Futility spending calibrated to assurance and probability of success

What is conditioned on?

Let \(C_i\) mean that the trial has crossed neither stopping boundary through interim \(i\), and let \(E_{>i}\) mean a later efficacy rejection. Conditional assurance is

\[ \Pr(E_{>i}\mid C_i) = \frac{\Pr(E_{>i}\cap C_i)}{\Pr(C_i)}. \]

Both probabilities average over the prespecified effect prior. Conditioning on continuation implicitly changes the relative plausibility of the possible effects. This is the quantity calculated by gsCPOS(), including the full earlier stopping history.

In contrast, gsPOS() is the probability of any efficacy rejection before seeing data; it includes early efficacy. gsPP() conditions on a particular interim statistic and averages future success over its posterior effect distribution. Thus conditional assurance is not PP at the futility bound. A high assurance among trials allowed to continue does not say that a trial just at the boundary has high predictive power.

Without early efficacy, conditional assurance is POS divided by the prior probability of continuation. With early efficacy, its numerator must exclude those earlier successes. A point prior changes the averaging, but not this distinction between a continuation event and an exact observed statistic.

A Dragalin-related fixed information example

Dragalin’s Informative Futility Rules Based on Conditional Assurance motivates comparing assurance among continuing trials with the assurance of a larger, feasible fixed design. Here we use the paper’s normal outcome setting: interim sample size 72, maximum sample size 144, equal allocation, common standard deviation 70, and an effect prior \(N(33,16^2)\). These are total sample sizes across both treatment groups.

For a difference of normal means with equal allocation, the standardized effect is the mean difference divided by \(2\,SD=140\). We use that scale throughout. This is the paper’s informative prior, not the broader default prior in gsBoundSummary(). We supply it explicitly to every prior-based calculation.

effect_sd <- 70
effect_h1 <- 33
prior <- normalGrid(mu = effect_h1 / (2 * effect_sd),
                    sigma = 16 / (2 * effect_sd))
prior$wgts <- prior$wgts / sum(prior$wgts)

reference <- gsDesign(
  k = 2, test.type = 4, delta = effect_h1 / (2 * effect_sd),
  n.I = c(72, 144), maxn.IPlan = 144,
  testUpper = c(FALSE, TRUE), sfl = sfHSD, sflpar = 0, r = 32
)

The benchmark has a single final analysis at \(N=230\) and one-sided alpha 0.025. Since gsDesign() constructs sequential designs, we specify the one-look boundaries directly for probability evaluation. Both terminal bounds are the same: every trial has a final decision.

fixed_benchmark <- structure(
  list(k = 1L, test.type = 4L, n.I = 230,
       upper = list(bound = qnorm(.975)),
       lower = list(bound = qnorm(.975)), r = 32L, overrun = 0),
  class = "gsDesign"
)
benchmark <- gsPOS(fixed_benchmark, theta = prior$z, wgts = prior$wgts)
benchmark
#> [1] 0.7901662

Compute this benchmark once, not again from each candidate design.

fit_dragalin <- gsCPOSFutilitySpending(
  reference, target_cpos = benchmark, i = 1, prior = prior,
  mode = "fixed_information"
)
dragalin_results <- c(
  Target_assurance = benchmark,
  Achieved_assurance = gsCPOS(1, fit_dragalin, prior$z, prior$wgts),
  Interim_N = fit_dragalin$n.I[1],
  Final_N = fit_dragalin$n.I[2],
  Effect_at_futility = fit_dragalin$lower$bound[1] *
    2 * effect_sd / sqrt(fit_dragalin$n.I[1]),
  Overall_power_H1 = sum(fit_dragalin$upper$prob[, 2]),
  Fitted_beta = fit_dragalin$beta,
  Fitted_sflpar = fit_dragalin$cposFutilitySpending$sflpar
)
data.frame(Quantity = names(dragalin_results), Value = unname(dragalin_results)) |>
  knitr::kable(digits = 4)

The target is approximately 0.7902. Solving equality gives an effect threshold of approximately 7.274, rather than the illustrative threshold of 10 in the paper (which exceeds the benchmark). The sample sizes remain 72 and 144; the achieved overall power at effect 33 is about 79.3%.

This is a spending-based implementation of the fixed information assurance criterion. It is not a reproduction of an entire sample size re-estimation procedure or an inverse normal combination test.

Compare all three methods on the same reference

For a broader comparison, retain the same effect prior, SD and planned effect, but use a reference with 85% power, information fractions 0.4 and 0.75, and one active futility analysis at the first interim. There are efficacy analyses at all three looks. A single early futility analysis is often sufficient; a one-parameter spending function can also be adequate with additional futility looks after inspection of every bound.

