Bayesian decisions with piecewise-exponential hazards

library(goldilocks)

The “Example: Two-armed RCT” vignette uses method = "logrank" and a single constant hazard per arm. That is convenient when the proportional-hazards assumption is reasonable and the underlying event rate is well-approximated by an exponential distribution. In practice, neither assumption always holds. This vignette shows how to set up a Goldilocks design with:

a piecewise-exponential hazard, in which the constant hazard rate changes at one or more cut-points; and
a Bayesian decision rule (method = "bayes") using a posterior probability threshold on the cumulative-failure-probability scale.

We use a small example so that the simulation can be run while reading the vignette; the live survival_adapt() chunk takes around 10–20 seconds and is cached on first knit.

When piecewise hazards help

Survival in many settings – post-surgical mortality, transplant outcomes, oncology with early treatment toxicity – shows a higher hazard early and a lower steady-state hazard later. A single exponential rate cannot capture both regimes; using the average rate over-states early survival and under-states late survival. A piecewise-exponential model with a single internal cut-point is often a good compromise, fitting one hazard for the early interval and another for everything afterwards.

The package handles piecewise hazards through two related arguments:

cutpoints: the times at which the hazard changes. The first element must be 0 and represents the start of the first interval; subsequent elements are the internal cut-points. With J intervals there are J elements in cutpoints and J hazard pieces per arm. (So cutpoints = c(0, 6) produces two intervals, [0, 6) and [6, \infty), in contrast to the default cutpoints = 0 used in the “Two-armed RCT” vignette, which produces a single interval and a non-piecewise exponential.)
hazard_treatment and hazard_control: vectors of length J giving the constant hazard for each interval.

The helper prop_to_haz() maps cumulative event probabilities at given times into the corresponding piecewise hazards.

Setting up the design

Suppose the primary endpoint is overall survival at 24 months. Based on prior data, we expect the control arm to have:

30% mortality by 6 months (an “early high-risk” window), and
50% mortality by 24 months overall.

The treatment is hypothesised to attenuate the early-period hazard, leaving the late-period hazard unchanged. Concretely, we target:

18% mortality by 6 months in the treatment arm, and
40% mortality by 24 months overall.

We set the cut-point at 6 months and translate these proportions into piecewise hazards:

cutpoints <- c(0, 6)         # one internal cut at 6 months -> two intervals
end_of_study <- 24

hc <- prop_to_haz(probs = c(0.30, 0.50), cutpoints = cutpoints, endtime = end_of_study)
ht <- prop_to_haz(probs = c(0.18, 0.40), cutpoints = cutpoints, endtime = end_of_study)

round(rbind(control = hc, treatment = ht), 4)
#>             [,1]   [,2]
#> control   0.0594 0.0187
#> treatment 0.0331 0.0174

The first column is the hazard during [0, 6) months and the second column is the hazard from 6 months onward. Both arms have a higher hazard in the early window. We can sanity-check that the implied survival probabilities at 24 months match what we specified by running the cumulative incidence computation back through ppwe():

ppwe(hazard       = matrix(hc, nrow = 1),
     cutpoints    = cutpoints,
     end_of_study = end_of_study)
#> [1] 0.5
ppwe(hazard       = matrix(ht, nrow = 1),
     cutpoints    = cutpoints,
     end_of_study = end_of_study)
#> [1] 0.4

These should be 0.50 and 0.40 respectively (modulo rounding).

The Bayesian decision rule

With method = "bayes", survival_adapt() puts independent Gamma $(\alpha, \beta)$ priors on each piecewise hazard rate (one per interval, per arm) and combines them with the observed exposure time and event counts to obtain a closed-form Gamma posterior on each $\lambda_j$ . Posterior draws of $\lambda_j$ are pushed through the piecewise-exponential cumulative incidence function to obtain posterior draws of the cumulative-failure probability $p$ at end_of_study for each arm.

The decision rule is one-sided. The treatment effect is defined as

$\Delta = p_{\text{treatment}} - p_{\text{control}},$

i.e. the difference in failure (not survival) probabilities at end_of_study. On the survival scale this is equivalent to $S_{\text{control}} - S_{\text{treatment}}$ , with a negative $\Delta$ corresponding to the treatment having higher survival. With alternative = "less" and a margin h0 (default 0), the trial declares success at the final analysis when

$\Pr(\Delta < h_0 \mid \text{data}) \;>\; \texttt{prob\_ha}.$

Because a beneficial treatment has a lower failure probability, alternative = "less" is the appropriate choice here. (method = "bayes" does not allow alternative = "two.sided" – it raises an error.)

