Describing Prior Assumptions in Trial Results

Why Prior Assumptions Belong in a Trial Results Report

A posterior summary is conditional on a model that includes both the likelihood for the observed data and a prior distribution for the unknown parameter. Readers therefore need to know what prior was used to understand the assumptions behind the reported posterior results. A statement such as “a weak prior was used” is not enough: it does not identify the parameterization, distribution, or amount and direction of prior information.

A clear account does not suggest that the prior alone establishes whether a treatment works. The posterior combines prior assumptions with the trial data through the likelihood. When data are informative, the likelihood can substantially update the prior; when data are limited, prior assumptions may have greater influence. Reporting the prior makes that relationship visible so the posterior can be interpreted in context.

Definition: A prior distribution is a probability distribution for an unknown parameter, denoted here by $\theta$, before conditioning on the current trial data. The posterior distribution is the distribution for $\theta$ after conditioning on those data. In a Bayesian model, $$ p(\theta\mid y) \propto p(y\mid\theta)\,p(\theta), $$ where $y$ denotes the observed data, $p(y\mid\theta)$ is the likelihood, and $p(\theta)$ is the prior density or probability mass function.

The expression does not mean that a prior and data contribute equally, or that the prior can be interpreted separately as the trial result. The influence of each depends on the model and the information in the data. A report should describe the prior sufficiently for a reader to understand its implications, then report posterior results and conclusions separately.

What to State About a Prior

For each key parameter, report the prior in a form that can be identified and interpreted. At minimum, give the parameter's meaning, its scale and units, the prior family, and the distribution's parameter values. If a joint prior is used, describe any dependence assumptions that are important to the analysis. If the prior is applied after a transformation, name the transformed parameter and explain how it maps back to the clinical scale.

The rationale is also part of the report. Explain whether the distribution reflects a design assumption, external evidence, a prespecified modeling choice, or another stated consideration. Where a prior has a clinically interpretable implication, give it. For example, describe the center and spread of a prior for a treatment difference in points, or translate a prior for a log odds ratio to odds-ratio values. Do not use adjectives such as “vague” or “noninformative” as substitutes for this information: the effect of a prior depends on the parameterization and model.

Name the parameter.
Define what it represents, including the treatment contrast, direction of benefit, and clinical units where relevant.

Specify the distribution.
Give its family and parameter values, and state whether it is marginal or part of a joint prior.

Explain the scale and implications.
Describe the center and spread in interpretable terms, translating a transformed parameter when needed.

Connect the prior to the result.
Present the posterior result as conditional on the model and data; if a sensitivity analysis is reported, identify what changed and what remained fixed.

The same named distribution can imply different assumptions under different parameterizations. For example, a beta distribution may be written using shape parameters, while a normal distribution is commonly written using a mean and variance. State the actual values and parameterization, rather than relying on the family name alone. For a normal prior, report the standard deviation or variance unambiguously. For a prior on a transformed scale, report both the transformation and enough information to interpret its implications on the clinical scale.

Worked Example: Reporting Priors for Response Probabilities

Suppose a hypothetical randomized trial observes responses in 15 of 40 participants assigned to treatment and 9 of 40 assigned to control. Let $p_T$ and $p_C$ denote the response probabilities in the two arms. The analysis uses independent $\mathrm{Beta}(2,2)$ priors for $p_T$ and $p_C$. Under the beta-binomial model, a $\mathrm{Beta}(a,b)$ prior and $x$ responses among $n$ participants give a $\mathrm{Beta}(a+x,b+n-x)$ posterior.

A complete description would identify each parameter as an arm-specific response probability, give the beta shape parameters, and state that the priors are independent. Each prior has mean $2/(2+2)=0.5$. Its concentration is modest: the total of its shape parameters is 4, though that number should not be presented as a universal measure of prior influence independent of the model.

For treatment, the posterior is $$ p_T\mid y \sim \mathrm{Beta}(2+15,\,2+40-15) =\mathrm{Beta}(17,27), \qquad E(p_T\mid y)=\frac{17}{44}=0.3864. $$ For control, it is $$ p_C\mid y \sim \mathrm{Beta}(2+9,\,2+40-9) =\mathrm{Beta}(11,33), \qquad E(p_C\mid y)=\frac{11}{44}=0.2500. $$

The difference between these posterior means is $0.3864-0.2500=0.1364$, or 13.64 percentage points. This is a difference of posterior means, not by itself a credible interval or a posterior probability of benefit. The report should give the desired posterior summary for the treatment contrast separately. The prior description helps explain the analysis, but the observed response counts also contribute to these posterior results.

