Technical details of the Goldilocks design

3.1 Independent dropout and administrative censoring

For treatment arm \(z\), let \(p_z\) be prop_loss and let \(\tau = \texttt{end\_of\_study}\) be the planned follow-up duration per subject, measured from enrollment. The simulator independently draws

\[D_i \mid Z_i=z \sim \operatorname{Exponential}(\eta_z), \qquad \eta_z = -\log(1-p_z)/\tau,\]

independently of event times and enrollment within each arm. Thus \(\Pr(D_i \leq \tau \mid Z_i=z)=p_z\). With prop_loss = 0, \(D_i=\infty\) and no dropout random numbers are drawn. Values must lie in \([0,1)\); a probability of one would require an infinite exponential hazard and is rejected. A scalar applies the same dropout distribution to both arms; a vector named control and treatment supplies separate arm probabilities.

The complete simulated observation has

\[C_i=\min(D_i,\tau), \qquad T_i=\min(T_i^*,D_i,\tau), \qquad \delta_i=I\{T_i^*\leq\min(D_i,\tau)\}.\]

The loss_to_fu flag is true only if \(D_i<\min(T_i^*,\tau)\). Events before dropout remain observed. Administrative censoring is not counted as dropout, and dropout after an observed event does not make that endpoint missing. Exact ties retain the event or administrative censoring; ties have probability zero under the continuous model. At an interim calendar cut, these observations are additionally limited to each enrolled subject’s available follow-up.

prop_loss describes the dropout distribution, not the observed fraction censored by dropout. If \(S_z(t)\) is the underlying event-free survival curve, the probability of observed dropout by the full follow-up horizon is

\[q_z=\Pr(D_i<T_i^*,D_i<\tau\mid Z_i=z) =\int_0^\tau S_z(t)\eta_z e^{-\eta_z t}\,dt \leq p_z.\]

For a constant event hazard \(\lambda_z\), this reduces to

\[q_z=\frac{\eta_z}{\lambda_z+\eta_z} \{1-e^{-(\lambda_z+\eta_z)\tau}\}.\]

For example, with 30% event probability by 12 months and prop_loss = 0.05, the dropout hazard is \(-\log(0.95)/12\) and the expected observed dropout proportion is about 4.21%. The realized count varies across simulated trials; five losses are not forced in a 100-subject trial. Subjects censored by dropout still contribute their observed event-free follow-up to survival analysis. Equal dropout distributions in arms with different event hazards can yield different observed dropout proportions. The fraction observed at an interim look also depends on enrollment and available follow-up.

To retain an externally specified probability \(q\) at reference time \(t_0\) (for example, 5% annually), supply \(p=1-(1-q)^{\tau/t_0}\) at the package horizon. Changing end_of_study while holding prop_loss fixed changes the implied dropout hazard. All event, dropout, enrollment, and administrative times must use the same unit. Treatment discontinuation is not separately modeled: it is not loss to follow-up when endpoint collection continues.

This mechanism replaces the previous fixed-count selection with uniform censoring before each selected subject’s potential event or administrative time. That previous censoring depended on the latent event time. Positive prop_loss values now change seeded results and operating characteristics; design simulations should be rerun. Zero-dropout simulations are unchanged.

6.1 Predictive probability at the current sample size

The current-sample-size predictive probability \(P_{n_\ell}\) is estimated by:

1. draw one set of hazard parameters from the interim posterior;
2. impute remaining endpoint outcomes for enrolled subjects only;
3. apply the prespecified final analysis rule to the completed data; and
4. record whether that completed trial is successful.

For Bayesian survival, step 1 uses prior_surv and step 3 updates prior_surv_final with the completed events and exposure. The posterior used to generate outcomes is not substituted for the completed-data analysis prior. The maximum-sample calculation uses this same separation of priors.

For example, write \(\pi_{\mathrm{pred}}\) for prior_surv and \(\pi_{\mathrm{ana}}\) for prior_surv_final. For alternative = "less", the predictive target is

\[P_{n_\ell} = \operatorname{E}_{\mathrm{pred}}\left[ I\left\{ \Pr_{\mathrm{ana}}\left(\Delta < h_0 \mid \mathcal{D}^{\mathrm{obs}}_\ell,\mathcal{D}^{\mathrm{mis}}\right) > c \right\} \mid \mathcal{D}^{\mathrm{obs}}_\ell \right].\]

The outer expectation generates missing outcomes from the posterior predictive distribution under \(\pi_{\mathrm{pred}}\). The inner probability analyzes each completed dataset under \(\pi_{\mathrm{ana}}\), with \(c=\) prob_ha. In a single-arm design, replace \(\Delta\) with the treatment event probability. Thus the hypothetical final success test and the actual final success test use the same analysis prior, even when the predictive prior differs.

