Calculate Stage Information for an Adaptive Design

Calculate the additional information required after an interim analysis, or its unconditional expectation, for an optimal conditional error design.

Usage

getStageInformation(
  design,
  stage = 2,
  type = c("conditional", "expected"),
  pValue = NULL,
  likelihoodRatioDistribution = NULL,
  ...
)

Arguments

design

An object of class TrialDesignOptimalConditionalError created by getDesignOptimalConditionalError(). Contains all necessary arguments to calculate the optimal conditional error function for the specified case.

stage

Target stage for which additional information is required. Currently only 2 is supported; the interim p-value comes from stage one.

type

"conditional" (default) for information given pValue, or "expected" for the unconditional expected additional information.

pValue

First-stage p-value or p-values. Must be a numeric vector between 0 and 1.

likelihoodRatioDistribution

The distribution to be used for the effect size of the likelihood ratio in the calculation of the expected second-stage information. Options are "fixed", "normal", "exp", "unif" for fixed effect size, normally distributed, exponentially distributed, and uniformly distributed prior of the effect size, respectively. Each case requires different additional specifications:

likelihoodRatioDistribution="fixed" uses one (or more) fixed effect sizes for the likelihood ratio and requires the parameter thetaLR which provides the mean difference under which to calculate the likelihood ratio. If thetaLR contains multiple values, they may be weighted using an additional argument weightsLR. Omitting weightsLR automatically leads to equal weighting.
likelihoodRatioDistribution="normal" uses a normal prior for the effect size and requires parameters thetaLR and stDevLR for the mean and standard deviation of the normal distribution (both on mean difference scale).
likelihoodRatioDistribution="exp" uses an exponential prior for the effect size and requires the parameter kappaLR which specifies the rate on the non-centrality scale divided by sqrt(firstStageInformation). The mean effect on the mean difference scale is 1 / (kappaLR * firstStageInformation).
likelihoodRatioDistribution="unif" uses a uniform prior for the effect size and requires the specification of maxThetaLR, which is the maximum of the support for the uniform likelihood ratio distribution (on the mean difference scale).

The default is likelihoodRatioDistribution=NULL. In this case, the likelihood ratio distribution under which the expected second-stage information is calculated is taken directly from the design object.

...

Distribution parameters for type = "expected": thetaLR, weightsLR, stDevLR, kappaLR, and maxThetaLR, as applicable.

Value

For type = "conditional", a numeric vector with one information value per p-value, including zero after early stopping. For type = "expected", a numeric scalar including zero information for trials stopped at stage one. Neither result includes first-stage information. Designs using "maxlr" require an explicit probability distribution for the expectation.

Conditional information

The second-stage information $I_{2}$ is calculated given a first-stage p-value $p_1$ as: $$I_{2}(p_1) = \frac{(\Phi^{-1}(1-\alpha_2(p_1)) + \Phi^{-1}(CP))^2}{\Delta_1^2} = \frac{\nu(\alpha_2(p_1))}{\Delta_1^2},$$ where

$\alpha_2(p_1)$ is the conditional error function
$CP$ is the target conditional power
$\Delta_1$ is the assumed treatment effect (expressed as a mean difference).

The conditional error is calculated according to the specification provided in the design argument. For p-values smaller or equal to the first-stage efficacy boundary as well as p-values greater than the first-stage futility boundary, the returned information is 0 (since the trial is ended early in both cases). When efficacyBounds = 0, early efficacy stopping is disabled, including at a p-value of zero.

Expected information

The expected second-stage information is calculated as: $$\mathbb{E}(I_{2})=\int_{\alpha_1}^{\alpha_0}\frac{\nu(\alpha_2(p_1)) \cdot l(p_1)}{\Delta_1^2} dp_1,$$ where

$\alpha_1, \alpha_0$ are the first-stage efficacy and futility boundaries
$\alpha_2(p_1)$ is the optimal conditional error calculated for $p_1$
$l(p_1)$ is the "true" likelihood ratio under which to calculate the expected sample size. This can be different from the likelihood ratio used to calibrate the optimal conditional error function.
$\Delta_1$ is the assumed treatment effect to power for, expressed as a mean difference. It may depend on the interim data (i.e., $p_1$) in case useInterimEstimate = TRUE was specified for the design object.
$\nu(\alpha_2(p_1)) = (\Phi^{-1}(1-\alpha_2(p_1))+\Phi^{-1}(CP))^2$ is a factor calculated for the specific assumptions about the optimal conditional error function and the target conditional power $CP$.

Add design$firstStageInformation to obtain expected total information. This expectation is unconditional, not conditional on reaching stage two. Changing the evaluation distribution here does not recalibrate the design.

References

Brannath, W. & Bauer, P. (2004). Optimal conditional error functions for the control of conditional power. Biometrics, 60(3), 715-723. doi:10.1111/j.0006-341X.2004.00221.x

Brannath, W., zur Verth, J., Dreher, M. & Scharpenberg, M. (2024; revised 2026). Optimal monotone conditional error functions. doi:10.48550/arXiv.2402.00814

The related R package optconerrf: Optimal Monotone Conditional Error Functions provides a standalone implementation of this methodology. The rpact implementation uses rpact design objects and interfaces; optconerrf is not required to run these functions.

Examples

# Get a design
design <- getDesignOptimalConditionalError(
    alpha = 0.025, efficacyBounds = 0.001, futilityBounds = 0.5, 
    conditionalPower = 0.9, thetaH1 = 0.25, 
    likelihoodRatioDistribution = "fixed", thetaLR = 0.25,
    firstStageInformation = 80, useInterimEstimate = FALSE
)
# Calculate expected information under correct specification
getStageInformation(type = "expected", design = design)
#> [1] 59.45876

# Compare operating characteristics under a different true effect.
getStageInformation(type = "expected", design = design, 
    likelihoodRatioDistribution = "fixed", thetaLR = 0.15)
#> [1] 108.4258

# Calculate expected information under the null hypothesis
getStageInformation(type = "expected",
    design = design, likelihoodRatioDistribution = "fixed", thetaLR = 0
)
#> [1] 94.61795

# Required additional information given the interim results
getStageInformation(design, pValue = c(0.05, 0.1, 0.3))
#> [1] 101.8229 140.8483 217.6371