Create an Optimal Conditional Error Design

This function returns a design object which contains all important parameters for the specification of the optimal conditional error function. The returned object is of class TrialDesignOptimalConditionalError and can be passed to other package functions.

Usage

getDesignOptimalConditionalError(
  alpha,
  efficacyBounds,
  futilityBounds,
  conditionalPower = NA_real_,
  thetaH1 = NA_real_,
  minThetaH1 = NA_real_,
  maxThetaH1 = Inf,
  useInterimEstimate = TRUE,
  firstStageInformation,
  likelihoodRatioDistribution,
  minInformationPerStage = 0,
  maxInformationPerStage = Inf,
  minConditionalError = 0,
  maxConditionalError = 1,
  conditionalPowerFunction = NULL,
  minLevelConstant = 0,
  maxLevelConstant = 10,
  enforceMonotonicity = TRUE,
  efficacyBoundsScale = "pValue",
  futilityBoundsScale = "pValue",
  nonCentralityParameterH1 = NULL,
  minNonCentralityParameterH1 = NULL,
  maxNonCentralityParameterH1 = Inf,
  ...
)

Arguments

alpha

The overall type I error rate \(\alpha\) of the design. Must be a numeric value between 0 and 1.

efficacyBounds

Stage 1 efficacy boundary \(\alpha_1\) (p-value scale). Must be a numeric value between 0 and 1. Should be smaller than futilityBounds.

futilityBounds

Binding stage 1 futility boundary \(\alpha_0\) (p-value scale). Must be a numeric value between 0 and 1. Should be greater than efficacyBounds. Specify futilityBounds=1 to disable early stopping for futility.

conditionalPower

The target conditional power \(CP\) of the design. Must be a numeric scalar strictly between 0 and 1. Takes precedence over conditionalPowerFunction.

thetaH1

Fixed effect assumption at which the conditional power should be achieved, expressed on the mean difference scale. Is only used if useInterimEstimate=FALSE. Must be a numeric value greater than 0.

minThetaH1

The minimum for an interim estimate of the treatment effect, specified on the mean difference scale. If the interim estimate (on the mean difference scale) yields a value smaller than minThetaH1, minThetaH1 is used for it. Is only used if useInterimEstimate=TRUE. Must be a numeric value.

maxThetaH1

The maximum for an interim estimate of the treatment effect, specified on the mean difference scale. If the interim estimate (on the mean difference scale) yields a value larger than maxThetaH1, maxThetaH1 is used for it. Is only used if useInterimEstimate=TRUE. Must be a numeric value. Default value is Inf, i.e., no upper restriction.

useInterimEstimate

Logical. Defines whether or not an interim estimate should be used for conditional power. If TRUE, a lower cut-off for the interim estimate must be specified by minThetaH1 or minNonCentralityParameterH1. An upper cut-off may also be specified by maxThetaH1 or maxNonCentralityParameterH1. If FALSE, the fixed effect size must be specified by thetaH1 or nonCentralityParameterH1.

firstStageInformation

Information of the first stage of the trial. Must be a positive numeric value.

likelihoodRatioDistribution

The distribution to be used for the effect size of the likelihood ratio in the optimal conditional error function. Options are "fixed", "normal", "exp", "unif", "maxlr" for fixed effect size, normally distributed, exponentially distributed, uniformly distributed prior of the effect size and maximum likelihood ratio, 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).

• likelihoodRatioDistribution="maxlr" estimates the non-centrality parameter to be used for the likelihood ratio from the data. No additional parameters must be specified.

minInformationPerStage

The minimum additional information allowed in stage two. A single numeric value; does not apply to firstStageInformation. Default value is 0, i.e., no restriction.

maxInformationPerStage

The maximum additional information allowed in stage two. A single numeric value; does not apply to firstStageInformation. Default value is Inf, i.e., no restriction.

minConditionalError

Lower boundary for the optimal conditional error function. Default 0 (no restriction).

maxConditionalError

Upper boundary for the optimal conditional error function. Default value is 1, however, the optimal conditional error function is inherently bounded by the conditional power.

conditionalPowerFunction

A function accepting a single first-stage p-value and returning a numeric scalar strictly between 0 and 1. The function is evaluated separately for each p-value. This function should not be increasing in the first-stage p-value, otherwise monotonicity issues may occur. With a conditional power function, the resulting conditional error function is not necessarily optimal for expected second-stage information.

minLevelConstant

The minimum of the interval on which the value for the level constant should be searched. Default value is 0.

maxLevelConstant

The maximum of the interval on which the value for the level constant should be searched. Default value is 10.