Formally,

\[P_{n_\ell} = \operatorname{E}\{ \psi(\mathcal{D}_{n_\ell}^{\mathrm{comp}}) \mid \mathcal{D}_{\ell}^{\mathrm{obs}} \},\]

where \(\mathcal{D}_{n_\ell}^{\mathrm{comp}}\) is the completed dataset formed from the \(n_\ell\) enrolled subjects after imputing their remaining follow-up. Equivalently,

\[P_{n_\ell} = \int \psi(\mathcal{D}_{n_\ell}^{\mathrm{obs}}, \mathcal{D}_{n_\ell}^{\mathrm{mis}}) p(\mathcal{D}_{n_\ell}^{\mathrm{mis}} \mid \mathcal{D}_{\ell}^{\mathrm{obs}}) d\mathcal{D}_{n_\ell}^{\mathrm{mis}}.\]

Repeating the four-step procedure above for Monte Carlo replicate \(m = 1,\ldots,M\), where \(M\) is set by N_impute, gives

\[\widehat{P}_{n_\ell} = \frac{1}{M}\sum_{m=1}^{M} I\{\text{success in replicate } m\}.\]

For Bayesian completed-data analyses that use N_mcmc posterior draws, a replicate is counted as successful when its estimated posterior success probability exceeds prob_ha. Deterministic quadrature, approximation, and frequentist analyses use the same strict point-estimate comparison. Exact one-sided binomial bounds for posterior Monte Carlo estimates are retained as diagnostics but do not change this classification.

For the fixed-horizon binary methods, a completed replicate is summarized by \((x_0,n_0,x_1,n_1)\) in a two-arm design, or \((x_1,n_1)\) in a single-arm design, where \(x_z\) and \(n_z\) are the event count and denominator in arm \(z\). There is no loss of statistical information for the risk-difference or beta-binomial analysis because individual event times are not used once endpoint status is complete. With bin_method = "mc", each predictive replicate receives an independent set of N_mcmc beta-posterior draws, thereby preserving the two sources of Monte Carlo variation.

The upper decisions are evaluated in order. If

\[\widehat{P}_{n_\ell} > Q_\ell,\]

the trial stops and immediately declares official success. Otherwise, if

\[\widehat{P}_{n_\ell} > S_\ell,\]

accrual is stopped for expected success. Enrolled subjects are still followed to the planned final analysis time.

The decision trace also reports the Monte Carlo standard error and one-sided exact bounds for \(\widehat{P}_{n_\ell}\) at confidence level mc_conf_level. These quantify finite-imputation uncertainty and are diagnostic only.

6.2 Predictive probability at the maximum sample size

The maximum-sample-size predictive probability \(P_{\max,\ell}\) is estimated similarly, except that the completed trial includes both currently enrolled subjects and future subjects required to reach \(N_{\max}\). For each replicate, event times are imputed for the future subjects, the completed dataset is analyzed, and success is recorded:

\[P_{\max,\ell} = \operatorname{E}\{ \psi(\mathcal{D}_{N_{\max}}^{\mathrm{comp}}) \mid \mathcal{D}_{\ell}^{\mathrm{obs}} \}.\]

\[\widehat{P}_{\max,\ell} = \frac{1}{M}\sum_{m=1}^{M} I\{\text{success at } N_{\max} \text{ in replicate } m\}.\]

If neither upper decision is crossed and

\[\widehat{P}_{\max,\ell} < F_\ell,\]

the trial stops for binding futility and declares official failure. Otherwise, accrual continues to the next interim look.

The corresponding exact upper bound is reported in the trace as a diagnostic; it does not replace \(\widehat{P}_{\max,\ell}\) in the stopping rule.

