Specifying prior distributions is a crucial step in designing a Bayesian adaptive trial. Priors encapsulate existing knowledge (or lack thereof) before the trial begins and are combined with the data to form posterior distributions, which then guide adaptive decisions.

Here's a structured approach to specifying prior distributions, covering common parameter types and important considerations:

1. Core Principles for Prior Specification

Before diving into specific distributions, keep these principles in mind:

• Reflect Existing Knowledge: Priors should genuinely reflect what is known or believed about the parameters before seeing any trial data. This knowledge can come from:


• Previous Trials: Similar studies, meta-analyses.


• Historical Data: Real-world evidence, registries.


• Expert Opinion: Structured elicitation from clinicians, statisticians, and domain experts.


• Pilot Studies/Phase 1/2 Data: Earlier phase results for the same drug/intervention.


• Theoretical Considerations: Biological plausibility, known limits.


• Transparency: Clearly state your chosen priors and provide a robust justification for their selection. This is especially important for regulatory submissions.


• Conjugacy (Practicality): Whenever possible, choose a prior distribution that is conjugate to your likelihood function. This means the posterior distribution will belong to the same family as the prior, simplifying computations significantly – which is critical in adaptive designs where posteriors are updated frequently.


• Sensitivity Analysis: Always perform a sensitivity analysis by trying a range of plausible priors (e.g., more skeptical, more enthusiastic, less informative). This helps demonstrate that your conclusions are robust and not overly dependent on a specific prior choice.

2. Common Parameter Types and Recommended Prior Distributions

Let's look at typical parameters encountered in clinical trials and suitable prior distributions.

A. For Proportions/Rates (e.g., Response Rate, Success Rate, AE Rate)

• Parameter: $p \in [0, 1]$ (e.g., $pT$ for treatment, $pC$ for control).


• Likelihood: Bernoulli or Binomial.


• Conjugate Prior: Beta distribution ($\text{Beta}(\alpha, \beta)$).


• Interpretation: $\alpha$ can be thought of as the number of "prior successes" and $\beta$ as the number of "prior failures." The mean of a Beta distribution is $\alpha / (\alpha + \beta)$.


• How to Choose $\alpha$ and $\beta$:


• Non-Informative Priors:


• Uniform Prior: $\text{Beta}(1, 1)$. This assumes all values of $p$ are equally likely. It's often chosen when there's truly no prior information, or to demonstrate that results are data-driven.


• Jeffreys Prior: $\text{Beta}(0.5, 0.5)$. This is a common choice for its theoretical properties (e.g., invariance under reparameterization). It places more weight on values near 0 and 1 than the uniform prior.


• "Slightly Informative" (Weakly Informative): $\text{Beta}(2, 2)$ or $\text{Beta}(0.5, 0.5)$ for "success" and "failure" arms often used in early phases to ensure stable posteriors, or when you have a very vague belief that the rates aren't extreme.


• Informative Priors (Most Common in Adaptive Trials):


• From Historical Data: If a previous trial found a response rate of 30% in 100 patients (30 successes, 70 failures), you might use $\text{Beta}(30, 70)$ or scale it down for less influence (e.g., $\text{Beta}(3, 7)$ for a "prior effective sample size" of 10).


• From Expert Elicitation:


• Ask experts: "What's the most likely response rate?" (gives you the mean, $\mu = \alpha/(\alpha+\beta)$).


• Ask: "What's a plausible range (e.g., 95% credible interval)?" This helps determine the "concentration" or effective sample size ($\alpha+\beta$).


• Example: An expert believes the response rate is around 40% (mean = 0.4) and is fairly certain it's between 30% and 50%. You can use tools or iterative methods to find $\alpha, \beta$ that match this. A common approach: specify a mean and an effective sample size.


• $\alpha = \text{mean} \times \text{effective sample size}$


• $\beta = (1 - \text{mean}) \times \text{effective sample size}$


• e.g., mean = 0.4, effective sample size = 20. Then $\alpha = 0.4 \times 20 = 8$, $\beta = 0.6 \times 20 = 12$. So $\text{Beta}(8, 12)$.

