# Single-arm designs with a performance goal

    library(goldilocks)

Several other vignettes focus on two-arm randomized designs, although the Bayesian binary outcome vignette also includes a single-arm example. Single-arm trials – in which every subject receives the experimental therapy and the comparator is an external benchmark, often called a performance goal (PG) or objective performance criterion (OPC) – are common in early-phase oncology, rare-disease, and proof-of-concept studies. This vignette shows how to set up a Goldilocks single-arm design with `survival_adapt()`.

Two practical constraints on single-arm designs in this package:

- A single-arm trial is signaled by setting `hazard_control = NULL`.
- `method = "bayes-surv"` supports single-arm survival analyses with piecewise-exponential event-time modeling. `method = "bayes-bin"` supports single-arm analyses of complete binary outcomes. The frequentist methods (`logrank`, `cox`, `riskdiff`) require two arms and will raise an error if used in this mode.

## 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. In clinical-trial terminology this benchmark may be referred to as a performance goal (PG) or objective performance criterion (OPC). 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-surv"`.

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        = NULL,
      N_total          = 80,
      lambda           = 5,                 # enrollments per month (constant)
      lambda_time      = NULL,
      interim_look     = 50,
      end_of_study     = end_of_study,
      prior_surv            = c(0.1, 0.1),       # Gamma(0.1, 0.1) on the hazard
      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-surv")

    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        0.992 0.1825442        0.08             0                     0

There is no need to supply `block` or `rand_ratio`: they are redundant in a single-arm design because no randomization is performed.

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 \(p_{\text{treatment}} < h_0\).

## 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        = NULL,
      N_total          = 80,
      lambda           = 5,
      lambda_time      = NULL,
      interim_look     = 50,
      end_of_study     = end_of_study,
      prior_surv            = c(0.1, 0.1),
      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-surv",
      return_trace     = TRUE,
      seed             = 3082)

    # Type I error: simulate under the null (true rate = benchmark/PG/OPC = 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        = NULL,
      N_total          = 80,
      lambda           = 5,
      lambda_time      = NULL,
      interim_look     = 50,
      end_of_study     = end_of_study,
      prior_surv            = c(0.1, 0.1),
      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-surv",
      seed             = 3083)

    oc <- summarise_sims(list(
      "target event probability" = out_power$sims,
      "benchmark event probability" = out_t1error$sims
    ))
    oc$true_event_probability <- c(target, benchmark)

    oc
    plot_sim_ocs(
      oc,
      effect = "true_event_probability",
      xlab = "True treatment event probability"
    )
    plot_sim_stopping(out_power)
    plot_sim_decisions(out_power)

The operating-characteristic plot compares scenarios on the clinically natural event-probability scale. The stopping plot then expands one scenario to show where enrollment ends and why. Finally, the decision map uses the retained traces to show how interim predictive probabilities fall relative to the expected-success and futility thresholds. For large calibration grids, retain traces only for scenarios whose interim behavior needs closer inspection.

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` (the PG or OPC) 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 randomization would otherwise neutralize. 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 characterize its sensitivity.

## See also

- The “Two-arm randomized trials” vignette covers the corresponding two-arm randomized design with a log-rank decision rule.
- The “Bayesian piecewise-exponential designs” 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.