Technical details of the Goldilocks design • goldilocks

Overview

The goldilocks package implements the Goldilocks adaptive sample-size design of Broglio, Connor, and Berry (2014) for time-to-event and fixed-time binary endpoints. This vignette outlines the technical details of the design, including notation, the continuous-time enrollment process, the piecewise-exponential event-time model, Gamma posterior updating, posterior predictive probabilities, interim decision rules, final analysis options, and simulation-based calibration. The package vignettes “Two-arm randomized trials”, “Bayesian piecewise-exponential designs”, “Bayesian binary outcome designs”, and “Single-arm designs with a performance goal” provide more application-focused examples in R.

1. Design and notation

Consider a trial with maximum sample size N_{\max}, planned endpoint time \tau, and interim sample-size selection analyses after

n_1 < n_2 < \cdots < n_L < N_{\max}

subjects have been enrolled. The package argument N_total corresponds to N_{\max}, end_of_study to \tau, and interim_look to (n_1,\ldots,n_L).

Let Z_i \in \{0,1\} denote treatment assignment for subject i, matching the package data column treatment: Z_i = 1 for the treatment arm and Z_i = 0 for the control arm. In a single-arm design, all Z_i = 1 and there is no concurrent control. The package assumes that randomization occurs at enrollment; in practice those times can differ, but the distinction is not represented in the simulation model. We therefore use enrollment time throughout. Let E_i denote enrollment time from first patient in, T_i^* the true event time from enrollment, C_i the administrative or loss-to-follow-up censoring time, and

T_i = \min(T_i^*, C_i), \qquad \delta_i = I(T_i^* \le C_i).

At the interim analysis held when n_\ell subjects have been enrolled, the analysis calendar time is E_{n_\ell}. Subject i’s current observed follow-up is

u_{i\ell} = \max\{0, E_{n_\ell} - E_i\}.

For enrolled subjects, the interim data are masked to the information available at this calendar time. If an event has not yet occurred by u_{i\ell}, the subject is treated as censored at u_{i\ell}. Subjects with i > n_\ell are not yet enrolled and contribute only to the maximum-sample-size prediction.

At each interim analysis the design estimates two predictive probabilities:

P_{n_\ell}, the probability of final success if accrual stops at the current enrolled sample size and all enrolled subjects complete follow-up;
P_{\max,\ell}, the probability of final success if the trial continues to N_{\max}.

The current-sample-size probability is compared first with the immediate-success threshold Q_\ell and then with the expected-success threshold S_\ell. The maximum-sample-size probability is compared with the futility threshold F_\ell. These correspond to the package arguments Qn, Sn, and Fn. The package requires Q_\ell \ge S_\ell; the default Q_\ell = 1 disables immediate-success stopping because decisions use strict inequalities.

2. Continuous-time enrollment process

The simulation model places the first enrolled subject at time zero. This is a first-patient-in origin, not an earlier protocol-approval, site-activation, or recruitment-opening date. Let E_1 = 0 and let E_i for i > 1 denote the calendar time of the ith enrollment measured from first patient in.

Enrollment after the first patient follows a non-homogeneous Poisson process with a piecewise-constant intensity. Write the K internal enrollment-rate knots as

0 < a_1 < a_2 < \cdots < a_K,

and define a_0 = 0 and a_{K+1} = \infty. These internal knots are supplied as lambda_time; zero is implicit and is not included. There are K+1 positive rates

\boldsymbol{\rho} = (\rho_1,\ldots,\rho_{K+1}),

supplied as lambda, with

\rho(t) = \rho_j, \qquad a_{j-1} \le t < a_j.

This is a convenient right-continuous representation of the intensity; its value at an isolated knot does not change the Poisson process. When a realized enrollment time is assigned to a rate interval, goldilocks uses the survival counting-process convention (a_{j-1},a_j]. An enrollment exactly at a_j therefore belongs to the interval ending there. The first patient at time zero is fixed as the calendar origin rather than generated by this process.

