Bayesian integration G-formula for platform SMART designs allowing for adding new treatments

Abstract: Dynamic treatment regimes (DTRs) are sequences of decision rules to guide treatment assignments in response to a patient's evolving, time-varying disease status. Sequential multiple assignment randomized trials (SMARTs) are considered the gold standard experimental design for evaluating DTRs. However, SMARTs often require more time to complete compared with a single stage RCT and new candidate treatments may become available or feasible during the trial. Platform trials are an adaptive trial design that allow new treatments to be added to the ongoing study according to a prespecified master protocol. In this paper, we introduce a novel platform SMART that integrates features from both platform trials and SMARTs, allowing new treatments to be added during the trial. Additionally, we propose the Bayesian integration G-formula (BIG) estimators for platform SMARTs to account for non-concurrent treatment comparisons. Extensive simulations are conducted to evaluate the performance of different BIG estimators against benchmark methods. We demonstrate the proposed BIG estimators based on the S. aureus Network Adaptive Platform (SNAP) trial.

• The paper introduces Bayesian integration G-formula (BIG) estimators that borrow information across pre- and post-adaptation cohorts through adaptive priors on shared model coefficients.
• Simulations show that pooling is efficient without time effects but becomes substantially biased with non-concurrent time effects, whereas log-distance and commensurate BIG methods provide a stronger bias–variance balance with near-nominal coverage.
• The framework enables posterior estimation and comparison of dynamic treatment regimes in platform SMARTs, demonstrated through SNAP-calibrated simulations, but currently covers only one added first-stage treatment and depends on generalized linear models.

Motivation and problem setting

Dynamic treatment regimes (DTRs) are sequences of decision rules that tailor treatment to a patient's evolving status, and sequential multiple assignment randomized trials (SMARTs) are the gold-standard design for estimating them. A practical drawback of SMARTs is their extended duration relative to single-stage randomized controlled trials, which increases the likelihood that new candidate treatments become available mid-trial. Platform trials address this by admitting new treatments under a master protocol, but the statistical literature on platform trials has focused on stage-specific treatment effects rather than DTR estimation. The paper identifies a gap: no existing analytical methods handle SMARTs that allow new treatments to be added during the trial — designs the authors call "platform SMARTs" — motivated by the S. aureus Network Adaptive Platform (SNAP) trial, in which responders to initial therapy are re-randomized and treatments can be added or dropped over time.

The core statistical difficulty is the non-concurrent comparison problem. When a new first-stage treatment $a_{13}$ is added, participants split into a pre-adaptation cohort $c_1$ (original treatments only) and a post-adaptation cohort $c_2$ (all treatments). Estimating the mean of an embedded DTR involving $a_{13}$ requires deciding how much, if any, information from cohort $c_1$ should be borrowed.

Design and estimands

The authors formalize a two-stage SMART based on SNAP: patients are randomized among first-stage treatments $a_{1j}$; responders are then randomized between continuing standard care or switching to oral antibiotics ($a_{2k}$), while non-responders continue initial treatment. Each design yields embedded DTRs $d_{jk}$ ("treat with $a_{1j}$; if response, switch to $a_{2k}$; otherwise continue"), with DTR mean $c_1$0 estimated via inverse probability weighting (IPW), with closed-form asymptotic variances and covariances for DTRs sharing an initial treatment.

Adding $c_1$1 produces six embedded DTRs across two concurrently randomized cohorts. Two benchmark strategies are considered:

• Separate approach: use only cohort $c_1$2 data. Unbiased but discards cohort $c_1$3 information — wasteful given both cohorts share a protocol.
• Pooling approach: combine cohort estimates weighted by sample size. Efficient, but unbiased only under an exchangeability assumption ($c_1$4), which the authors note is unlikely to hold in practice due to population shifts, changes in standard care, and other time effects.

The BIG estimators

The proposed Bayesian integration G-formula (BIG) estimators build on Bayesian G-computation (2604.25252). Response and outcome models are specified as generalized linear models over cohort $c_1$5 data. Crucially, priors are placed on model coefficients rather than on arm-level means, distinguishing this from most historical-control borrowing methods. Coefficients separate into $c_1$6 (shared with cohort $c_1$7, hence prior-informable) and $c_1$8 (involving the newly added treatment, given weakly informative priors such as $c_1$9).

