DTE-BOP2: Bayesian Phase II Design for Delayed Effects

• The paper introduces DTE-BOP2, a Bayesian phase II design that explicitly models delayed treatment effects through an uncertain separation time S with a truncated-Gamma prior.
• It employs adaptive group-sequential monitoring using posterior probabilities to guide interim decisions while balancing power and controlling type I error.
• Simulation results show that DTE-BOP2 achieves higher power and requires smaller sample sizes compared to traditional piecewise log-rank methods under delayed-effect scenarios.

Searching arXiv for papers on DTE-BOP2 and related Bayesian phase II designs. Bayesian Optimal Phase II Design with delayed treatment effects, usually denoted DTE-BOP2, is a Bayesian framework for randomized phase-II clinical trials with time-to-event endpoints in settings where delayed treatment effects are anticipated, such as immuno-oncology. It extends the BOP2 framework by explicitly modeling uncertainty in the treatment separation timepoint $S$, the time from randomization until the survival curves of the investigational and control arms begin to separate. The design uses a truncated-Gamma prior for $S$, retains adaptive Bayesian group-sequential monitoring, incorporates type I error control, maintains power, and is implemented in an open-source R package, DTEBOP2 (Cai et al., 29 Aug 2025).

1. Conceptual position within Bayesian phase-II design

DTE-BOP2 belongs to a broader class of Bayesian early-phase designs in which a likelihood $f(y \mid x,\theta)$ is combined with a prior $p(\theta \mid \xi)$, updated sequentially as data accrue, and linked to adaptive decision rules for continuation, stopping, or treatment choice (Thall, 2010). Within that general tradition, the BOP2 line emphasizes posterior-probability-based monitoring and design calibration across diverse endpoint structures.

Several recent extensions clarify the scope of that line. BOP2-DC incorporates both statistical significance and clinical relevance into a go/consider/no-go framework and accommodates binary, continuous, time-to-event, multiple, and co-primary endpoints in single-arm and randomized trials (Zhao et al., 2021). BOP2-TE uses a Dirichlet-multinomial model to jointly monitor efficacy and toxicity, with go/no-go decisions based on posterior probability of toxicity and futility, and emphasizes rigorous type I error control when the treatment is toxic and futile, effective but toxic, or safe but futile (Chen et al., 2024). DTE-BOP2 extends this BOP2 tradition to randomized immunotherapy trials with delayed treatment effects by focusing on non-proportional hazards induced by an uncertain separation time (Cai et al., 29 Aug 2025).

A common source of confusion is the acronym “BOP2” itself. Recent Bayes factor papers use “Bayesian optimal two-stage design” for simulation-free, Bayes factor-based phase II designs in single-arm and two-arm settings (Kelter et al., 28 Nov 2025, Kelter, 1 Jun 2026). This suggests that, in current usage, “BOP2” names more than one methodological branch, and DTE-BOP2 should be interpreted specifically through its posterior-probability delayed-effect formulation rather than through the Bayes factor branch.

2. Delayed-effect model and prior specification

The defining innovation of DTE-BOP2 is explicit modeling of the delay time $S$ as a random variable rather than a fixed, known parameter. The treatment separation timepoint is constrained to an interval $[L,U]$ and assigned a truncated-Gamma prior,

$$\pi(S) \propto \mathrm{Gamma}(s_1,s_2)\cdot I(L,U),$$

with parameters that can be elicited from experts or inferred from historical data; default settings are available when prior knowledge is scarce (Cai et al., 29 Aug 2025).

The event-time model is piecewise exponential. For the control arm, $T_{i0} \sim \mathrm{Exp}(\mu_0)$. For the experimental arm, hazards are assumed equal before $S$ and then allowed to differ after $S$, so that survival curves are identical up to time $S$0 and separate thereafter (Cai et al., 29 Aug 2025). In this formulation, $S$1 and $S$2 are assigned weakly informative inverse-Gamma priors, and the likelihood is built from all observed, possibly censored, data up to the current analysis.

The model is connected to clinically interpretable medians through $S$3, and the post-separation median $S$4 is mapped to the overall clinical median $S$5 by

$S$6

This mapping allows clinical hypotheses stated in terms of overall medians to be translated into model parameters (Cai et al., 29 Aug 2025).