Consequently, length(lambda) must be exactly length(lambda_time) + 1. A constant enrollment rate is represented by lambda_time = NULL and a scalar lambda. The final rate continues beyond the last knot until the requested N_total is reached; enrollment() does not impose a finite recruitment horizon.

The cumulative enrollment intensity is

A(t) = \int_0^t \rho(u)\,du = \sum_{j=1}^{K+1} \rho_j\{\min(t,a_j)-a_{j-1}\}_+.

To generate the process exactly, draw independent variables X_2,\ldots,X_{N_{\max}} \sim \operatorname{Exponential}(1) and form

Q_i = \sum_{k=2}^{i} X_k.

The enrollment times are then obtained through the inverse cumulative intensity,

E_1 = 0, \qquad E_i = A^{-1}(Q_i), \quad i=2,\ldots,N_{\max}.

More explicitly, if A(a_{j-1}) \le Q_i < A(a_j), then

E_i = a_{j-1} + \frac{Q_i-A(a_{j-1})}{\rho_j}.

This time-rescaling construction is exact: after the patient anchored at zero, the number of arrivals in any interval (s,t] is Poisson with mean A(t)-A(s), and counts over disjoint intervals are independent. Under a constant rate \rho, the successive gaps are independent \operatorname{Exponential}(\rho) variables and E_n \sim \operatorname{Gamma}(n-1,\text{rate}=\rho). In particular, \operatorname{E}(E_n)=(n-1)/\rho. The fixed first patient accounts for the n-1, rather than n, random gaps.

This differs from generating Poisson counts in unit-time bins and adding uniform jitter afterward. Such binning is sensitive to the arbitrary width of a time unit and cannot represent a rate change inside a bin. Cumulative- intensity inversion instead preserves the prespecified continuous-time accrual model for integer and fractional change-points alike.

For example,

enrollment(
  lambda = c(2, 5, 8),
  lambda_time = c(3.5, 9),
  N_total = 100
)

assigns positive realized enrollment times in (0,3.5] to rate 2, times in (3.5,9] to rate 5, and later times to rate 8. All rates are enrollments per common time unit. lambda_time, enrollment times, event times, cutpoints, generation_cutpoints, and end_of_study should therefore all use the same unit, such as days or months.

Although lambda_time, cutpoints, and generation_cutpoints are all entered as interior change-points, they operate on different time scales. Enrollment change-points are trial-calendar times from first patient in. Both event-time partitions use follow-up from each participant’s enrollment. Their values and lengths are unrelated unless the scientific design makes them coincide.

The package does not currently model site activations, site-specific random rates, pauses, recruitment caps, or uncertainty in the supplied rates. Those operational features require a richer site-level accrual model; lambda represents the trial-level rate schedule assumed for a simulation scenario.

3. Event-time model

The data-generating and predictive-analysis models are both piecewise exponential, but their partitions may differ. For generation, let

0 = g_0 < g_1 < \cdots < g_{K-1}.

The interior values (g_1,\ldots,g_{K-1}) are supplied through generation_cutpoints. For treatment value z \in \{0,1\}, the corresponding hazard_treatment or hazard_control vector supplies one hazard \theta_{zk} per generating interval:

h^{\mathrm{gen}}_z(t) = \theta_{zk}, \qquad g_{k-1} \le t < g_k,

where g_K = \infty. This is PWEALL’s representation of the continuous generating hazard.

For posterior estimation, predictive imputation, and final piecewise-exponential analysis, let

0 = s_0 < s_1 < \cdots < s_{J-1}.

The interior cutpoints (s_1,\ldots,s_{J-1}) are supplied through cutpoints; the initial boundary s_0 = 0 is implicit. The analysis model has one hazard \lambda_{zj} per interval. Realized event times are assigned to analysis intervals using

(s_0, s_1], (s_1, s_2], \ldots, (s_{J-1}, \infty).

