Single-arm trials

The decision rule

In a single-arm trial there is no concurrent control, so the “treatment effect” is replaced by the cumulative event probability on the treatment arm itself:

\(\text{effect} \;=\; p_{\text{treatment}} \;=\; \Pr(\text{event by end\_of\_study} \mid \text{data}).\)

The argument h0 plays the role of a benchmark on this scale: a target failure probability (or, equivalently, \(1 - h_0\) is a target survival probability) drawn from external evidence such as a published rate, registry, or historical cohort. With alternative = "less" and prob_ha, the trial declares success when

\[\Pr(p_{\text{treatment}} < h_0 \mid \text{data}) \;>\; \texttt{prob\_ha},\]

i.e. when the posterior assigns enough mass to “the experimental therapy has a lower failure rate than the benchmark”. Choosing alternative = "greater" reverses the direction; alternative = "two.sided" is not allowed for method = "bayes".

The same posterior is used at each interim look to compute the predictive probability of eventual success, which drives the futility (Fn) and expected-success (Sn) stopping rules. Predictive probabilities are obtained by imputing remaining follow-up from the posterior predictive distribution of the (piecewise-)exponential model and re-evaluating the success criterion on each completed dataset.

Setting up the design

Suppose the existing standard of care has a 30% event probability by 24 months, and we are testing a new agent that we hope will reduce this to 20%. We use an interim look at 50 of 80 enrolled subjects:

end_of_study <- 24
benchmark <- 0.30                       # external standard-of-care failure rate
target    <- 0.20                       # rate we hope the new therapy achieves

# Convert the target failure rate into a constant hazard (so we can simulate)
ht <- prop_to_haz(probs = target, endtime = end_of_study)
ht
#> [1] 0.009297648

Now we run survival_adapt():

out <- survival_adapt(
  hazard_treatment = ht,
  hazard_control   = NULL,              # single-arm
  cutpoints        = 0,
  N_total          = 80,
  lambda           = 5,                 # enrolments per month (constant)
  lambda_time      = 0,
  interim_look     = 50,
  end_of_study     = end_of_study,
  prior            = c(0.1, 0.1),       # Gamma(0.1, 0.1) on the hazard
  block            = 2,                 # default; inert in single-arm mode
  rand_ratio       = c(1, 1),           # default; inert in single-arm mode
  prop_loss        = 0.05,
  alternative      = "less",
  h0               = benchmark,         # benchmark failure probability
  Fn               = 0.05,
  Sn               = 0.95,
  prob_ha          = 0.95,
  N_impute         = 50,
  N_mcmc           = 2000,
  method           = "bayes")

out
#>   prob_threshold margin alternative N_treatment N_control N_enrolled N_max
#> 1           0.95    0.3        less          80         0         80    80
#>   post_prob_ha est_final ppp_success stop_futility stop_expected_success
#> 1       0.9985 0.1689997        0.86             0                     0

A few points to highlight in the output:

• N_control = 0: no concurrent control was simulated.
• margin = 0.30: this is the value of h0 that the trial is testing against. Note that it is on the cumulative-failure scale, not the survival scale.
• est_final is the posterior mean of \(p_{\text{treatment}}\) at end_of_study, not a treatment effect relative to control.
• post_prob_ha is the posterior probability that \(p_{\text{treatment}} < h_0\).

Why block and rand_ratio still appear

survival_adapt() shares its trial-data simulator with the two-arm case. In single-arm mode the simulator skips randomization() entirely and assigns every subject to the treatment arm; block and rand_ratio are therefore inert and can be left at their defaults. The minimum-interim_look rule (interim_look >= max(block)) only applies to two-arm designs, so a single-arm trial can use any interim_look strictly less than N_total.

Operating characteristics

A single trial does not tell you whether the design is well-calibrated. To estimate power and type I error, we run the design under each scenario using sim_trials(). The chunks below are not run when knitting (each takes a few minutes) but illustrate the workflow:

# Power: simulate under the alternative (true rate = 0.20)
out_power <- sim_trials(
  N_trials         = 1000,
  hazard_treatment = ht,
  hazard_control   = NULL,
  cutpoints        = 0,
  N_total          = 80,
  lambda           = 5,
  lambda_time      = 0,
  interim_look     = 50,
  end_of_study     = end_of_study,
  prior            = c(0.1, 0.1),
  block            = 2,
  rand_ratio       = c(1, 1),
  prop_loss        = 0.05,
  alternative      = "less",
  h0               = benchmark,
  Fn               = 0.05,
  Sn               = 0.95,
  prob_ha          = 0.95,
  N_impute         = 50,
  N_mcmc           = 2000,
  method           = "bayes")

# Type I error: simulate under the null (true rate = benchmark = 0.30)
ht_null <- prop_to_haz(probs = benchmark, endtime = end_of_study)
out_t1error <- sim_trials(
  N_trials         = 1000,
  hazard_treatment = ht_null,
  hazard_control   = NULL,
  cutpoints        = 0,
  N_total          = 80,
  lambda           = 5,
  lambda_time      = 0,
  interim_look     = 50,
  end_of_study     = end_of_study,
  prior            = c(0.1, 0.1),
  block            = 2,
  rand_ratio       = c(1, 1),
  prop_loss        = 0.05,
  alternative      = "less",
  h0               = benchmark,
  Fn               = 0.05,
  Sn               = 0.95,
  prob_ha          = 0.95,
  N_impute         = 50,
  N_mcmc           = 2000,
  method           = "bayes")

summarise_sims(list(out_power$sims, out_t1error$sims))

Calibration proceeds the same way as for two-arm designs: if the type I error under the null (where the true rate equals the benchmark) is above the desired level, raise prob_ha; if power is too low, increase N_total or relax the Fn/Sn thresholds.

A practical caveat on benchmarks

The validity of a single-arm Goldilocks trial rests entirely on the benchmark h0 being a fair representation of the population the trial is enrolling. Drift in standard of care, differences in patient mix, and unmeasured confounding all bias the comparison in a way that randomisation would otherwise neutralise. A Bayesian framework can incorporate uncertainty about the benchmark itself – e.g. by replacing a fixed h0 with a prior distribution informed by historical data – but this is outside the scope of the simple h0 scalar that survival_adapt() exposes, and would require a custom analysis. When in doubt, simulating the design under several plausible values of the true rate (including ones near the benchmark) is a useful way to characterise its sensitivity.

See also

• The “Example: Two-armed RCT” vignette covers the corresponding two-arm randomised design with a log-rank decision rule.
• The “Bayesian decisions with piecewise-exponential hazards” vignette covers the same decision rule used here, but in a two-arm setting and with non-constant hazards. The piecewise machinery applies directly to single-arm trials too (just keep hazard_control = NULL and pass a per-interval hazard_treatment vector).
• ?survival_adapt documents all arguments.