enforceMonotonicity

Logical. Determines whether or not the optimal conditional error function should automatically be modified to be non-increasing. Default is TRUE.

efficacyBoundsScale, futilityBoundsScale

Scale of the interim boundaries. Currently only "pValue" is supported. These are p-value cutoffs, not the test-statistic boundaries used by conventional group sequential designs.

nonCentralityParameterH1

Fixed conditional-power effect on the non-centrality scale; alternative to thetaH1 when useInterimEstimate = FALSE.

minNonCentralityParameterH1, maxNonCentralityParameterH1

Lower and upper limits for the interim effect on the non-centrality scale; alternatives to minThetaH1 and maxThetaH1 when useInterimEstimate = TRUE. Non-centrality equals the mean difference times sqrt(firstStageInformation). Mean-difference arguments take precedence when both scales are supplied. The default upper limit is Inf.

...

Distribution parameters thetaLR, weightsLR, stDevLR, kappaLR, and maxThetaLR, as described under Likelihood ratio distribution.

Value

An object of class TrialDesignOptimalConditionalError, which can be passed to getConditionalError(), getStageInformation(), and getDesignCharacteristics(). This adaptive design is not interchangeable with a conventional TrialDesign.

Details

The design object contains the information required to determine the specific setting of the optimal conditional error function and can be passed to other package functions. From the given user specifications, the constant to achieve level condition for control of the overall type I error rate as well as the constants to ensure a non-increasing optimal CEF (if required) are automatically calculated.

Hypotheses and trial decisions

The one-sided hypotheses are \(H_0: \Delta \leq 0\) versus \(H_1: \Delta > 0\). With a first-stage p-value \(p_1\), stop for efficacy if \(p_1 \leq \alpha_1\) and \(\alpha_1 > 0\); stop for futility if \(p_1 > \alpha_0\). Futility is binding: continuing beyond this boundary is not part of the calibrated design. In the continuation region, reject at stage two if its p-value is no larger than getConditionalError(). The second-stage p-value must be based on the new, independent second-stage data, not the cumulative data. Setting efficacyBounds = 0 disables early efficacy and futilityBounds = 1 disables early futility. Here, \(\alpha_1\) denotes efficacyBounds and \(\alpha_0\) denotes futilityBounds; both are single interim cutoffs on the p-value scale. The calibration satisfies $$\alpha = \alpha_1 + \int_{\alpha_1}^{\alpha_0} \alpha_2(p_1) \, dp_1.$$ Target conditional power refers to a specified planning effect after continuation; it is not the overall power returned by getDesignCharacteristics().

Likelihood ratio distribution

To calculate the optimal conditional error function, an assumption about the true parameter under which the second-stage information is to be minimised is required. Various options are available and can be specified via the argument likelihoodRatioDistribution:

• likelihoodRatioDistribution="fixed": calculates the likelihood ratio for a fixed \(\Delta\). The non-centrality parameter of the likelihood ratio \(\vartheta\) is then computed as thetaLR*sqrt(firstStageInformation) and the likelihood ratio is calculated as: $$l(p_1) = e^{\Phi^{-1}(1-p_1)\vartheta - \vartheta^2/2}.$$ thetaLR may also contain multiple elements, in which case a weighted likelihood ratio is calculated for the given values. Unless positive weights that sum to 1 are provided by the argument weightsLR, equal weights are assumed.

• likelihoodRatioDistribution="normal": calculates the likelihood ratio for a normally distributed prior of \(\vartheta\) with mean thetaLR*sqrt(firstStageInformation) (\(\mu\)) and standard deviation stDevLR*sqrt(firstStageInformation) (\(\sigma\)). The parameters thetaLR and stDevLR must be specified on the mean difference scale. $$l(p_1) = (1+\sigma^2)^{-\frac{1}{2}}\cdot e^{-(\mu/\sigma)^2/2 + (\sigma\Phi^{-1}(1-p_1) + \mu/\sigma)^2 / (2\cdot (1+\sigma^2))}$$

• likelihoodRatioDistribution="exp": calculates the likelihood ratio for an exponentially distributed prior of \(\vartheta\) with rate kappaLR*sqrt(firstStageInformation) (\(\eta\)). The likelihood ratio is then calculated as: $$l(p_1) = \eta \cdot \sqrt{2\pi} \cdot e^{(\Phi^{-1}(1-p_1)-\eta)^2/2} \cdot \Phi(\Phi^{-1}(1-p_1)-\eta)$$