Here z = 1 denotes the treatment arm and z = 0 denotes the control arm. The analysis hazard is

h_z(t) = \lambda_{zj}, \qquad s_{j-1} \le t < s_j,

where the last interval has no finite upper endpoint. With cutpoints = NULL, J = 1 and the analysis uses an ordinary exponential model. With generation_cutpoints = NULL, generation uses one constant hazard. By default, generation_cutpoints = cutpoints, which reproduces the historical shared partition; explicitly supplying both arguments permits model misspecification scenarios without transforming either partition.

The cumulative hazard and event-time distribution are continuous at each generating cutpoint, so changing the hazard at the cutpoint itself does not change generated event times. For observed-data sufficient statistics, goldilocks follows Surv(start, stop, event) and survSplit(): an event recorded exactly at s_j is counted in the analysis interval ending at s_j. This explicit rule matters for rounded or discrete recorded times even though the boundary has probability zero under a continuous generating model.

The cumulative hazard for treatment value z at time t is

H_z(t) = \sum_{j=1}^{J} \lambda_{zj} \{ \min(t, s_j) - s_{j-1} \}_+,

where s_J = \infty and \{x\}_+ = \max(x,0). The corresponding survival and cumulative event probability are

S_z(t) = \exp\{-H_z(t)\}, \qquad p_z(t) = 1 - S_z(t).

For simulation planning, prop_to_haz() returns the piecewise hazards implied by event probabilities at prespecified time points. ppwe() evaluates p_z(\tau), and haz_to_prop() applies this transformation to posterior hazard draws.

For observed follow-up (T_i, \delta_i, Z_i), the piecewise-exponential likelihood can be written in terms of interval-specific event counts and exposure times. Define

d_{zj} = \sum_i I(Z_i = z)\delta_i I(T_i \in (s_{j-1}, s_j]),

and

y_{zj} = \sum_i I(Z_i = z) \{ \min(T_i, s_j) - s_{j-1} \}_+ I(T_i > s_{j-1}).

Up to factors not involving \lambda_{zj}, the likelihood contribution for treatment value z is

L_z(\boldsymbol{\lambda}_z; \mathcal{D}) \propto \prod_{j=1}^{J} \lambda_{zj}^{d_{zj}} \exp(-\lambda_{zj}y_{zj}),

where \boldsymbol{\lambda}_z = (\lambda_{z1},\ldots,\lambda_{zJ})^\top. This sufficient-statistic form is what makes the Gamma posterior update available in closed form.

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.

4. Posterior distribution of hazards

For each treatment value z and interval j, goldilocks assumes an independent Gamma prior

\lambda_{zj} \sim \operatorname{Gamma}(\alpha_{0zj}, \beta_{0zj}),

where \alpha_{0zj} is the shape and \beta_{0zj} is the rate for arm z and interval j. This follows the stats::rgamma() shape-rate parameterization. The specification prior_surv = c(alpha0, beta0) applies one shape-rate pair to every arm and interval. Alternatively, a two-row matrix specifies interval-specific shapes in row 1 and rates in row 2, with intervals in chronological order, and applies them to both arms. Independent arm- specific priors are supplied as a list named control and treatment; each element may be a length-two vector or a two-row interval-specific matrix. Both arms must be specified, and no borrowing occurs between them. prior_surv_final accepts the same specifications and defaults to prior_surv; priors may therefore differ by analysis stage, arm, and interval. For method = "bayes-surv", prior_surv_final is used during interim calculations as well as at the actual final analysis:

Calculation	Prior used
Generate outstanding outcomes at interim, for enrolled and future participants	`prior_surv`
Test each hypothetical completed trial at the current or maximum sample size	`prior_surv_final`
Impute missing outcomes at final analysis, if `imputed_final = TRUE`	`prior_surv_final`
Analyze the actual final data	`prior_surv_final`

An informative predictive prior can bring in external evidence while a separate weak analysis prior defines the success criterion. Both must be specified explicitly for this separation: omitting prior_surv_final makes it equal to prior_surv, including when prior_surv is informative. These roles also apply to evaluate_interim(), which performs the two interim calculations. See the worked example.

The predictive prior can still influence the stopping decision and selected sample size. Calibrate the design with both priors fixed to their intended values. This table applies to Bayesian survival: Bayesian binary completed-data analyses use prior_bin, and frequentist completed-data tests have no analysis prior.