The elicitation step is central. Experts are asked for bounds on $S$7, expected or median values, and plausible ranges, and prior parameters are fitted by weighted least squares to align the truncated-Gamma distribution with those summaries. The paper’s example states that if experts place $S$8 in $S$9 months with median about $f(y \mid x,\theta)$0, the resulting prior may be $f(y \mid x,\theta)$1 truncated to $f(y \mid x,\theta)$2 (Cai et al., 29 Aug 2025).

3. Posterior monitoring and decision rule

DTE-BOP2 uses adaptive Bayesian group-sequential monitoring with posterior-probability-based futility assessment. At each interim or final analysis, the key quantity is the posterior probability that the experimental-arm median is inferior to the control-arm median, conditional on the data and $f(y \mid x,\theta)$3:

$f(y \mid x,\theta)$4

The trial is stopped for futility if

$f(y \mid x,\theta)$5

where $f(y \mid x,\theta)$6 is the current sample size and the boundary is

$f(y \mid x,\theta)$7

Here $f(y \mid x,\theta)$8 and $f(y \mid x,\theta)$9 are tuning parameters calibrated during design optimization (Cai et al., 29 Aug 2025).

The paper reports a computationally efficient formula for the posterior probability using the Beta distribution. It is expressed in terms of total time-on-test before and after $p(\theta \mid \xi)$0, event counts in the corresponding intervals, and a Beta-distributed latent quantity $p(\theta \mid \xi)$1; this avoids a fully simulation-based posterior calculation at each monitoring look (Cai et al., 29 Aug 2025). The practical implication is that the design preserves the operational simplicity associated with BOP2 while replacing the proportional-hazards assumption with a delayed-effect model.

This posterior-threshold logic is consistent with other Bayesian phase II designs, but the estimand differs. In the binary-outcome randomized treatment-selection design of Zaslavsky and colleagues, for example, decisions are based on posterior interval probabilities such as

$p(\theta \mid \xi)$2

an ambiguity probability $p(\theta \mid \xi)$3, and the combined criterion

$p(\theta \mid \xi)$4

with treatment selection when $p(\theta \mid \xi)$5 (Komaki et al., 14 May 2025). DTE-BOP2 uses an analogous posterior-decision philosophy, but its endpoint is time-to-event with delayed separation rather than binary response.

4. Calibration, type I error control, and sample-size planning

Under DTE-BOP2, type I error depends on $p(\theta \mid \xi)$6 because $p(\theta \mid \xi)$7 is random. The design objective is therefore to control the average type I error over the prior for $p(\theta \mid \xi)$8, with an option to enforce control at the interval boundaries of the prior range for conservatism. The formulation is

$p(\theta \mid \xi)$9

and similarly

$S$0

Calibration proceeds by searching a grid of $S$1 values to maximize average power while ensuring that type I error remains below the prespecified threshold $S$2 on average and, if desired, at the boundaries of $S$3 (Cai et al., 29 Aug 2025).

Sample-size determination is handled through a two-stage design algorithm that iterates on interim and final per-arm sample sizes $S$4 to minimize expected sample size under the null, satisfy power under the alternative for all $S$5 in $S$6, and allow control of the minimum early stopping probability under $S$7 (Cai et al., 29 Aug 2025). The paper describes this as a genuinely Bayesian approach to interim/final sample-size planning under complex non-proportional hazard models. It also notes a pragmatic option that is less computationally demanding but nearly optimal.

This calibration problem is structurally related to the broader BOP2 literature, where posterior boundaries are optimized to balance power and type I error. In BOP2-DC, for example, posterior cutoffs are optimized either to maximize the correct go rate when the drug is effective or to minimize expected sample size when the drug is futile, under constraints on false go, false no-go, and false consider rates (Zhao et al., 2021). DTE-BOP2 inherits that design-calibration ethos but replaces the fixed endpoint effect parameter with a delayed-effect survival structure indexed by $S$8.

5. Operating characteristics and comparative performance

The simulation results reported for DTE-BOP2 are organized around three recurrent findings. First, the design uniformly controls type I error at the nominal level across a wide range of treatment effect separation timepoint $S$9. Second, power decreases monotonically as $[L,U]$0 increases. Third, power is primarily driven by the relative magnitude of treatment benefit before and after the separation time, that is, the ratio of medians, rather than their absolute values (Cai et al., 29 Aug 2025).

