# Conditional power and conditional error

## Introduction

We provide a simple plot of conditional power at the time of interim analysis. While group sequential boundaries should be designed to be the primary decision boundaries, conditional power evaluations can be useful supportive information. In addition to conditional power, we provide a predictive power estimate that averages conditional power based on a flat prior updated using an interim analysis result.

## Design

We consider the default design from `gsSurv()`, only altering the targeted hazard ratio to `hr = 0.7`, trial duration of `T = 36` and minimum follow-up of `minfup = 24`. This implies expected enrollment duration of 12 months at a constant rate. We note that the expected interim analysis timing leaves enough time for a data monitoring committee (DMC) to review and make recommendations to the trial sponsor between analyses. We use the `toInteger()` function to round sample size and event counts to appropriate integers resulting in a slightly larger power (90.05% versus the targeted 90%) and slightly altered interim timing (117/353 = 0.3314, 235/353 = 0.6657) compared to plan (0.3333, 0.6667).

    design <- gsSurv(hr = 0.7, lambdaC = log(2) / 12, minfup = 24, T = 36) |> toInteger()
    design |> gsBoundSummary()
    #>     Analysis              Value Efficacy Futility
    #>    IA 1: 33%                  Z   3.0092  -0.2322
    #>       N: 466        p (1-sided)   0.0013   0.5918
    #>  Events: 118       ~HR at bound   0.5746   1.0437
    #>    Month: 12   P(Cross) if HR=1   0.0013   0.4082
    #>              P(Cross) if HR=0.7   0.1427   0.0149
    #>    IA 2: 67%                  Z   2.5484   0.9406
    #>       N: 466        p (1-sided)   0.0054   0.1735
    #>  Events: 235       ~HR at bound   0.7171   0.8845
    #>    Month: 21   P(Cross) if HR=1   0.0062   0.8346
    #>              P(Cross) if HR=0.7   0.5813   0.0436
    #>        Final                  Z   1.9991   1.9991
    #>       N: 466        p (1-sided)   0.0228   0.0228
    #>  Events: 353       ~HR at bound   0.8083   0.8083
    #>    Month: 36   P(Cross) if HR=1   0.0233   0.9767
    #>              P(Cross) if HR=0.7   0.9005   0.0995

We also provide a textual summary.

    cat(design |> summary())

Asymmetric two-sided group sequential design with non-binding futility bound, 3 analyses, time-to-event outcome with sample size 466 and 353 events required, 90 percent power, 2.5 percent (1-sided) Type I error to detect a hazard ratio of 0.7. Enrollment and total study durations are assumed to be 12 and 36 months, respectively. Efficacy bounds derived using a Hwang-Shih-DeCani spending function with gamma = -4. Futility bounds derived using a Hwang-Shih-DeCani spending function with gamma = -2.

## Update design at time of interim analysis

Assume when the first interim is performed, there are 125 instead of the planned 117 endpoints included in the analysis. We update the bounds as follows:

    update <- gsDesign(
      k = design$k,
      test.type = design$test.type,
      alpha = design$alpha,
      beta = design$beta,
      sfu = design$upper$sf,
      sfupar = design$upper$param,
      sfl = design$lower$sf,
      sflpar = design$lower$param,
      n.I = c(117, design$n.I[2:3]),
      maxn.IPlan = design$n.I[design$k],
      delta = design$delta,
      delta1 = design$delta1,
      delta0 = design$delta0
    )
    gsBoundSummary(
      update,
      deltaname = "HR",
      logdelta = TRUE,
      Nname = "Events",
      digits = 4,
      ddigits = 2,
      tdigits = 1,
      exclude = c(
        "B-value", "CP", "CP H1", "PP",
        paste0("P(Cross) if HR=", round(c(design$hr0, design$hr), digits = 2))
      )
    )
    #> Warning: gsBoundSummary: hr0 is not present; using hr0 =
    #> 1 for HR at bound calculations.
    #>     Analysis        Value Efficacy Futility
    #>    IA 1: 33%            Z   3.0139  -0.2451
    #>  Events: 117  p (1-sided)   0.0013   0.5968
    #>              ~HR at bound   0.5728   1.0464
    #>                  Spending   0.0013   0.0147
    #>    IA 2: 67%            Z   2.5478   0.9413
    #>  Events: 235  p (1-sided)   0.0054   0.1733
    #>              ~HR at bound   0.7172   0.8844
    #>                  Spending   0.0049   0.0289
    #>        Final            Z   1.9991   1.9991
    #>  Events: 353  p (1-sided)   0.0228   0.0228
    #>              ~HR at bound   0.8083   0.8083
    #>                  Spending   0.0188   0.0564

## Testing and conditional power

We assume an interim p-value of 0.04, one-sided. This does not come close to the first efficacy or futility bound above. However, it is a trend in the right direction.

