Bayesian framework in clinical trials

The Bayesian model updates a prior distribution $p(\theta)$ on the treatment effect $\theta$ with the observed data $\mathcal{D}_n$ via Bayes’ theorem:

\[p(\theta \mid \mathcal{D}_n) = \frac{p(\mathcal{D}_n \mid \theta) \cdot p(\theta)}{p(\mathcal{D}_n)} \propto p(\mathcal{D}_n \mid \theta) \cdot p(\theta)\]

The posterior $p(\theta \mid \mathcal{D}_n)$ is the complete probabilistic summary of $\theta$ after $n$ observations. All inference and decisions flow from this posterior — point estimates (posterior mean, median), intervals (highest density intervals, HPD), and decision thresholds.

Key contrasts with frequentist trials:

Property	Frequentist	Bayesian
Parameters	Fixed unknowns	Random (distributed)
Data interpretation	What if $H_0$ true?	$P(\theta \in A \mid \text{data})$
Stopping rules	Pre-specified $\alpha$-spending	Posterior probability
Prior information	Ignored	Formally incorporated
Sequential analysis	Inflates $\alpha$	Naturally sequential

Prior specification

The prior $p(\theta)$ encodes information before the trial. The choice is the most contested aspect of Bayesian trials.

Conjugate priors: when the posterior has the same distributional form as the prior, updating is analytic.

For a binary outcome with event rate $\pi$:

• Prior: $\pi \sim \text{Beta}(\alpha_0, \beta_0)$
• Likelihood: $Y \sim \text{Binomial}(n, \pi)$
• Posterior: $\pi \mid Y=y \sim \text{Beta}(\alpha_0 + y, \beta_0 + n - y)$

The prior parameters $\alpha_0, \beta_0$ can be interpreted as pseudo-counts: $\alpha_0$ prior successes, $\beta_0$ prior failures. Effective sample size $= \alpha_0 + \beta_0$.

For a normal outcome with mean $\mu$ and known variance $\sigma^2$:

• Prior: $\mu \sim N(\mu_0, \tau_0^2)$
• Likelihood: $\bar{Y} \sim N(\mu, \sigma^2/n)$
• Posterior: $\mu \mid \bar{Y} \sim N(\mu_n, \tau_n^2)$ where:

\[\mu_n = \frac{\mu_0/\tau_0^2 + n\bar{Y}/\sigma^2}{1/\tau_0^2 + n/\sigma^2}, \quad \frac{1}{\tau_n^2} = \frac{1}{\tau_0^2} + \frac{n}{\sigma^2}\]

The posterior mean is a precision-weighted average of prior mean and data mean.

Non-informative (vague) priors: $\text{Beta}(0.5, 0.5)$ (Jeffreys), $\text{Beta}(1,1)$ (uniform). For regulatory purposes, vague priors are preferred unless strong prior data exists (e.g., historical control data for the same disease).

Power priors: borrow information from historical data $\mathcal{D}_0$ by weighting:

\[p(\theta \mid \mathcal{D}_0)^{a_0} \cdot p(\theta)\]

where $a_0 \in [0,1]$ controls borrowing. $a_0 = 0$: ignore history; $a_0 = 1$: full borrowing.

Bayesian stopping rules

Instead of frequentist $p < 0.05$, Bayesian trials stop when posterior probabilities exceed pre-specified thresholds.

Efficacy stopping: stop and declare treatment superior if:

\[P(\theta > \theta_0 \mid \mathcal{D}_n) > p_{\text{sup}}\]

where $\theta_0$ is the null value (e.g., 0 for difference, 1 for ratio) and $p_{\text{sup}} \approx 0.975$–$0.99$.

Futility stopping: stop for futility if:

\[P(\theta > \theta_0 \mid \mathcal{D}_n) < p_{\text{fut}}\]

or equivalently, if the predictive probability of success (see below) is too low.

Posterior probability threshold calibration: because Bayesian posterior probabilities need not control frequentist type I error, simulation under the null $H_0: \theta = \theta_0$ is used to choose $p_{\text{sup}}$ such that $P(\text{stop for efficacy} \mid H_0) \le \alpha$.

This requires Monte Carlo: simulate $N_{\text{sim}} = 10{,}000$–$100{,}000$ trials under $H_0$, count how often the stopping rule fires. Adjust $p_{\text{sup}}$ until the frequentist type I error rate is controlled.