Relative to the original BOP2, DTE-BOP2 is reported to be no worse when there is no delayed effect and superior when $[L,U]$1, because the original BOP2 assumes proportional hazards and loses power rapidly as $[L,U]$2 increases (Cai et al., 29 Aug 2025). Relative to piecewise weighted log-rank procedures, DTE-BOP2 differs conceptually because the frequentist procedures require a fixed, prespecified $[L,U]$3, whereas DTE-BOP2 models $[L,U]$4 as uncertain through its prior. The paper interprets this as a shift from sensitivity to misspecification toward design-stage averaging over plausible delay scenarios.

The numerical comparisons in the paper are explicit. For $[L,U]$5 patients per arm, DTE-BOP2 achieved $[L,U]$6–$[L,U]$7 power, compared with $[L,U]$8–$[L,U]$9 for piecewise log-rank and about $$\pi(S) \propto \mathrm{Gamma}(s_1,s_2)\cdot I(L,U),$$0–$$\pi(S) \propto \mathrm{Gamma}(s_1,s_2)\cdot I(L,U),$$1 for standard log-rank. To ensure $$\pi(S) \propto \mathrm{Gamma}(s_1,s_2)\cdot I(L,U),$$2 power for $$\pi(S) \propto \mathrm{Gamma}(s_1,s_2)\cdot I(L,U),$$3 in $$\pi(S) \propto \mathrm{Gamma}(s_1,s_2)\cdot I(L,U),$$4, DTE-BOP2 required only $$\pi(S) \propto \mathrm{Gamma}(s_1,s_2)\cdot I(L,U),$$5 per arm; piecewise log-rank methods required about $$\pi(S) \propto \mathrm{Gamma}(s_1,s_2)\cdot I(L,U),$$6–$$\pi(S) \propto \mathrm{Gamma}(s_1,s_2)\cdot I(L,U),$$7 per arm, and standard log-rank required $$\pi(S) \propto \mathrm{Gamma}(s_1,s_2)\cdot I(L,U),$$8–$$\pi(S) \propto \mathrm{Gamma}(s_1,s_2)\cdot I(L,U),$$9 per arm (Cai et al., 29 Aug 2025). Within the terms used in the paper, DTE-BOP2 therefore achieves higher power with smaller sample sizes while preserving type I error robustness across plausible delay scenarios.

These results place DTE-BOP2 alongside other recent BOP2 extensions that aim to improve operating characteristics under endpoint structures not well served by simpler rules. BOP2-TE, for instance, was proposed to improve operating characteristics of the original BOP2 under jointly monitored efficacy and toxicity, emphasizing type I error control under multiple null configurations while optimizing power when treatment is effective and safe (Chen et al., 2024). DTE-BOP2 plays an analogous role for delayed-effect survival endpoints.

6. Software, implementation, and relation to adjacent designs

Practical implementation is centered on the DTEBOP2 R package on CRAN. The package provides functions for sample size calculation, prior elicitation, design calibration, operating characteristic simulation, and interim/final analysis, together with vignettes that walk users through real-data examples such as the CheckMate 017 trial (Cai et al., 29 Aug 2025). The package also allows users to supply their own prior for $T_{i0} \sim \mathrm{Exp}(\mu_0)$0 or use default settings, and to choose the degree of conservatism in type I error control.

The broader software context matters because the BOP2 literature has developed as a suite of specialized but related tools. BOP2-DC is implemented through a freely available tool at www.trialdesign.org, and BOP2-TE is available as part of the BOP2 app on the same platform (Zhao et al., 2021, Chen et al., 2024). The treatment-selection design for randomized cancer trials with binary outcomes is accompanied by a user-friendly R Shiny application for rapid sample size calculation and post-trial analysis (Komaki et al., 14 May 2025). DTE-BOP2 extends that practical orientation to delayed-effect survival trials.

For investigators, the operational recommendations in the DTE-BOP2 paper are specific: engage clinical experts to specify a plausible range $T_{i0} \sim \mathrm{Exp}(\mu_0)$1 and summaries for $T_{i0} \sim \mathrm{Exp}(\mu_0)$2; state clinical hypotheses on overall medians and map them to model medians; use the package to obtain sample size, stage allocation, and boundary parameters; calibrate for type I error robustness if regulatory submission is planned; and apply the posterior-threshold rule at each interim and final look (Cai et al., 29 Aug 2025). A plausible implication is that DTE-BOP2 is designed not only as a methodological correction for non-proportional hazards, but also as a deployable phase-II workflow for trials in which the timing of benefit is uncertain.