Business Context

You’re a data scientist embedded with the clinical development team at OncoNova Therapeutics, a biotech running a Phase II randomized clinical trial for a new oncology drug intended to reduce severe neutropenia (a dangerous side effect). The trial is expensive and ethically sensitive: if the drug is clearly better, you want to stop early and move to Phase III; if it’s not, you want to stop to avoid exposing patients and burning runway.

The team is considering a Bayesian interim analysis after the first cohort completes follow-up. Regulators and clinicians care about interpretable statements like “there is a 97% probability the drug reduces severe neutropenia by at least 5 percentage points,” rather than only p-values.

Problem Statement

At the interim look, compute the Bayesian posterior for the severe neutropenia rate in each arm and answer whether the trial should stop early for efficacy under a pre-specified Bayesian rule.

Given Data

Severe neutropenia is a binary endpoint (yes/no) assessed within 30 days.

Prior beliefs (from historical studies and mechanism-of-action):

• Control event rate prior: $p_C \sim \text{Beta}(28, 72)$ (prior mean 0.28; “about 100 pseudo-patients”)
• Treatment event rate prior: $p_T \sim \text{Beta}(24, 76)$ (prior mean 0.24; also ~100 pseudo-patients)

Decision rule for early efficacy at interim:

• Stop for efficacy if $\Pr(p_T < p_C - \delta \mid \text{data}) \ge 0.975$, where $\delta = 0.05$ (an absolute 5 percentage point reduction).

Requirements

1. Explain (briefly, in your own words) what Bayesian updating is in this context (prior → likelihood → posterior) and why it’s attractive in clinical trials.
2. Compute the posterior distributions for $p_C$ and $p_T$.
3. Compute $\Pr(p_T < p_C - 0.05 \mid \text{data})$ using a reasonable numerical method (e.g., Monte Carlo).
4. Based on the decision rule, state whether you would stop early for efficacy at interim.
5. Provide one business/clinical interpretation and one caveat (e.g., prior sensitivity, operational bias, multiplicity).

Assumptions and Constraints

• Outcomes are independent across patients within each arm.
• Randomization is valid (no major baseline imbalance).
• A Beta-Binomial model is appropriate for each arm’s event probability.
• You may assume posterior independence between arms given separate priors and separate binomial likelihoods.
• You are not asked to adjust for covariates; treat this as a simple two-arm comparison at interim.