• likelihoodRatioDistribution="unif": calculates the likelihood ratio for a uniformly distributed prior of \(\vartheta\) on the support \([0, \Delta\cdot\sqrt{I_1}]\), where \(\Delta\) is specified as maxThetaLR and \(I_1\) is the firstStageInformation. $$l(p_1) = \frac{\sqrt{2\pi}}{\Delta\cdot\sqrt{I_1}} \cdot e^{\Phi^{-1}(1-p_1)^2/2} \cdot (\Phi(\Delta\cdot\sqrt{I_1} - \Phi^{-1}(1-p_1))-p_1)$$

• likelihoodRatioDistribution="maxlr": the non-centrality parameter \(\vartheta\) is estimated from the data and no additional parameters must be specified. The likelihood ratio is estimated from the data as: $$l(p_1) = e^{max(0, \Phi^{-1}(1-p_1))^2/2}$$ The maximum likelihood ratio is always restricted to effect sizes \(\vartheta \geq 0\) (corresponding to \(p_1 \leq 0.5\)).

Effect for conditional power

For the treatment effect at which the target conditional power should be achieved, either a fixed effect or an interim estimate can be used. The planning effect thetaH1 need not equal thetaLR: the former sets the conditional power target, whereas the latter specifies the effect or mixture under which expected second-stage information is minimised. The usage of a fixed effect is indicated by setting useInterimEstimate=FALSE, in which case the fixed effect is provided by thetaH1 on the mean difference scale. For an interim estimate, specified by useInterimEstimate=TRUE, a lower cut-off for the interim estimate must be provided, by minThetaH1 on the mean difference scale. In addition, an upper limit of the estimate may be analogously provided by maxThetaH1. These effects may alternatively be specified on the non-centrality parameter scale as nonCentralityParameterH1, minNonCentralityParameterH1, and maxNonCentralityParameterH1.

Sample size and information

The first-stage information of the trial design must be specified to allow for calculations between the mean difference and non-centrality parameter scale. It is provided to the design object via firstStageInformation.
Listed below are some examples for the calculation between information (\(I_1\)) and sample size:

• One-sample z-test with \(n\) total patients: \(I_1 = \frac{n}{\sigma^2}\), where \(\sigma^2\) is the variance of an individual observation

• Balanced two-sample z-test with \(n_1\) patients per group: \(I_1 = \frac{1}{2}\cdot\frac{n_1}{\sigma^2}\), where \(\sigma^2\) is the common variance

• General two-sample z-test with \(n_1\), \(n_2\) patients per group: \(I_1 = 1/(\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2})\), where \(\sigma_1^2\), \(\sigma_2^2\) are the group-wise variances

Monotonicity

By default, the function Q (likelihood ratio divided by the squared effect) is transformed to be non-increasing in the first-stage p-value. For constant conditional power this yields a non-increasing conditional error function. A conditional power callback combined with interim effect estimates and information constraints can still produce a non-monotone conditional error function; enforceMonotonicity does not guarantee monotonicity in that setting. The necessary intervals and constants for the transformation are calculated by an internal monotonisation routine. Although not recommended for the operating characteristics of the design, the transformation may be omitted by setting enforceMonotonicity=FALSE.

Constraints

In some applications, it may be feasible to restrict the optimal conditional error function by a lower and/or upper limit. These constraints can be directly implemented on the function by using the arguments minConditionalError and maxConditionalError. By default, minConditionalError=0 and maxConditionalError=1, i.e., no constraints are applied. The constraints may also be specified on the second-stage information via minInformationPerStage and maxInformationPerStage. If both minConditionalError and maxInformationPerStage respectively maxConditionalError and minInformationPerStage are provided, both constraints will be applied. Set these constraints when creating the design, so that calibration accounts for them. For fixed \(\Delta_1\) and conditional power \(CP\), an upper information bound corresponds to the lower conditional error bound \(\Phi(\Phi^{-1}(CP) - \Delta_1 \sqrt{I_{2,\max}})\); a lower information bound analogously corresponds to an upper conditional error bound. Information bounds apply only when the trial continues: information is zero after early stopping. They refer to additional stage-two information, not cumulative information. minInformationPerStage and maxInformationPerStage are scalar limits for stage two only; they do not constrain firstStageInformation. In the continuation region, conditional error cannot exceed the conditional power; an upper bound above it has no additional effect. Before calibration, the integrated lower and upper bounds are checked for compatibility with the overall alpha level.