Predictive probability of success

The predictive probability of success (PPS) is the probability that the trial will be declared a success at the final analysis, given current data $\mathcal{D}_n$:

\[\text{PPS}(n) = \int P(\text{success at final} \mid \mathcal{D}_n, \theta) \, p(\theta \mid \mathcal{D}_n)\, d\theta\]

This integrates over the predictive distribution of future data. PPS is used for:

• Interim futility: if $\text{PPS} < 0.05$, stopping is warranted
• Go/No-Go decisions: $\text{PPS} > 0.80$ to proceed to a confirmatory trial
• Dynamic sample size: enroll more if needed to achieve target PPS

Computing PPS requires simulation (for each posterior draw $\theta^{(s)}$, simulate the remaining $N-n$ observations and check the stopping criterion) — typically expensive but feasible with modern hardware.

Response-adaptive randomization

Response-adaptive randomization (RAR) updates allocation probabilities based on observed outcomes, steering more patients toward the better-performing arm.

Thompson sampling for $K$ arms with binary responses: at each allocation decision, draw $\pi_k^{(t)} \sim p(\pi_k \mid \mathcal{D}_{n_k})$ for each arm $k$ and allocate to $\arg\max_k \pi_k^{(t)}$.

With Beta-Binomial conjugacy, this reduces to: draw $\pi_k^{(t)} \sim \text{Beta}(\alpha_k + y_k, \beta_k + n_k - y_k)$ and assign to the arm with the highest draw.

Power-transformed Thompson: allocate arm $k$ with probability proportional to:

\[\rho_k \propto P(\pi_k = \max_j \pi_j \mid \mathcal{D}_n)^c \cdot \sqrt{n_k}\]

where $c < 1$ reduces variability and the $\sqrt{n_k}$ term ensures exploration. Large $c$ is more exploitative; small $c$ more exploratory.

Properties and concerns:

• RAR assigns more patients to the better arm — an ethical advantage in serious disease
• RAR can reduce trial power compared to equal allocation (more variance in allocation proportions)
• Drift bias: if the patient population changes over time, RAR can confound treatment comparison
• Regulatory agencies (FDA, EMA) accept RAR but require extensive simulation evidence

Platform, basket, and umbrella trials

Platform trials: multiple treatments evaluated simultaneously on a shared control arm (RECOVERY, REMAP-CAP). Treatments enter and leave the platform as evidence accumulates. Bayesian sharing of the control arm improves efficiency.

Basket trials: a single treatment tested across multiple tumor types (biomarker-defined “baskets”). Bayesian hierarchical models share information across baskets:

\[\theta_k \sim N(\mu, \tau^2), \quad \mu \sim N(\mu_0, \sigma_\mu^2), \quad \tau \sim \text{Half-Cauchy}(0, s)\]

Posterior for $\theta_k$ borrows strength from other baskets proportionally to $1/\tau^2$. When $\tau \to 0$ (homogeneous), full pooling; when $\tau \to \infty$, no borrowing.

Umbrella trials: multiple targeted therapies tested within a single disease (e.g., lung cancer), with patients screened for biomarkers and assigned to the matching arm. Bayesian biomarker subgroup models identify which biomarker-defined populations benefit.

Regulatory considerations

The FDA’s 2019 guidance on Adaptive Designs for Clinical Trials and the 2010 guidance on Bayesian Statistics specify requirements for Bayesian trials:

1. Pre-specification: the analysis model, prior, and stopping rules must be pre-specified in the protocol and SAP
2. Type I error control: frequentist type I error rate must be assessed by simulation and controlled at a pre-specified level ($\alpha = 0.05$ two-sided for most confirmatory trials)
3. Sensitivity analysis: results must be robust to prior choice — try weakly informative and vague priors
4. Operating characteristics: power, expected sample size, and decision probabilities under a range of scenarios must be reported

Posterior probability thresholds for confirmatory evidence are typically $\ge 0.975$ to achieve frequentist type I error $\le 0.025$ (one-sided). For exploratory basket or platform trials, $\ge 0.90$ may be accepted.

Computational tools: JAGS, Stan (HMC-based MCMC), and specialized software (EAST Bayes, FACTS) are used for posterior computation and operating characteristic simulation.