At an analysis, let d_{zj} be the number of observed events for treatment value z, interval j, and let y_{zj} be the total observed exposure time for that treatment value and interval. Gamma-exponential conjugacy gives

\lambda_{zj} \mid \mathcal{D} \sim \operatorname{Gamma}(\alpha_{0zj} + d_{zj}, \beta_{0zj} + y_{zj}).

The package obtains (d_{zj}, y_{zj}) by splitting each subject’s observed follow-up over the cut-point intervals. Posterior draws are generated independently for each treatment group and interval. In early interim analyses, later piecewise intervals may have no exposure. The empty_interval argument controls the policy for these intervals:

empty_interval = "prior" is the default. It leaves the interval with d_{zj} = 0 and y_{zj} = 0, so the posterior for that interval is exactly the specified Gamma prior. Interval-specific priors should therefore be checked by prior-predictive simulation when little late follow-up is expected.
empty_interval = "propagate" is an explicit legacy heuristic. It propagates exposure and event counts from the nearest non-empty interval within the same treatment group and emits a warning. This should not be interpreted as observed evidence about the empty interval. Set this option explicitly only to reproduce or sensitivity-check historical analyses.
empty_interval = "error" stops the analysis when any treatment-arm interval has no exposed subjects.

The posterior density factorizes as

\pi(\boldsymbol{\lambda} \mid \mathcal{D}) = \prod_z \prod_{j=1}^{J} \pi(\lambda_{zj} \mid d_{zj}, y_{zj}),

where each marginal factor is the Gamma distribution above. Posterior predictive calculations integrate over this density rather than conditioning on a single plug-in hazard estimate.

5. Predictive distribution for incomplete outcomes

At an interim analysis, subjects can be separated into three sets:

enrolled subjects with complete endpoint information;
enrolled subjects with partial follow-up; and
future subjects not yet enrolled.

The first set contributes observed events and exposure to the posterior. The second and third sets require prediction.

For an enrolled subject who is event-free through time u, a future event time is drawn from the conditional piecewise-exponential distribution

\Pr(T \le t \mid T > u) = \frac{F(t) - F(u)}{1 - F(u)}, \qquad t > u,

where F(t) = 1 - S(t). Equivalently, if U \sim \operatorname{Uniform}(0,1), then

T = F^{-1}\{F(u) + U[1 - F(u)]\}.

pwe_impute() samples from this conditional distribution. For future participants in the maximum-sample-size calculation, pwe_sim() draws unconditional event times from the same piecewise-exponential model. The number of future participants is N_{\max}-n_\ell; their individual future enrollment times are not needed for this predictive calculation.

Let \mathcal{D}_{\ell}^{\mathrm{obs}} denote the data observed at look \ell, and let \mathcal{D}^{\mathrm{mis}} denote unobserved event times and future censoring indicators. The posterior predictive density is

p(\mathcal{D}^{\mathrm{mis}} \mid \mathcal{D}_{\ell}^{\mathrm{obs}}) = \int p(\mathcal{D}^{\mathrm{mis}} \mid \boldsymbol{\lambda}) \pi(\boldsymbol{\lambda} \mid \mathcal{D}_{\ell}^{\mathrm{obs}}) d\boldsymbol{\lambda}.

The package approximates this integral by Monte Carlo simulation. Each replicate draws \boldsymbol{\lambda} from its interim posterior, completes the unobserved outcomes conditional on those hazards, and applies the prespecified analysis. Treatment assignments remain fixed. Bayesian survival, log-rank, Cox, and RMST analyses use completed follow-up and event outcomes; fixed-horizon binary analyses use completed event counts and denominators by arm, which are sufficient statistics for the risk-difference and beta-binomial methods.

6. Interim decision rule

At interim look \ell, the package first estimates the posterior of the hazard parameters from the currently observable data. It then uses Monte Carlo integration to approximate P_{n_\ell} and P_{\max,\ell}.