Level constant

The level constant is determined by an internal root-finding routine. It is identified using the uniroot() function and by default, the interval between 0 and 10 is searched for the level constant. In specific settings, the level constant may lie outside of this interval. In such cases, the search interval can be changed by altering the parameters minLevelConstant and maxLevelConstant.
If inappropriate constraints to the optimal conditional error function are provided via minConditionalError and maxConditionalError or minInformationPerStage and maxInformationPerStage, it may be impossible to find a level constant which exhausts the full alpha level.

Numerical integration uses an adapted routine for piecewise constant functions where applicable. Set options(rpact.design.optimal.enforce.basic.integration = TRUE) to use standard adaptive integration for comparison.

Generic functions

The print() and plot() functions are available for objects of class TrialDesignOptimalConditionalError. For details, see ?print.TrialDesignOptimalConditionalError and ?plot.TrialDesignOptimalConditionalError.

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.

See also

Optimal Conditional Error Designs with rpact for a worked example with stopping rules, information constraints, interim estimates, and conditional power functions.

Examples

# Create a single-arm design with fixed parameter for the likelihood ratio
# and a fixed effect for conditional power. 80 patients are observed in the
# first-stage (firstStageInformation = 80 in the one-sample test, variance 1).
# The second-stage information is restricted to be between 40 and 160.
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,
    minInformationPerStage = 40, maxInformationPerStage = 160
)
#> Optimal Conditional Error Design: 
#>  
#> General design parameters: 
#>   Overall significance level: 0.025 
#>   First-stage efficacy boundary (p-value scale): 0.001 
#>   Binding first-stage futility boundary (p-value scale): 0.5 
#>   Constraints on second-stage information: [40, 160] 
#> 
#> Conditional power specification: 
#>   Target conditional power: 0.9 
#>   Alternative: 0.25 
#>   First-stage non-centrality parameter: 2.236068 
#>   First-stage information: 80 
#> 
#> Likelihood ratio specification: 
#>   Fixed parameter(s) in likelihood ratio:  0.25 
#>   Parameter weights:  1 
#> 
#> Level constant: 
#>   Constant: 7.61398 
#>   Searched on interval: [0, 10] 

# Create a design comparing two groups using the maximum likelihood ratio
# and an interim estimate for the effect for conditional power.
# 160 patients per arm are observed in the first stage
# (firstStageInformation = 80 in the balanced two-sample test, variance 1).
getDesignOptimalConditionalError(
    alpha = 0.025, efficacyBounds = 0.001, futilityBounds = 0.5, conditionalPower = 0.9,
    minThetaH1 = 0.25, likelihoodRatioDistribution = "maxlr",
    firstStageInformation = 80, useInterimEstimate = TRUE
)
#> Optimal Conditional Error Design: 
#>  
#> General design parameters: 
#>   Overall significance level: 0.025 
#>   First-stage efficacy boundary (p-value scale): 0.001 
#>   Binding first-stage futility boundary (p-value scale): 0.5 
#> 
#> Conditional power specification: 
#>   Target conditional power: 0.9 
#>   Alternative: interim estimate restricted to [0.25, Inf] 
#>   First-stage non-centrality parameter restricted to [2.23606797749979, Inf] 
#>   First-stage information: 80 
#> 
#> Likelihood ratio specification: 
#>   Maximum likelihood ratio 
#> 
#> Level constant: 
#>   Constant: 7.68814 
#>   Searched on interval: [0, 10] 

# Weight several effects for the optimisation; retain a separate power target.
weightedDesign <- getDesignOptimalConditionalError(
    alpha = 0.025, efficacyBounds = 0.001, futilityBounds = 0.5, conditionalPower = 0.9,
    thetaH1 = 0.25, useInterimEstimate = FALSE, firstStageInformation = 80,
    likelihoodRatioDistribution = "fixed", thetaLR = c(0, 0.25, 0.5),
    weightsLR = c(0.25, 0.5, 0.25)
)
getStageInformation(pValue = c(0.05, 0.1, 0.3), design = weightedDesign)
#> [1] 110.6716 142.3435 185.2514

# Allow a lower conditional power target for less promising interim results.
# Leave conditionalPower unspecified when supplying a callback.
flexibleDesign <- getDesignOptimalConditionalError(
    alpha = 0.025, efficacyBounds = 0.001, futilityBounds = 0.5,
    conditionalPowerFunction = function(p) pnorm(1 - p),
    thetaH1 = 0.25, useInterimEstimate = FALSE, firstStageInformation = 80,
    likelihoodRatioDistribution = "maxlr"
)
getConditionalError(pValue = c(0.05, 0.1, 0.3), design = flexibleDesign)
#> [1] 0.08343571 0.04767596 0.02220325