Thus the interim action at look \(\ell\) can be represented as

\[A_\ell = \begin{cases} \text{stop and declare immediate success}, & \widehat{P}_{n_\ell} > Q_\ell,\\ \text{stop accrual for expected success}, & S_\ell < \widehat{P}_{n_\ell} \le Q_\ell,\\ \text{stop and declare binding futility}, & \widehat{P}_{n_\ell} \le S_\ell \text{ and } \widehat{P}_{\max,\ell} < F_\ell,\\ \text{continue accrual}, & \text{otherwise}. \end{cases}\]

The first look producing a non-continuation decision defines the adaptive stopping look. Because \(Q_\ell \ge S_\ell\), the immediate-success region is contained in the upper stopping region, so this can be written as

\[L^* = \inf\{\ell : \widehat{P}_{n_\ell} > S_\ell \text{ or } \widehat{P}_{\max,\ell} < F_\ell\},\]

with \(L^* = L + 1\) if no interim stopping condition is met and the design continues to \(N_{\max}\).

7.1 Frequentist final tests

For method = "logrank", success is based on a log-rank test. For method = "cox", success is based on the Wald test from a Cox proportional hazards regression. For method = "rmst", success uses a Wald test of the treatment-control RMST difference through rmst_tau. For a treatment-control difference in binary event risks at end_of_study, method = "riskdiff-wald" uses a Wald test and method = "riskdiff-fm" uses a Farrington-Manning score test. For these methods, goldilocks stores \(1-p\) in post_prob_ha; this is not a posterior probability, but it puts frequentist and Bayesian rules on a common “larger is stronger evidence” scale. For example, a one-sided test at \(\alpha = 0.025\) corresponds to prob_ha = 0.975.

For backward compatibility, method = "riskdiff" is accepted as an alias for "riskdiff-wald" and produces a warning. Results identify the analysis as "riskdiff-wald".

For the log-rank option, let \(Z_{\mathrm{LR}}\) denote the signed log-rank statistic, with positive values corresponding to excess events in the control arm under the package convention. Let \(p_{\mathrm{LR}}\) denote the traditional two-sided log-rank test P-value. The log-rank method supports only h0 = 0, denoting equal survival distributions; a nonzero margin is rejected rather than silently ignored. For the Cox option, let \(\widehat{\eta}\) be the estimated log hazard ratio for treatment versus control. For Cox analysis, h0 is on the log-hazard-ratio scale, so the package uses

\[Z_{\mathrm{Cox}} = \frac{\widehat{\eta} - h_0} {\operatorname{se}(\widehat{\eta})}.\]

When h0 = 0, this is the usual hazard-ratio-equals-1 null. A non-inferiority margin specified as a hazard ratio can be supplied as h0 = log(margin). Here, lower treatment hazard corresponds to \(Z_{\mathrm{Cox}} < 0\). Let \(p_{\mathrm{Cox}}\) denote the two-sided Wald-test P-value relative to h0. The package uses the following method-specific definitions of \(Q(\mathcal{D})\):

For method = "rmst", let \(\tau_R\) denote rmst_tau, a prespecified restriction time in \((0,\tau]\) that defaults to end_of_study. The effect is

\[\widehat\Delta_R = \int_0^{\tau_R}\widehat S_1(t)\,dt - \int_0^{\tau_R}\widehat S_0(t)\,dt.\]

The curves are Kaplan-Meier estimates. For arm \(z\), the Greenwood plug-in variance is

\[U_z = \sum_{j:t_{zj}<\tau_R} \left\{\int_{t_{zj}}^{\tau_R}\widehat S_z(t)\,dt\right\}^{2} \frac{d_{zj}}{Y_{zj}(Y_{zj}-d_{zj})},\]

where \(Y_{zj}\) is the risk set immediately before event time \(t_{zj}\) and \(d_{zj}\) is the number of events there. A terminal jump to zero contributes zero tail area and zero variance term. The test statistic is \(Z_R=(\widehat\Delta_R-h_0)/\sqrt{U_0+U_1}\). Positive effects mean longer time without the event on treatment. Both the estimated effect and h0 use time units; h0 must lie in \([-\tau_R,\tau_R]\). For an adverse event, superiority uses alternative = "greater", and non-inferiority allowing a loss of \(m\) time units uses the same alternative with h0 = -m.

The horizon is identical across arms, looks, and completed replicates; it is never selected from the observed follow-up. Positive estimated survival at an arm’s last follow-up before \(\tau_R\) makes the analysis non-estimable. A curve reaching zero before that horizon is allowed. The test requires positive total variance, although one arm may have zero variance. RMST inference does not require proportional hazards, but predictive imputation still depends on the piecewise-exponential model. Reducing rmst_tau does not reduce end_of_study or change the calendar-time calculations.