B. For Means (e.g., Change in Biomarker, Pain Score, Continuous Outcome)

• Parameter: $\mu \in (-\infty, \infty)$.


• Likelihood: Normal distribution ($\text{N}(\mu, \sigma^2)$), where $\sigma^2$ (variance) might be known or unknown.


• Conjugate Prior:


• If $\sigma^2$ is known: Normal distribution ($\text{N}(\mu_0, \tau^2)$).


• $\mu_0$: Prior mean (your best guess for the true mean).


• $\tau^2$: Prior variance (reflects your uncertainty about $\mu_0$). A smaller $\tau^2$ means more certainty.


• If $\sigma^2$ is unknown: A Normal-Inverse-Gamma prior for $(\mu, \sigma^2)$ is conjugate. This models $\mu$ conditionally on $\sigma^2$ as Normal, and $\sigma^2$ marginally as Inverse-Gamma.


• Often, you can set a prior on $\mu$ and a separate prior on $\sigma^2$ and use MCMC (e.g., in Stan, JAGS, or WinBUGS) if analytical conjugacy is too complex.


• How to Choose $\mu_0$ and $\tau^2$ (for $\mu$):


• Non-Informative: A very large $\tau^2$ (e.g., $10^6$ or $10^8$) effectively makes the prior flat over a wide range. Or use a scale-invariant prior like Jeffreys for mean/variance.


• Informative:


• From Historical Data/Pilot: $\mu_0$ would be the observed mean from a prior study. $\tau^2$ could be derived from the standard error of that mean, reflecting your belief in how precisely that prior estimate represents the true mean.


• From Expert Elicitation:


• Ask: "What's the most likely value for the mean change?" (gives $\mu_0$).


• Ask: "What range would contain the true mean 95% of the time?" (e.g., from $X$ to $Y$). You can then approximate $\tau^2$ such that $\mu_0 \pm 1.96 \sqrt{\tau^2}$ covers this range.

C. For Variances/Standard Deviations ($\sigma^2$ or $\sigma$)

• Parameter: $\sigma^2 \in (0, \infty)$ or $\sigma \in (0, \infty)$.


• Likelihood: Normal.


• Conjugate Prior (for $\sigma^2$): Inverse-Gamma distribution ($\text{InvGamma}(\alpha, \beta)$).


• This is the conjugate prior for the variance of a Normal distribution (when the mean is unknown). The mean of $\text{InvGamma}(\alpha, \beta)$ is $\beta/(\alpha-1)$ for $\alpha > 1$.


• Alternative (for $\sigma$): Half-Normal, Half-Cauchy, or Lognormal for the standard deviation ($\sigma$). These are often easier to interpret and elicit.


• How to Choose Parameters:


• Non-Informative:


• For Inverse-Gamma: $\text{InvGamma}(0.001, 0.001)$ or $\text{InvGamma}(0.5, 0.5)$ (Jeffreys' for scale). These are often called "weakly informative" but can sometimes be too influential for small sample sizes.


• Informative:


• From Historical Data: If a prior study had a sample standard deviation $s$, you might center your prior around $s^2$. The precision of this prior (e.g., parameters of the Inverse-Gamma) would depend on your certainty in that historical estimate.


• From Expert Elicitation: Ask about the likely range of the standard deviation or variance. It's often difficult for experts to think directly in terms of variance, so focusing on standard deviation or range is better.

D. For Hazard Ratios (HR) in Survival Analysis

• Parameter: $\text{HR} \in (0, \infty)$. Often, it's easier to model $\log(\text{HR}) \in (-\infty, \infty)$.


• Likelihood: Cox Proportional Hazards model or parametric survival models.


• Prior (for $\log(\text{HR})$): Normal distribution ($\text{N}(\mu{\text{logHR}}, \sigma^2{\text{logHR}})$).


• Prior (for HR directly): Lognormal distribution (if $\log(\text{HR})$ is Normal).