Reporting Priors on Transformed Scales

In clinical analyses, a parameter may be modeled on a scale convenient for the likelihood even though readers interpret an effect on another scale. A report should not leave the transformation implicit. State the modeled parameter, its relationship to the clinical measure, and an interpretable summary of what the prior permits. This is especially important when a symmetric prior on one scale becomes asymmetric after transformation.

Worked Example: Describing a Prior for a Log Odds Ratio

Suppose a hypothetical trial models $\theta=\log(\mathrm{OR})$, where $\mathrm{OR}$ is the treatment-versus-control odds ratio for response. Positive values of $\theta$ indicate higher response odds under treatment. The analysis uses $\theta\sim N(0,0.5^2)$, with the second normal-distribution argument denoting the variance.

The prior is centered at no difference in odds because $\exp(0)=1$. Its standard deviation is 0.5 on the log-odds-ratio scale. To describe its spread on the odds-ratio scale, an approximate central 95% prior interval for $\theta$ is $$ 0 \pm 1.96(0.5) = [-0.98,\,0.98]. $$ Exponentiating the endpoints gives an approximate interval from $\exp(-0.98)=0.375$ to $\exp(0.98)=2.664$ for the odds ratio.

A concise report could say that the log odds ratio had a normal prior with mean 0 and standard deviation 0.5, corresponding to a prior centered on an odds ratio of 1 and an approximate central 95% prior interval of 0.375 to 2.664. This interval describes the prior distribution, not the posterior treatment effect and not a clinical conclusion. The report should also identify the outcome and odds-ratio direction so that “treatment versus control” is unambiguous.

Showing How Prior Assumptions Relate to Posterior Results

A useful report distinguishes the prior specification from the evidence provided by the trial. Posterior summaries answer questions about the parameter conditional on both the specified model and the observed data. When the result is sensitive to a plausible change in prior assumptions, that sensitivity is relevant to interpretation and should be described rather than hidden. Conversely, a single posterior result should not be called “driven by the prior” without examining how it changes under reasonable alternatives.

A compact sensitivity comparison can hold the likelihood, endpoint, and other modeling choices fixed while changing a prespecified prior feature. The purpose is to show whether the reported inference is robust to that choice, not to select whichever prior produces the preferred conclusion. Sensitivity results do not replace the primary analysis; identify the primary prior and label alternative analyses clearly.

Worked Example: Comparing Prior Influence in a Continuous-Outcome Model

In a hypothetical trial, define $\delta$ as the mean outcome in the control arm minus the mean outcome in the treatment arm, measured in symptom-score points; positive values favor treatment. Suppose the trial summary is $y=0.12$ points with known standard error $0.10$ points, and the model is $y\mid\delta\sim N(\delta,0.10^2)$. Compare two centered normal priors, $N(0,0.20^2)$ and $N(0,0.05^2)$. These are illustrative alternative assumptions, not a claim that either is universally appropriate.

For a normal observation with known variance and a normal prior, the posterior variance is the reciprocal of the sum of prior and likelihood precisions. The posterior mean is the precision-weighted average of the prior mean and observed estimate: $$ V_{\text{post}}=\left(\frac{1}{\tau^2}+\frac{1}{SE^2}\right)^{-1}, \qquad m_{\text{post}}=V_{\text{post}}\left(\frac{\mu_0}{\tau^2}+\frac{y}{SE^2}\right), $$ where $\mu_0$ and $\tau^2$ are the prior mean and variance.

With $\mu_0=0$, $\tau=0.20$, and $SE=0.10$, the prior precision is $1/0.20^2=25$ and the likelihood precision is $1/0.10^2=100$. Thus, $$ m_{\text{post}}=\frac{0(25)+0.12(100)}{25+100}=0.096, \qquad V_{\text{post}}=\frac{1}{125}=0.008, \qquad SD_{\text{post}}=\sqrt{0.008}=0.08944. $$

With $\tau=0.05$, the prior precision is $1/0.05^2=400$, so $$ m_{\text{post}}=\frac{0(400)+0.12(100)}{400+100}=0.024, \qquad V_{\text{post}}=\frac{1}{500}=0.002, \qquad SD_{\text{post}}=\sqrt{0.002}=0.04472. $$

The narrower prior pulls the posterior mean closer to zero and reduces posterior spread in this model. The example shows why the prior's scale matters, but neither posterior mean alone establishes a trial conclusion. A results report would identify the prior used for the primary analysis, present the relevant posterior summaries for $\delta$, and explain whether the alternative assumption changes the interpretation in clinical units.