n_fixed <- nNormal(delta1 = effect_h1, sd = effect_sd, beta = .15)
x <- gsDesign(
  k = 3, test.type = 4, n.fix = n_fixed, beta = .15,
  timing = c(.4, .75), delta1 = effect_h1,
  sfu = sfLDOF, sfl = sfHSD, sflpar = -2,
  testLower = c(TRUE, FALSE, FALSE)
)
data.frame(Fixed_design_N = n_fixed, Sequential_reference_N = tail(x$n.I, 1)) |>
  knitr::kable(digits = 2)

fit_cpos <- gsCPOSFutilitySpending(x, target_cpos = .80, i = 1, prior = prior)
fit_ca <- gsCPOSFutilitySpending(x, target_cpos = .80, i = 1, prior = prior,
                               mode = "fixed_information")

stopifnot(identical(fit_ca$n.I, x$n.I),
          identical(fit_ca$upper$bound, x$upper$bound))

fit_pos <- gsPOSFutilitySpending(
  x, target_pos = .7206, prior = prior,
  control = list(start = 0, lower = -2, upper = 2, pos_tol = 1e-7)
)

verification <- data.frame(
  Method = c("Power-preserving CA", "Fixed information CA", "Unconditional POS"),
  Target = c(.80, .80, .7206),
  Achieved = c(gsCPOS(1, fit_cpos, prior$z, prior$wgts),
               gsCPOS(1, fit_ca, prior$z, prior$wgts),
               gsPOS(fit_pos, prior$z, prior$wgts))
)
knitr::kable(verification, digits = 6)

stopifnot(max(abs(verification$Achieved - verification$Target)) < 1e-4)

# Verify that total beta and the fitted lower spending agree.
desired_beta <- fit_ca$lower$sf(
  fit_ca$beta, fit_ca$lower$sTime, fit_ca$lower$param
)$spend
# At the inactive second futility look, spending is deferred.
desired_beta[2] <- desired_beta[1]
stopifnot(max(abs(cumsum(fit_ca$lower$prob[, 2]) - desired_beta)) < 1e-5)

replay <- gsDesign(
  k = 3, test.type = 4, n.fix = n_fixed, beta = .15,
  timing = c(.4, .75), delta1 = effect_h1,
  sfu = sfLDOF, sfl = sfHSD,
  sflpar = fit_cpos$cposFutilitySpending$sflpar,
  testLower = c(TRUE, FALSE, FALSE)
)
gsCPOS(1, replay, prior$z, prior$wgts)
#> [1] 0.8

Operating characteristics, not just a target

designs <- list(Reference = x, "Power-preserving CA" = fit_cpos,
                "Fixed information CA" = fit_ca, "Unconditional POS" = fit_pos)
operating <- do.call(rbind, lapply(names(designs), function(label) {
  d <- designs[[label]]
  p <- gsProbability(d = d, theta = c(0, x$delta))
  data.frame(
    Method = label, Max_N = tail(d$n.I, 1),
    Inflation_percent = 100 * (tail(d$n.I, 1) / n_fixed - 1),
    Actual_type_I = sum(p$upper$prob[, 1]),
    Power_H1 = sum(p$upper$prob[, 2]),
    POS = gsPOS(d, prior$z, prior$wgts),
    CA_at_IA1 = gsCPOS(1, d, prior$z, prior$wgts),
    Futility_H0 = sum(p$lower$prob[-d$k, 1]),
    Futility_H1 = sum(p$lower$prob[-d$k, 2]),
    EN_H0 = p$en[1], EN_H1 = p$en[2]
  )
}))
knitr::kable(operating[c("Method", "Max_N", "Inflation_percent",
                        "Actual_type_I", "Power_H1")], digits = 4)

knitr::kable(operating[c("Method", "POS", "CA_at_IA1",
                        "Futility_H0", "Futility_H1")], digits = 4)

knitr::kable(operating[c("Method", "EN_H0", "EN_H1")], digits = 2)

The fixed design requires about 161.6 participants. Sample sizes are continuous planning values; rounding requires another check of the resulting operating characteristics and target probabilities. The inflation column uses that fixed design, not the sequential reference, as its denominator. Much of the inflation in these examples is the cost of preserving power despite futility stopping. A practical maximum, such as 20% inflation, should be considered alongside the probability target; none of these functions automatically enforces such a cap.

For nonbinding futility, the nominal efficacy-only alpha remains 0.025, while actual type I error with futility enforced may be smaller. With a binding reference (test.type = 3), freezing efficacy while changing futility can increase actual type I error above the reference alpha. The fixed information method reports that probability; it does not guarantee alpha preservation.

effect <- seq(-10, 66, length.out = 101)
curves <- do.call(rbind, lapply(names(designs), function(label) {
  p <- gsProbability(d = designs[[label]], theta = effect / (2 * effect_sd))
  data.frame(Effect = effect, Power = colSums(p$upper$prob), Method = label)
}))
ggplot(curves, aes(Effect, Power, colour = Method, linetype = Method)) +
  geom_line(linewidth = .8) +
  geom_vline(xintercept = effect_h1, colour = "grey60", linetype = 3) +
  labs(x = "True difference in means", y = "Probability of efficacy rejection") +
  theme_minimal() + theme(legend.position = "bottom") +
  guides(colour = guide_legend(nrow = 2), linetype = guide_legend(nrow = 2))