Let \psi(\mathcal{D}) be the final success indicator for a completed analysis dataset:

\psi(\mathcal{D}) = I\{Q(\mathcal{D}) > c\},

where Q(\mathcal{D}) is the final analysis quantity and c is the success threshold. In package notation, c is set by prob_ha. The final analysis quantity Q(\mathcal{D}) depends on the analysis method:

Design setting	`method`	Q(\mathcal{D})	Supported alternatives
Two-arm randomized trial	`logrank`	1-p(\mathcal{D}), where p(\mathcal{D}) is the traditional log-rank test P-value, with one-sided variants defined in Section 7.1	`"less"`, `"greater"`, `"two.sided"`
Two-arm randomized trial	`rmst`	1-p(\mathcal{D}) for a Wald test of the treatment-control RMST difference through `rmst_tau`	`"less"`, `"greater"`, `"two.sided"`
Two-arm randomized trial	`cox`	1-p(\mathcal{D}), where p(\mathcal{D}) is the traditional Wald-test P-value, with one-sided variants defined in Section 7.1	`"less"`, `"greater"`, `"two.sided"`
Two-arm randomized trial	`riskdiff-wald`	1-p(\mathcal{D}), where p(\mathcal{D}) is the Wald-test P-value for the treatment-control event-risk difference	`"less"`, `"greater"`, `"two.sided"`
Two-arm randomized trial	`riskdiff-fm`	1-p(\mathcal{D}), where p(\mathcal{D}) is the Farrington-Manning score-test P-value for the treatment-control event-risk difference	`"less"`, `"greater"`, `"two.sided"`
Two-arm randomized trial	`bayes-surv`	\Pr(\Delta < h_0 \mid \mathcal{D}) or \Pr(\Delta > h_0 \mid \mathcal{D})	`"less"`, `"greater"`
Single-arm trial	`bayes-surv`	\Pr(p_1(\tau) < h_0 \mid \mathcal{D}) or \Pr(p_1(\tau) > h_0 \mid \mathcal{D})	`"less"`, `"greater"`
Two-arm randomized trial	`bayes-bin`	\Pr(\Delta_{\mathrm{bin}} < h_0 \mid \mathcal{D}) or \Pr(\Delta_{\mathrm{bin}} > h_0 \mid \mathcal{D})	`"less"`, `"greater"`
Single-arm trial	`bayes-bin`	\Pr(\pi_1 < h_0 \mid \mathcal{D}) or \Pr(\pi_1 > h_0 \mid \mathcal{D})	`"less"`, `"greater"`

For frequentist analyses, prob_ha is therefore a transformed P-value threshold. For example, prob_ha = 0.975 corresponds to a one-sided \alpha = 0.025 rule. The value should be chosen during design calibration to control the desired type I error rate across relevant null scenarios.

6.1 Predictive probability at the current sample size

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

draw one set of hazard parameters from the interim posterior;
impute remaining endpoint outcomes for enrolled subjects only;
apply the prespecified final analysis rule to the completed data; and
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. Final analysis

After an expected-success stop or completion of enrollment, the final analysis is conducted when the relevant follow-up has completed for the enrolled cohort, subject to the handling of loss to follow-up described below. Immediate success and binding futility are terminal official decisions at the interim look and do not require a later analysis to determine trial_success. The final rule below also supplies the binary success indicator used inside the predictive probability calculations.

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.