Three informative priors are developed:

• Log-distance prior: centered at the cohort $c_2$0 MLE with variance equal to the maximum of the squared between-cohort MLE difference and the cohort $c_2$1 estimation variance. This adapts prior sharpness to observed commensurability; the authors deliberately omit the $c_2$2 scaling term from the original formulation to avoid over-shrinking.
• Commensurate prior: introduces a precision parameter $c_2$3 linking cohort $c_2$4 coefficients to cohort $c_2$5 estimates, integrated out jointly with the historical parameters.
• Mixed commensurate prior: a mixture over fixed $c_2$6 values (here $c_2$7, $c_2$8 with equal weights), avoiding posterior sampling of $c_2$9 while enabling partial borrowing.

DTR means are obtained by Monte Carlo: draw $a_{13}$0 parameter vectors from the posterior, simulate a population of size $a_{13}$1 under each target DTR, and average outcomes per draw to approximate the posterior of $a_{13}$2. Posterior draws also yield variance estimates for DTR mean contrasts directly, sidestepping the need for IPW covariance formulas.

The meta-analytic-predictive prior is excluded because it is overly sensitive to variance-parameter priors with a single historical study, and the power prior is excluded on computational grounds — both concessions worth noting when generalizing these results.

Simulation evidence

Simulations cover five scenarios crossing three factors: presence/absence of time effects (violating or satisfying exchangeability), different optimal DTRs, and homogeneous versus heterogeneous time effects. Sample sizes $a_{13}$3 and pre-adaptation fractions $a_{13}$4 are varied, with 1,000 replicates each. Performance is judged by probability of selecting the true optimal DTR, bias, variance, MSE, and coverage for the contrast $a_{13}$5.

Key findings:

Method	No time effects	Time effects present
Pooling	Lowest variance, efficient	Substantial bias (>0.4 absolute, omitted from plots), poor coverage
Separate / BIGweak	Unbiased but highest variance/MSE	Unbiased but inefficient
BIGlogdis, BIGcomP, BIGcommP	Low bias, moderate variance	Best bias–variance trade-off

When Assumption 1 holds, pooling is competitive; when it fails, its apparent advantage in identifying the optimal DTR is driven by biased estimates favoring the truth — a point the authors flag explicitly rather than presenting as genuine superiority. Coverage remains near nominal (0.95) for all approaches except pooling under time effects. As $a_{13}$6 increases (later addition of the new treatment), the probability of correct DTR selection declines for all methods, and pooling's bias grows — implying that delayed adaptation amplifies non-concurrent confounding.

Application to SNAP

The SNAP trial demonstration uses simulated virtual trials calibrated to the trial's design parameters (binary endpoint: 90-day all-cause mortality in MSSA patients; reference mortality risks of 16.8% for non-responders and 15.0% for responders to $a_{13}$7), with $a_{13}$8 and $a_{13}$9. Three scenarios are examined: null, alternative without time effects, and alternative with a time effect of odds ratio 1.5 between cohorts. Results mirror the simulation study: under no time effect, pooling, BIGlogdis, BIGcomP, and BIGcommP achieve lower variance and MSE than separate/BIGweak with minimal bias; under time effects, pooling exhibits the highest bias, lowest coverage, and lowest probability of identifying the optimal DTR, while the adaptive-prior BIG variants retain low bias with improved efficiency. Because SNAP is ongoing, the demonstration relies entirely on simulated data rather than actual trial observations — a limitation on the empirical validation of the method.

Limitations and open questions

The authors restrict attention to a single added first-stage treatment; extensions to multiple simultaneous additions at either stage remain unaddressed. Both response and outcome models are generalized linear models, so model misspecification is a live concern, and more flexible modeling is left as future work. Additionally, the log-distance prior implementation departs from its original formulation by omitting the $c_1$0 term, and the mixed commensurate prior requires prespecifying $c_1$1 values and weights — choices whose sensitivity is not systematically explored. Whether the favorable operating characteristics persist under more complex time-effect structures or with covariate adjustment beyond the simplified setting ($c_1$2, $c_1$3) is not established here.

Conclusion

This paper formalizes the platform SMART design and supplies the first analytical framework — the BIG estimators — for comparing embedded DTRs when treatments are added mid-trial. By placing adaptive priors (log-distance, commensurate, mixed commensurate) on shared model coefficients, the methods recover much of the efficiency of full pooling while remaining robust to the time effects that invalidate exchangeability. Simulations and a SNAP-calibrated case study consistently show that adaptive-prior BIG estimators dominate the separate and weakly-informative alternatives on efficiency and the pooling alternative on robustness, making them a practical default for platform SMART analyses.