For the risk-difference options, let \(\widehat p_1\) and \(\widehat p_0\) be the observed event proportions in the treatment and control arms, with sample sizes \(n_1\) and \(n_0\). Both report the estimated effect

\[\widehat\Delta = \widehat p_1 - \widehat p_0,\]

and test \(H_0: p_1-p_0=h_0\). The Wald method uses the unpooled plug-in variance

\[U_{\Delta} = \frac{\widehat p_1(1-\widehat p_1)}{n_1} + \frac{\widehat p_0(1-\widehat p_0)}{n_0}.\]

The complete-data Wald statistic is

\[Z_{\mathrm{Wald}} = \frac{\widehat\Delta-h_0}{\sqrt{U_{\Delta}}}.\]

For riskdiff-fm, let \(\widetilde p_0\) and \(\widetilde p_1\) be the joint binomial maximum likelihood estimates constrained so that \(\widetilde p_1-\widetilde p_0=h_0\). The package obtains these estimates by maximizing the one-dimensional likelihood over

\[\max(0,-h_0) \leq \widetilde p_0 \leq \min(1,1-h_0),\]

with \(\widetilde p_1=\widetilde p_0+h_0\). The Farrington-Manning statistic is

\[Z_{\mathrm{FM}} = \frac{\widehat\Delta-h_0} {\sqrt{\widetilde p_1(1-\widetilde p_1)/n_1 + \widetilde p_0(1-\widetilde p_0)/n_0}}.\]

No continuity correction is applied. If the constrained variance is zero, the statistic is defined as zero when \(\widehat\Delta=h_0\), positive infinity when \(\widehat\Delta>h_0\), and negative infinity otherwise. Consequently, equal-arm all-zero and all-one tables under \(h_0=0\) give a neutral one-sided result rather than an error.

The one-sided directions differ between the log-rank rows and the model-based rows because of the sign convention of the package’s log-rank statistic.

The risk-difference analysis discards event-time information and requires complete binary endpoint status. est_final reports \(\widehat\Delta\).

When a Cox, RMST, or riskdiff-wald final analysis uses multiple imputation, the analysis is applied separately to each completed dataset. Let \(\widehat{\theta}_m\) and \(U_m\) be the scalar effect estimate and its estimated variance from imputation \(m = 1,\ldots,M\). For Cox regression \(\widehat{\theta}_m\) is the log hazard ratio; for risk difference it is \(\widehat\Delta_m\). For RMST it is \(\widehat\Delta_{R,m}\), with \(U_m\) the sum of the arm-specific Greenwood variances. Rubin’s scalar pooling rules give

\[\bar{\theta} = \frac{1}{M}\sum_{m=1}^{M}\widehat{\theta}_m, \qquad \bar{U} = \frac{1}{M}\sum_{m=1}^{M}U_m,\]

\[B = \frac{1}{M-1}\sum_{m=1}^{M} (\widehat{\theta}_m - \bar{\theta})^2, \qquad T = \bar{U} + \left(1 + \frac{1}{M}\right)B.\]

The pooled Wald statistic is \((\bar{\theta} - h_0) / \sqrt{T}\). Its P-value uses a \(t\) reference distribution with Rubin’s large-sample degrees of freedom

\[\nu = (M-1)\left(1 + \frac{1}{r}\right)^2, \qquad r = \frac{(1 + 1/M)B}{\bar{U}}.\]

When \(B = 0\) and \(\bar U > 0\), \(\nu = \infty\) and the reference distribution reduces to the standard normal distribution. If \(\bar U = 0\) and \(B > 0\), the package uses \(\nu = M-1\). If both variance components are zero, the pooled analysis is non-estimable and raises an error. At least two imputations and positive total variance are required. The returned est_final is \(\bar{\theta}\), while post_prob_ha is \(1-p\) from this pooled test with the direction determined by alternative.

If final outcomes are complete, the selected test is applied directly with either imputation flag. Genuine final imputation is unsupported for riskdiff-fm because no validated FM pooling rule is implemented; the package does not substitute a Wald test. Simulations combining FM with imputed_final = TRUE require zero prop_loss in both arms.

7.2 Bayesian survival final test

For method = "bayes-surv", the analysis posterior updates prior_surv_final with the dataset’s observed or imputed events and exposure. This applies both to hypothetical completed trials tested inside an interim prediction and to the actual final analysis. Posterior hazard draws are mapped to cumulative event probabilities at \(\tau\). In a two-arm design the treatment effect is

\[\Delta = p_1(\tau) - p_0(\tau),\]

where \(p_1(\tau)\) is the treatment-arm event probability and \(p_0(\tau)\) is the control-arm event probability. The effect is on the event scale, not the survival scale. For an adverse event, benefit usually means \(\Delta < 0\).