    # Nominal 1-sided p-value
    p <- 0.04

This translates to a first order approximation of the Cox regression estimate with the Schoenfeld (1981) approximation:

    zn2hr(-qnorm(p), n = update$n.I[1])
    #> [1] 0.7234658

Typically, conditional power is reported based on 3 different assumptions about the future treatment effect:

1. The observed HR; here we base this on the above approximation.
2. No treatment effect; this translates to conditional error.
3. The originally targeted treatment effect in the design.

These are displayed below, translated to the hazard ratio scale:

    cp <- gsCP(x = update, i = 1, zi = -qnorm(p))
    # 3 treatment effects as outlined above
    # design$ratio is the experimental:control randomization ratio
    exp(-cp$theta * sqrt((1 + design$ratio)^2 / design$ratio))
    #> [1] 0.7234658 1.0000000 0.6995487

Now we display the probability of crossing an efficacy boundary before a futility boundary conditional on the above observed p-value at the first interim analysis from the above call to `gsCP()`. The columns of the resulting matrix correspond to the above treatment effects. The rows correspond to the second interim and final analyses. Adding the numbers in the first column, we get a conditional power of 0.899 for the treatment effect observed at the first interim. The third column yields a conditional power of 0.937 for the originally targeted hazard ratio of 0.7. Finally, the second column yields a conditional error under the assumption of a future hazard ratio of 1 (no underlying treatment effect) of 0.121. This could be used with the conditional error method of Müller and Schäfer (2004) to adapt the design for endpoints other than the time-to-event example used here (e.g., a binary outcome).

    cp$upper$prob
    #>           [,1]       [,2]      [,3]
    #> [1,] 0.4625266 0.03199697 0.5352688
    #> [2,] 0.4364291 0.08886339 0.4017165

We demonstrate a conditional power plot that may be of some use. We will assume a wide range of potential underlying hazard ratios for future events.

    hr <- seq(.6, 1.1, .01)

We compute conditional probabilities based on the observed interim 1 p-value over this range:

    # Translate hazard ratio to standardized effect size
    theta <- -log(hr) * sqrt(design$ratio / (1 + design$ratio)^2)
    cp <- gsCP(x = update, i = 1, zi = -qnorm(p), theta = theta)

Finally, we plot conditional power as a function of future HR. Note the use of the power plot option for the gsDesign plot function. The `offset = 1` argument changes the legend to be labeled with “Future Analysis” 2 and 3. The solid black line shows the conditional probability of crossing any future bound prior to crossing a lower bound. In general, black lines on this plot will show the cumulative conditional probability of crossing an efficacy bound prior to crossing a lower bound by the time of any given future analysis by different underlying treatment effect assumptions. The red lines show 1 minus the cumulative conditional probability of crossing a lower bound prior to crossing an efficacy bound by any given future analysis by different underlying treatment effect (HR) assumptions.

    plot(cp, xval = hr, xlab = "Future HR", ylab = "Conditional Power/Error",
         main="Conditional probability of crossing future bound", offset = 1)

## Predictive power

Assuming a flat prior distribution and a constant treatment underlying treatment effect throughout the trial, we can compute a Bayesian predictive power conditioning on the interim \(p\)-value above. This takes into account the uncertainty of the underlying treatment effect, updating the flat prior based on the interim result and using the resulting posterior distribution to average across total conditional probabilities from the conditional power plot above. This provides a single number to summarize the conditional probability of success given the interim result.

    # set up a flat prior distribution for the treatment effect
    # that is normal with mean .5 of the design standardized effect and
    # a large standard deviation.
    mu0 <- .5 * design$delta
    sigma0 <- design$delta * 2
    prior <- normalGrid(mu = mu0, sigma = sigma0)
    gsPP(x = update, i = 1, zi = -qnorm(p), theta = prior$z, wgts = prior$wgts)
    #> [1] 0.7631346

While the conditional power based on the observed effect was essentially 90%, we now have an estimate that effectively shrinks towards a lesser effect based on the uncertainly of the observed treatment effect. The predictive probability is similar whether `mu0` above is `0` or `design$delta`.

## References

Müller, Hans-Helge, and Helmut Schäfer. 2004. “A General Statistical Principle for Changing a Design Any Time During the Course of a Trial.” Statistics in Medicine 23 (16): 2497–2508.
Schoenfeld, David. 1981. “The Asymptotic Properties of Nonparametric Tests for Comparing Survival Distributions.” Biometrika 68 (1): 316–19.