Method	Alternative	Q(\mathcal{D})
`logrank`	`"less"`	\Phi(Z_{\mathrm{LR}})
`logrank`	`"greater"`	1 - \Phi(Z_{\mathrm{LR}})
`logrank`	`"two.sided"`	1 - p_{\mathrm{LR}}
`rmst`	`"less"`	1-\Phi(Z_R)
`rmst`	`"greater"`	\Phi(Z_R)
`rmst`	`"two.sided"`	1-2\Phi(-\|Z_R\|)
`cox`	`"less"`	1 - \Phi(Z_{\mathrm{Cox}})
`cox`	`"greater"`	\Phi(Z_{\mathrm{Cox}})
`cox`	`"two.sided"`	1 - p_{\mathrm{Cox}}
`riskdiff-wald`	`"less"`	1 - \Phi(Z_{\mathrm{Wald}})
`riskdiff-wald`	`"greater"`	\Phi(Z_{\mathrm{Wald}})
`riskdiff-wald`	`"two.sided"`	1 - 2\Phi(-\|Z_{\mathrm{Wald}}\|)
`riskdiff-fm`	`"less"`	1 - \Phi(Z_{\mathrm{FM}})
`riskdiff-fm`	`"greater"`	\Phi(Z_{\mathrm{FM}})
`riskdiff-fm`	`"two.sided"`	1 - 2\Phi(-\|Z_{\mathrm{FM}}\|)

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. Operating characteristics

A Goldilocks design is calibrated by simulation. A single simulated trial describes one possible path; the design is characterized by repeated simulation over clinically relevant scenarios.

For a candidate design, sim_trials() generates repeated trials and summarise_sims() estimates:

power, or type I error under a null scenario;
probability of declaring immediate success;
probability of stopping for expected success;
probability of stopping for futility;
probability of reaching N_{\max};
mean and standard deviation of enrolled sample size; and
probability of stopping for expected success but failing at the final analysis.

Let R = 1,\ldots,R_{\max} index simulated trials under a scenario \theta, where \theta denotes the data-generating parameters such as control hazard, treatment hazard, accrual rate, loss-to-follow-up rate, and follow-up duration. The trial-level random variables are:

Symbol	Meaning
N_R	enrolled sample size in simulated trial R
I_R	indicator that trial R declared immediate success
E_R	indicator that trial R stopped accrual for expected success
F_R	indicator that trial R stopped for futility
Z_R	official success indicator in trial R

Let \Theta_0 denote the null parameter space, i.e. the set of data-generating scenarios in which the treatment does not satisfy the alternative hypothesis. For a two-arm superiority trial this includes scenarios with no beneficial treatment effect; in practice, it should be explored across plausible nuisance parameters such as control event rates and accrual rates. The main operating characteristics are

\operatorname{Power}(\theta) = \Pr_\theta(Z_R = 1),

\operatorname{Type\ I\ error}(\theta_0) = \Pr_{\theta_0}(Z_R = 1), \qquad \theta_0 \in \Theta_0,

\Pr_\theta(\text{declare immediate success}) = \Pr_\theta(I_R = 1),

\Pr_\theta(\text{stop for expected success}) = \Pr_\theta(E_R = 1),

\Pr_\theta(\text{stop for futility}) = \Pr_\theta(F_R = 1),

and

\operatorname{E}_\theta(N_R), \qquad \operatorname{Var}_\theta(N_R).

The stop_and_fail summary estimates

\Pr_\theta(E_R = 1, Z_R = 0),

which is the probability that accrual stops for expected success but the final analysis does not meet the success criterion.

One simulation specification is:

out <- sim_trials(
  hazard_treatment = ht,
  hazard_control   = hc,
  cutpoints        = cutpoints,
  N_total          = N_total,
  lambda           = lambda,
  lambda_time      = lambda_time,
  interim_look     = interim_look,
  end_of_study     = end_of_study,
  prior_surv       = prior_surv,
  Fn               = Fn,
  Sn               = Sn,
  Qn               = Qn,
  prob_ha          = prob_ha,
  N_impute         = N_impute,
  N_mcmc           = N_mcmc,
  N_trials         = N_trials,
  method           = method,
  seed             = 12345)

summarise_sims(
  out,
  max_mcse = c(power = 0.005, stop_futility = 0.01, mean_N = 1)
)

summarise_sims() reports a Monte Carlo standard error and 95% Monte Carlo confidence interval for each probability and for mean sample size. Wilson intervals are used for probabilities so that zero or one observed outcome does not produce a misleading zero-width interval; mean sample size uses a t interval. These quantify repeated-simulation error under fixed design and data-generating assumptions. They are not clinical confidence intervals for a treatment effect and do not include uncertainty about the assumed outcome, accrual, or missing-data models. Requested, analyzed, failed, and used counts make the denominator explicit. Failed simulations are reported separately and are not treated as unsuccessful clinical trials. The optional named max_mcse targets produce a warning when the achieved Monte Carlo precision is too weak for the intended comparison.