Because \(p_a(\tau) = 1 - \exp\{-H_a(\tau)\}\), posterior draws of \(\Delta\) are obtained by transforming posterior draws of \(\boldsymbol{\lambda}_1\) and \(\boldsymbol{\lambda}_0\):

\[\Delta^{(b)} = \left[1 - \exp\{-H_1^{(b)}(\tau)\}\right] - \left[1 - \exp\{-H_0^{(b)}(\tau)\}\right], \qquad b = 1,\ldots,B.\]

The fixed analysis-interval widths span time zero through end_of_study; they do not shorten to the maximum follow-up observed at an interim or final data cut. Observed events and person-time determine the Gamma posterior in each interval. Posterior hazards are transformed to \(H_a(\tau)\) and then to the event probability at the prespecified endpoint horizon.

The Monte Carlo estimate of the posterior probability for alternative = "less" is

\[\widehat{\Pr}(\Delta < h_0 \mid \mathcal{D}) = \frac{1}{B}\sum_{b=1}^{B} I(\Delta^{(b)} < h_0),\]

where \(B\) is set by N_mcmc.

With alternative = "less", success is declared when

\[\Pr(\Delta < h_0 \mid \mathcal{D}) > \texttt{prob\_ha}.\]

With alternative = "greater", success is declared when

\[\Pr(\Delta > h_0 \mid \mathcal{D}) > \texttt{prob\_ha}.\]

The Bayesian final test is one-sided in the package; alternative = "two.sided" is not supported.

In a single-arm design there is no \(p_0(\tau)\). The estimand becomes \(p_1(\tau)\), and \(h_0\) is an external benchmark event probability. In clinical-trial terminology this benchmark is often called a performance goal (PG) or objective performance criterion (OPC). Consequently, single-arm survival designs in goldilocks require method = "bayes-surv"; complete binary single-arm designs can use method = "bayes-bin".

7.3 Bayesian binary final test

For method = "bayes-bin", each analysis dataset is reduced to the binary indicator of whether the endpoint has occurred by \(\tau\). Subjects with right-censored follow-up before \(\tau\) must be imputed or excluded before this final test is applied, as described below.

Let \(x_z\) be the number of events and \(n_z\) the number of subjects in treatment group \(z\). With prior_bin = c(a, b), the event probability in arm \(z\) has posterior distribution

\[\pi_z \mid \mathcal{D} \sim \operatorname{Beta}(a + x_z, b + n_z - x_z).\]

In a two-arm design, the binary treatment effect is

\[\Delta_{\mathrm{bin}} = \pi_1 - \pi_0,\]

the treatment-arm event probability minus the control-arm event probability. For an adverse binary event, benefit usually means \(\Delta_{\mathrm{bin}} < 0\). With alternative = "less", success is declared when

\[\Pr(\Delta_{\mathrm{bin}} < h_0 \mid \mathcal{D}) > \texttt{prob\_ha}.\]

With alternative = "greater", success is declared when

\[\Pr(\Delta_{\mathrm{bin}} > h_0 \mid \mathcal{D}) > \texttt{prob\_ha}.\]

In a single-arm design, the estimand is \(\pi_1\) and h0 is the external benchmark event probability. Thus, alternative = "less" declares success when

\[\Pr(\pi_1 < h_0 \mid \mathcal{D}) > \texttt{prob\_ha}.\]

The posterior probability can be computed in three ways. With bin_method = "mc", the package draws from the beta posterior directly. With bin_method = "normal", it approximates the posterior event probability or treatment-control difference by a normal distribution with matching mean and variance. With bin_method = "quadrature", it evaluates the Beta CDF directly for a single arm and uses numerical integration for the two-arm posterior difference. The argument N_mcmc controls the number of Monte Carlo beta draws only when bin_method = "mc".

With sparse events or non-events and posterior event probabilities near 0 or 1, the normal approximation can misrepresent tail probabilities and change success or interim stopping decisions. Increasing N_mcmc does not correct this error. Use bin_method = "quadrature" or sufficiently precise Monte Carlo Beta draws when the approximation is unsuitable.

The completed-data calculation depends on \((x_z,n_z)\) directly. Participant- level data are reduced to these sufficient statistics after complete endpoint ascertainment has been verified. Normal-approximation, quadrature, and risk- difference results are deterministic conditional on these counts. With bin_method = "mc", posterior simulation is repeated independently for every predictive replicate.