bounds <- do.call(rbind, lapply(names(designs), function(label) {
  d <- designs[[label]]
  lower_effect <- d$lower$bound * 2 * effect_sd / sqrt(d$n.I)
  lower_effect[!d$testLower] <- NA_real_
  data.frame(Method = label, Analysis = seq_len(d$k), N = d$n.I,
             Efficacy_effect = d$upper$bound * 2 * effect_sd / sqrt(d$n.I),
             Futility_effect = lower_effect)
}))
knitr::kable(bounds, digits = 3)

gsBoundSummary(fit_ca, prior = prior, exclude = "B-value", digits = 4)
#>   Analysis                Value Efficacy Futility
#>  IA 1: 40%                    Z   3.3569   0.5803
#>      N: 68          p (1-sided)   0.0004   0.2808
#>                 ~delta at bound  57.1812   9.8856
#>                        Spending   0.0004   0.0874
#>                              CP   1.0000   0.0811
#>                           CP H1   0.9964   0.6017
#>                              PP   0.9966   0.3454
#>             P(Cross) if delta=0   0.0004   0.7192
#>            P(Cross) if delta=33   0.0779   0.0874
#>  IA 2: 75%                    Z   2.3449       NA
#>     N: 127          p (1-sided)   0.0095       NA
#>                 ~delta at bound  29.1705       NA
#>                        Spending   0.0093       NA
#>                              CP   0.9178       NA
#>                           CP H1   0.9416       NA
#>                              PP   0.9076       NA
#>             P(Cross) if delta=0   0.0094       NA
#>            P(Cross) if delta=33   0.6145       NA
#>      Final                    Z   2.0125       NA
#>     N: 169          p (1-sided)   0.0221       NA
#>                 ~delta at bound  21.6812       NA
#>                        Spending   0.0154       NA
#>             P(Cross) if delta=0   0.0222       NA
#>            P(Cross) if delta=33   0.8192       NA

The final specification is still a spending function

times <- seq(0, 1, length.out = 301)
spending <- do.call(rbind, lapply(names(designs), function(label) {
  d <- designs[[label]]
  data.frame(Time = times, Spend = d$lower$sf(d$beta, times, d$lower$param)$spend,
             Method = label)
}))
ggplot(spending, aes(Time, Spend, colour = Method, linetype = Method)) +
  geom_line(linewidth = .8) +
  geom_vline(xintercept = c(.4, .75), colour = "grey70", linetype = 3) +
  labs(x = "Information fraction", y = "Cumulative beta spending") +
  theme_minimal() + theme(legend.position = "bottom") +
  guides(colour = guide_legend(nrow = 2), linetype = guide_legend(nrow = 2))

These curves need not cross at the calibration time: the target is conditional assurance or POS, not cumulative beta spending. The fixed information design can also have a different total beta. The curves describe the spending family; actual spending is deferred at inactive futility analyses.

Spending functions retain the original motivation of the Lan–DeMets approach: the specification can accommodate different information times and, under an appropriate monitoring plan, a different number of analyses. They provide a continuous rule between planned looks, a compact reproducible specification, and a framework for recalculating boundaries while respecting spending already used. This flexibility is particularly useful when actual information accrues differently from the original plan.

It does not mean that one should refit parameters freely after seeing treatment effects, or that changing the monitoring schedule preserves the original assurance target or power exactly. Re-evaluate the resulting design and its operating characteristics. Timing changes informed by unblinded outcomes need additional justification.

Building on established computational tools with AI

The calibration framework builds on existing gsDesign tools rather than new probability integration machinery. Most of the initial CP/PP development was done interactively with AI using GPT-6 Astra High in Codex over a long weekend. The later assurance extensions follow the same principle: rapidly build new design interfaces on established computational tools.

Validation need not rest solely on line-by-line inspection of generated code. Reconstruct fitted designs and check their properties with gsDesign(), gsProbability(), gsCPOS(), gsPOS() and gsBoundSummary(). Unit tests also check independent bivariate normal integration, the numerical Dragalin benchmark, parameter replay, input errors and numerical failure handling. These checks support the statistical results; they do not establish that every possible software defect has been excluded.

Scope and practical limits

• Assurance/POS calibration supports gsDesign and direct survival objects with test types 3 and 4. Fixed information calibration retains the survival plan and efficacy bounds; power-preserving calibration recalculates enrollment.
• Harm bound designs are excluded: current gsCPOS() does not include harm stopping in its continuation denominator.
• Conditional assurance fits support the existing one-/two-parameter families and piecewise linear spending, with one target per free parameter.
• POS has only one design-wide target and fits one free parameter.
• Prior weights are normalized, and the prior is fixed throughout calibration. Continuation probabilities at or below sqrt(.Machine$double.eps) are rejected for conditional assurance.
• A failed numerical search is not a proof of mathematical infeasibility. A matched target is not proof that the design is clinically attractive.
• Review overall power, actual type I error, expected and maximum sample size, stopping probabilities, effect sizes at bounds, and sensitivity to the prior and monitoring schedule before choosing a design.

References

Dragalin V. Informative Futility Rules Based on Conditional Assurance. Statistics in Medicine (2026). doi:10.1002/sim.70330.

Lan KKG, DeMets DL. Discrete sequential boundaries for clinical trials. Biometrika (1983), 70:659–663. doi:10.1093/biomet/70.3.659.