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.

9. Threshold selection

The thresholds Q_\ell, S_\ell, and F_\ell may be constant across looks or may vary by look. They interact with the final analysis threshold, number and timing of looks, endpoint delay, accrual rate, loss to follow-up, prior distribution, and maximum sample size.

If type I error is too high, possible remedies include increasing prob_ha, increasing Qn or Sn, reducing the number of looks, or altering follow-up requirements. If power is too low, the maximum sample size, futility threshold, upper thresholds, or final analysis threshold may need reconsideration. Each change should be rechecked under null and alternative scenarios.

The quantity stop_and_fail is particularly useful when tuning S_\ell. It estimates how often a trial stops accrual for expected success but does not meet the final success criterion after follow-up is complete. If this is too large, the expected-success threshold is usually too permissive for the amount of uncertainty present at interim looks.

10. Relation to group-sequential designs

A group-sequential design usually indexes interim analyses by information, such as the number of observed events, and may stop immediately for efficacy when a boundary is crossed. A Goldilocks design indexes sample-size selection analyses by enrolled sample size and explicitly incorporates future follow-up of the currently enrolled cohort.

This distinction matters for delayed outcomes. A trial may enroll many subjects before accumulating enough events for an event-driven interim analysis. Goldilocks uses the partial information available during accrual to decide whether additional subjects are needed. With the default Qn = 1, success is determined at a single preplanned final analysis after the enrolled subjects have completed follow-up. A design with Qn < 1 can instead declare terminal success at an interim look and therefore requires explicit operating- characteristic calibration of that additional decision.

11. Scope and statistical limitations

The time-to-event example in Broglio et al. (2014) uses a Gamma-exponential prediction model. goldilocks extends this to a piecewise-exponential model, so the hazard may change at prespecified cut-points. This can be useful when there is a clinically plausible early-risk period, but each additional interval adds parameters and can make interim posteriors more diffuse.

The package also supports single-arm Bayesian Goldilocks designs by replacing the concurrent control with an external benchmark h_0, often referred to as a performance goal (PG) or objective performance criterion (OPC). This is convenient for early-phase, rare-disease, or proof-of-concept settings, but validity then depends on the benchmark being transportable to the enrolled population.

For complete binary endpoints, method = "bayes-bin" replaces the piecewise- exponential completed-data analysis with a conjugate beta-binomial analysis. Pending endpoint statuses are still imputed from the piecewise-exponential event-time model. These are separate models rather than components of one joint Bayesian model: the Gamma hazard prior used for imputation does not determine the Beta event-probability prior used for completed-data analysis. Both priors can affect predictive decisions when outcomes are pending. The analysis output records prior_surv, prior_surv_final, prior_bin, and the imputation horizon end_of_study so that the complete statistical specification can be reviewed.

References

Broglio KR, Connor JT, Berry SM. Not too big, not too small: a Goldilocks approach to sample size selection. Journal of Biopharmaceutical Statistics, 2014; 24(3): 685-705. doi:10.1080/10543406.2014.888569.

Farrington CP, Manning G. Test statistics and sample size formulae for comparative binomial trials with null hypothesis of non-zero risk difference or non-unity relative risk. Statistics in Medicine, 1990; 9: 1447-1454. doi:10.1002/sim.4780091208.

U.S. Food and Drug Administration. Adaptive Design Clinical Trials for Drugs and Biologics Guidance for Industry. December 2019. https://www.fda.gov/regulatory-information/search-fda-guidance-documents/adaptive-design-clinical-trials-drugs-and-biologics-guidance-industry.

U.S. Food and Drug Administration. Guidance for the Use of Bayesian Statistics in Medical Device Clinical Trials. February 5, 2010. https://www.fda.gov/media/71512/download.