variance convention explicitly. The elicitation record should preserve the hypothetical experts’ questions and the stated central value and range, along with the approximation used to translate them. The clinical rationale should explain why the range includes both potentially important benefit and some possibility of harm.

Worked Example: A Prior for a Continuous Treatment Difference

In a hypothetical trial, the outcome is a biomarker measured in mg/dL at a prespecified follow-up visit. Define \(\delta=\mu_T-\mu_C\), the treatment-arm mean minus the control-arm mean. Lower values are clinically favorable, so negative \(\delta\) favors treatment. Suppose an expert group’s documented judgment is represented by a Normal prior centered at no average difference, with standard deviation 5 mg/dL.

The protocol specification is

\delta\sim N(0,(5\ \text{mg/dL})^2).

Here the first parameter is the prior mean, 0 mg/dL, and the second is the variance, 25 \((\text{mg/dL})^2\). Under the Normal model, the approximate central 95% prior interval is \(0\pm1.96(5)=[-9.8,9.8]\) mg/dL. This calculation makes the scale of the judgment visible: it allows differences in either direction, and the negative direction represents benefit under the stated outcome convention.

The record should identify the follow-up visit, the units, the subtraction order, and the fact that lower biomarker values are favorable. It should cite the hypothetical expert elicitation record, explain how the experts were asked to judge plausible between-arm differences, and state why a spread of 5 mg/dL is clinically defensible. It should not say merely “a Normal prior was used,” because that omits the parameter, units, and numerical values needed to reproduce the analysis.

Why the Specification Must Be Complete

The examples illustrate a general requirement: the same verbal description can correspond to different probability models. A Normal distribution on a risk ratio is not the same as a Normal distribution on its logarithm. A prior for a treatment-arm risk is not interchangeable with a prior for a treatment contrast. A mean difference changes meaning if the subtraction order changes. Units matter because a standard deviation of 5 mg/dL is not a standard deviation of 5 mmol/L.

For a vector of parameters, listing separate marginal distributions may still fail to define the joint prior. If two treatment effects are correlated, state the joint distribution and its correlation or covariance parameters. If independence is assumed, state that explicitly and explain why it is appropriate for the model. A dependence assumption is part of the prior specification, not an implementation detail.

A short justification can be structured as a chain of reasoning: identify the clinical question, explain which evidence or judgments inform the parameter, describe how those inputs map to the chosen distribution, and state the limitations of that mapping. This makes it possible for reviewers to distinguish a genuine clinical assumption from a convenient mathematical choice. It also helps identify when a prior is not transportable to a new population, endpoint definition, or treatment context.

Review, Diagnostics, and Alternatives

Before the protocol is finalized, check the specification against the model and analysis code. Confirm that each prior parameter has the same definition, scale, and units in both places. Check that a transformed parameter is described on the correct scale, that all distribution parameters are present, and that the stated treatment direction agrees with the model’s sign convention. For a joint model, verify that every dependence or independence assumption is recorded. A missing time point, ambiguous reference arm, or unspecified variance convention is a warning that two analysts could implement different priors while claiming to follow the same protocol.

Protocol documentation complements, but does not replace, other approaches to communicating prior choices. A fully numerical prior is more reproducible than a qualitative label such as “skeptical,” but numerical detail alone does not explain its clinical basis. Citing historical evidence can improve traceability, but a citation alone does not show how that evidence was translated or adjusted. A set of alternative priors can show how conclusions depend on assumptions, but it does not remove the need to specify the primary prior. These approaches answer different questions and can be used together.

The record also has limits. A clear rationale does not prove that a prior is correct, and a well-documented elicitation can still be affected by expert disagreement, anchoring, or limited applicability of external evidence. Documentation makes these assumptions inspectable; it does not eliminate uncertainty. Separate assessments of implications for trial data and sensitivity to prior choices are useful, but they should not be confused with the protocol entry itself.

Define the parameter.
Name the estimand, outcome, time point, comparison, scale, direction, and units before writing a distribution.

State the mathematics.
Give the distribution family and every parameter, clarify the parameterization, and specify transformations and joint dependence.

Preserve provenance.
Identify the source, elicitation procedure, translation to distribution parameters, and any adjustments or assumptions.

Explain and govern the choice.
Give the clinical rationale, record reviewers and version dates, and define how changes before or during the trial will be documented.

Common Mistakes

Using a label instead of a distribution. “Weakly informative” does not define a prior. Give the mathematical form and all numerical parameters.
Omitting the parameter scale. A distribution on a log ratio and one on a ratio have different meanings. Name the scale and any transformation.
Leaving out direction or units. State which treatment contrast is used and which values favor treatment; include units for continuous outcomes.
Treating expert judgment as self-explanatory. Record who supplied it, what they were asked, when, and how their responses informed the parameters.
Listing only marginal priors. State whether parameters are independent or describe their joint distribution and dependence structure.
Changing a prior without a record. Any amendment should identify what changed, why, when, who approved it, and whether trial outcome data had been accessed.

Key takeaway: A defensible protocol specification lets a second analyst reproduce the prior and understand its clinical basis. Define the parameter first; then record its scale, distribution, source, elicitation path, rationale, dependence assumptions, and version history.

Check Your Understanding

Answer each question, then compare your response with the brief solutions.

A protocol says, “The treatment effect has a Normal skeptical prior.” Name three essential pieces of information still needed.
A prior is assigned to the log risk ratio for an adverse event, with values below 1 favoring treatment. What scale and direction must the protocol state?
For a continuous outcome, why must a protocol define whether the contrast is treatment minus control or control minus treatment?
A model gives separate priors for two endpoints but says nothing about their relationship. What specification issue remains?

Solutions. (1) At minimum, define the treatment-effect parameter and its scale, give the Normal distribution’s numerical parameters and parameterization, and document the source and rationale. (2) State that the modeled quantity is the logarithm of the treatment-to-control risk ratio, and that negative log ratios (equivalently, ratios below 1) favor treatment for this adverse event. (3) Reversing the subtraction reverses the sign, so it changes which parameter values represent benefit; the contrast convention is needed to interpret and reproduce the prior. (4) The joint prior is not fully defined: the protocol must specify the dependence structure or explicitly state an independence assumption.