The same posterior is also used at each interim look to compute the predictive probability of eventual success. Imputed completions are drawn from the posterior predictive distribution of the piecewise-exponential model for subjects still under follow-up, and the analysis is repeated on each imputed dataset. The fraction of imputations that would declare success at the maximum sample size is the predictive probability of success, which drives the futility and expected-success stopping rules through Fn and Sn.

We use a weakly informative Gamma $(0.1, 0.1)$ prior on every hazard component:

prior <- c(0.1, 0.1)   # shape and rate of the Gamma prior on each lambda_j

A single simulated trial

We will run one trial under the alternative hypothesis to illustrate the mechanics. We choose interim_look = 60 (the minimum required is max(block) = 4) and a constant accrual rate of 5 enrolments per month (lambda_time = 0 keeps the rate constant; piecewise accrual is also supported).

out <- survival_adapt(
  hazard_treatment = ht,
  hazard_control   = hc,
  cutpoints        = cutpoints,
  N_total          = 100,
  lambda           = 5,                # enrolments per month
  lambda_time      = 0,                # constant accrual rate
  interim_look     = 60,
  end_of_study     = end_of_study,
  prior            = prior,
  block            = 4,
  rand_ratio       = c(1, 1),
  prop_loss        = 0.05,
  alternative      = "less",
  h0               = 0,
  Fn               = 0.05,
  Sn               = 0.95,
  prob_ha          = 0.975,
  N_impute         = 50,
  N_mcmc           = 2000,
  method           = "bayes")

out
#>   prob_threshold margin alternative N_treatment N_control N_enrolled N_max
#> 1          0.975      0        less          50        50        100   100
#>   post_prob_ha   est_final ppp_success stop_futility stop_expected_success
#> 1        0.556 -0.01536969         0.1             0                     0

The output reports the posterior probability of the alternative at the final (or stopped) analysis (post_prob_ha), the posterior mean treatment effect on the cumulative-failure scale (est_final), the predictive probability of success (ppp_success), and indicators for whether the trial stopped early for futility or expected success.

Notes on the piecewise model

Two practical considerations are worth flagging:

Empty intervals at interim looks. Early interim looks may have no subjects with follow-up reaching the later piecewise intervals. The package handles this by propagating exposure time and event counts from the nearest non-empty interval within the same arm, and emits a warning when it does so. This supplies a placeholder Gamma posterior for an interval with no information; it is a fallback when an interim look has not yet generated follow-up in later intervals, not a substantive estimate of those intervals’ hazards. By the final analysis, all intervals will typically be populated.
Number of cut-points. Each additional cut-point adds two hazard parameters to estimate (one per arm). With limited interim data this can make individual interval posteriors diffuse. In our experience, one or two well-motivated cut-points (e.g. tied to a clinical milestone) is usually sufficient; finer partitions tend to add variance without commensurate bias reduction.

Sensitivity to the cut-point specification

If you suspect a piecewise structure but are unsure where the cut-point should sit, a useful sensitivity check is to fix the data-generating hazards (the truth) and vary the analysis cut-points – that is, what survival_adapt() is told to model. The simulator generates trial data from hazard_treatment and hazard_control evaluated against the supplied cutpoints, so to compare analysis-model choices on a common ground you would run separate sim_trials() simulations under the same data-generating truth, varying the analytic cut-points each time. If the operating characteristics are similar across choices, the design is robust to the cut-point specification. If they differ markedly, the cut-point becomes a design decision worth justifying in the protocol.

A simpler – but very different – comparison is the equivalent design that is both simulated and analysed under a single constant hazard per arm matched to the overall 24-month proportions:

hc_flat <- prop_to_haz(0.50, endtime = end_of_study)   # control, single hazard
ht_flat <- prop_to_haz(0.40, endtime = end_of_study)   # treatment, single hazard

out_flat <- survival_adapt(
  hazard_treatment = ht_flat,
  hazard_control   = hc_flat,
  cutpoints        = 0,
  N_total          = 100,
  lambda           = 5,
  lambda_time      = 0,
  interim_look     = 60,
  end_of_study     = end_of_study,
  prior            = prior,
  block            = 4,
  rand_ratio       = c(1, 1),
  prop_loss        = 0.05,
  alternative      = "less",
  h0               = 0,
  Fn               = 0.05,
  Sn               = 0.95,
  prob_ha          = 0.975,
  N_impute         = 50,
  N_mcmc           = 2000,
  method           = "bayes")

Note that this changes both the simulated data-generating process and the analysis model, so any difference in operating characteristics conflates the two effects. It is most useful when the question is “how would the trial behave if the world really were a single exponential?” rather than “how robust is my analysis cut-point?”.