7.4 Loss to follow-up at the final analysis

Interim predictions impute outcomes that are not yet known. At the final analysis, imputed_final controls whether subjects lost to follow-up are also imputed.

If no final outcomes require imputation, the selected complete-data test is used directly with either flag. Otherwise, with imputed_final = TRUE, Bayesian methods (method = "bayes-surv" or method = "bayes-bin") analyze each imputed completed dataset and average the resulting posterior summaries. Cox regression, RMST, and riskdiff-wald instead pool completed-data scalar estimates and variances using Rubin’s rules as described above; N_impute must be at least two. Genuine final imputation is unsupported for riskdiff-fm. Imputed final analyses remain unavailable for method = "logrank" because no pooling rule is implemented for that test.

If imputed_final = FALSE, the final analysis uses observed right-censored data for methods that can handle censoring (logrank, cox, rmst, and bayes-surv). For riskdiff-wald, riskdiff-fm, and bayes-bin, lost-to-follow-up subjects are excluded because these methods require complete binary outcomes and have no mechanism for right-censored observations. Rubin pooling applies to imputed Cox, RMST, and riskdiff-wald final analyses; it does not alter the interim posterior-predictive calculation, where each simulated completed trial is tested separately before the success indicators are averaged.

The independent dropout mechanism in Section 3.1 supports right-censored survival inference within each arm. It does not imply unbiased complete-case binary inference: early events can be ascertained before dropout, while later endpoint outcomes can be missing. Excluding the latter can overestimate the fixed-time event probability and distort treatment comparisons. Binary designs with dropout should assess final imputation under a suitable event-time model and sensitivity to that model. Designs where dropout depends on prognosis require sensitivity analyses outside the default independent mechanism.

8.1 Graphical assessment

The simulation plots address three complementary statistical questions:

• plot_sim_ocs() compares official success, immediate-success and other stopping probabilities, and mean sample size across data-generating scenarios.
• plot_sim_stopping() expands one scenario into marginal, conditional, or cumulative stopping summaries, or a count-based flowchart through successive looks.
• plot_sim_decisions() examines the joint interim predictive probabilities and the decision thresholds at each look.

For operating-characteristic curves, first attach a numeric effect scale to the scenario summary. The package does not infer this automatically because the appropriate scale may be a hazard ratio, risk difference, RMST difference, survival probability, or event probability depending on the analysis. Match values by scenario name because the summary can reorder the input scenarios:

scenario_oc <- summarise_sims(list(
  "null" = null_sims,
  "moderate" = moderate_sims,
  "target" = target_sims
))
effect_by_scenario <- c(null = 0, moderate = -0.10, target = -0.20)
scenario_oc$true_effect <- unname(effect_by_scenario[scenario_oc$scenario])

plot_sim_ocs(
  scenario_oc,
  effect = "true_effect",
  xlab = "True treatment-control event-probability difference"
)
plot_sim_stopping(target_sims)

Decision maps require the optional simulation traces:

target_sims_traced <- update(target_sims, return_trace = TRUE)
plot_sim_stopping(target_sims_traced, type = "flowchart")
plot_sim_decisions(target_sims_traced)

Retaining interim histories does not change the simulated trials, but it requires additional storage. The recorded sample sizes allow conditional, cumulative, and flowchart displays to include looks reached without a stopping decision. Detailed histories are therefore usually most useful for selected scenarios after a broad operating-characteristic grid has been screened.

Broglio et al. emphasize that type I error for this class of adaptive design should be examined across the null space, not only at one convenient null scenario. For time-to-event endpoints, the relevant null space includes plausible control event rates and accrual rates. Accrual rate is especially important because rapid enrollment can leave little endpoint information available at interim looks, increasing the uncertainty in both \(\widehat{P}_{n_\ell}\) and \(\widehat{P}_{\max,\ell}\). The follow-up period after accrual stops also affects operating characteristics because it determines how much additional information is observed before the final analysis.

The Monte Carlo sizes N_impute, N_mcmc, and N_trials should be chosen so that simulation error is small relative to the design decision being made. goldilocks reports the estimate, Monte Carlo standard error, exact bounds, number of draws, and stopping reason in the interim history. Decisions use strict comparisons of the point estimates with their thresholds; the bounds quantify Monte Carlo uncertainty but do not alter those decisions. mc_conf_level = 0.95 sets the confidence level for these bounds. Small values are appropriate only for illustration; operating characteristics must be calibrated with the Monte Carlo settings prespecified for the design.