Justifying the Prior Description and Checking Its Clarity

A practical test of a prior description is whether another analyst could reconstruct the prior used in the reported model. The report should make clear which parameter receives the prior, whether the values are on the original or transformed scale, and whether multiple parameters are modeled independently or jointly. If a prior depends on external information, explain the source and how that information maps to the parameter. If the prior was specified without direct external-data input, state the modeling rationale rather than implying that it is assumption-free.

The posterior relationship also explains why a report must present prior and data information together. For the normal model in the preceding example, completing the square in the product of the likelihood and prior yields a quadratic in $\delta$. Its coefficient on $\delta^2$ is the sum of the two precisions, and its linear coefficient is the sum of the precision-weighted observed estimate and prior mean. Therefore, the posterior mean and variance depend on both inputs:

-\frac{1}{2}\left[ \frac{(y-\delta)^2}{SE^2} + \frac{(\delta-\mu_0)^2}{\tau^2} \right] = -\frac{1}{2}(\delta-m_{\text{post}})^2 \left(\frac{1}{SE^2}+\frac{1}{\tau^2}\right) +\text{constant}.

This identity justifies the precision-weighted expressions used above. It also makes the interpretive point precise: with the likelihood held fixed, changing the prior can change the posterior; with the prior held fixed, changing the data can also change it. The posterior conclusion is not determined by the prior alone.

Diagnostics for reporting are primarily checks for ambiguity or mismatch. Ask whether the stated units agree with the parameter, whether the sign of benefit is defined, whether a transformed-scale description matches the reported clinical effect, and whether independence or dependence assumptions are stated where material. A prior described only by a label, an omitted scale, or a directionally ambiguous contrast cannot be interpreted reliably. Computational assessment of posterior sampling is a separate matter.

Common Pitfalls and Alternatives

Calling a prior “noninformative” without specifying it. That label does not tell readers its distribution, scale, or implications. Give the actual family and parameter values, and explain the resulting spread in context.
Reporting a transformed prior only in transformed units. A normal prior on a log odds ratio may be difficult to interpret clinically. State the transformation and, where useful, show corresponding odds-ratio values.
Treating the prior as the trial result. A prior describes assumptions before conditioning on current trial data. Report posterior summaries and their clinical interpretation as results conditional on both the prior and data.
Hiding sensitivity to assumptions. If a reasonable alternative specification changes the interpretation, identify the alternative and describe the change. Do not present a single preferred result as if other assumptions could not matter.
Using “weak” as a guarantee of negligible influence. A prior's influence depends on its scale, parameterization, and the information in the likelihood. Describe the actual distribution instead of assuming the adjective settles the question.

The main alternatives are not competing ways to state the same prior. A report may give only the primary prior, or it may additionally show results under alternative plausible specifications. The latter is more informative when prior assumptions are consequential, but it must be clearly labeled and should not obscure the prespecified primary analysis. In either case, transparent parameter definitions and distribution details are essential.

Key takeaway: Report each prior by naming its parameter, distribution, scale, and values; explain its rationale and interpretable implications; and distinguish prior assumptions from posterior conclusions based on the trial data.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A report says only “a normal prior was used” for a treatment effect. Name three details needed to make this description interpretable.
A log odds ratio prior has mean 0 and standard deviation 0.4. What is the prior median odds ratio, and what is an approximate central 95% prior interval for the odds ratio?
In the continuous-outcome example, why does the posterior mean move closer to zero under the prior with standard deviation 0.05 than under the prior with standard deviation 0.20?
True or false: if the posterior treatment effect is close to the prior mean, the prior alone determined the trial conclusion. Explain.

Solutions. (1) Define the treatment-effect parameter and its direction, give the normal prior mean and variance or standard deviation, and state the units or scale; a rationale or clinical translation may also be needed. (2) The median odds ratio is $\exp(0)=1$. The approximate interval on the log scale is $0\pm1.96(0.4)=[-0.784,0.784]$, so the odds-ratio interval is approximately $[\exp(-0.784),\exp(0.784)]=[0.456,2.190]$. (3) The narrower prior has greater precision: $1/0.05^2=400$, compared with $1/0.20^2=25$. With the same likelihood, the precision-weighted posterior mean is therefore pulled more strongly toward the prior mean of zero. (4) False. A posterior combines the prior and the likelihood. Similarity to the prior mean may reflect prior influence, data compatible with that value, or both; the posterior alone does not establish that the prior determined the conclusion.