• How to Choose Parameters:


• Non-Informative: For $\log(\text{HR})$, a $\text{N}(0, \text{large variance})$ (e.g., $N(0, 100)$). This implies no prior belief that HR is above or below 1 (i.e., log(HR) is positive or negative).


• Informative:


• From Historical Data: If a prior trial showed an HR of 0.7 with a 95% CI of (0.5, 0.98), you can convert these to log scale, find the mean and standard error of $\log(\text{HR})$, and use those to parameterize a Normal prior.


• $\log(0.7) \approx -0.357$ (prior mean $\mu_{\text{logHR}}$).


• Width of CI on log scale: $\log(0.98) - \log(0.5) = -0.02 - (-0.69) = 0.67$.


• Standard deviation: $0.67 / (2 \times 1.96) \approx 0.17$. So $\sigma_{\text{logHR}}^2 \approx 0.17^2 \approx 0.029$.


• This would lead to a prior like $\text{N}(-0.357, 0.029)$.


• From Expert Elicitation: Ask experts for their best guess for the HR and a plausible range.

3. Strategies for Eliciting and Setting Priors

• Review Existing Literature: This is the starting point. What's known about the control arm, the effect size, and the variability?


• Formal Expert Elicitation:


• Sheffey-Tukey Method / Deliberation: Gather a panel of experts. Present them with hypothetical scenarios or questions about the parameter.


• "Prior Elicitation" Software/Tools: Programs exist that help experts specify a mean and various percentiles (e.g., 5th, 25th, 50th, 75th, 95th) of a parameter, and then fit a distribution (e.g., Beta, Normal, Gamma) to those specified points.


• Focus on Quantiles: Experts often find it easier to think about "I'm 90% sure the response rate is between 20% and 40%" rather than "the variance is 0.005."


• Use of "Skeptical" or "Enthusiastic" Priors:


• Skeptical Prior: Centered on a null effect (e.g., HR=1, difference=0) or a very small effect, often with high precision. This requires the data to provide very strong evidence to move the posterior away from the null. Useful for regulatory bodies who want to see convincing evidence.


• Enthusiastic Prior: Centered on a clinically meaningful effect, often with high precision. Useful for early phase development to quickly identify promising treatments.


• These are often used as part of sensitivity analyses.


• Borrowing Information: Priors are the formal mechanism to borrow information from external sources (historical controls, previous trials). For instance, if you have a control arm with ample historical data, you might use a tight, informative prior for its parameters, while using a less informative prior for the novel treatment arm. This is a common feature in platform trials or multi-arm multi-stage (MAMS) designs.


• Hierarchical Priors: If you have multiple related parameters (e.g., response rates for several different subgroups, or parameters for multiple arms in a platform trial), you can model them as coming from a common "hyper-prior" distribution. This allows for "borrowing strength" across parameters, especially useful when some groups have very little data.

4. Practical Steps for Your Design

• List All Parameters: Identify every parameter in your statistical model for which you need a prior. (e.g., $pC$, $pT$, $\sigma^2$).


• Determine Likelihood: For each parameter, determine the likelihood function of the data given that parameter.


• Select Prior Family: Choose a conjugate prior distribution family where possible.


• Parameterize Priors:


• Start with non-informative priors for initial exploration if unsure.


• Gather all available external data and expert opinion.


• Use elicitation strategies to specify parameters ($\alpha, \beta, \mu_0, \tau^2$, etc.).


• Document your rationale for each parameter.


• Visualize Priors: Plot your prior distributions. Do they look reasonable? Do they cover the plausible range of values?


• Perform Sensitivity Analysis (Crucial):


• Run simulations of your adaptive trial using different sets of priors (e.g., your chosen prior, a more skeptical prior, a less informative prior).


• Observe how the choice of prior impacts operating characteristics (e.g., Type I error, power, sample size, stopping probabilities, probability of selecting the best arm).


• If the results change dramatically with minor changes in priors, your design might be too sensitive and require stronger justification for your prior choice or larger sample sizes.