Title:	Optimal Monotone Conditional Error Functions
Version:	1.0.0
Description:	Design and analysis of confirmatory adaptive clinical trials using the optimal conditional error framework according to Brannath and Bauer (2004) <doi:10.1111/j.0006-341X.2004.00221.x>. An extension to the optimal conditional error function using interim estimates as described in Brannath and Dreher (2024) <doi:10.48550/arXiv.2402.00814> and functions to ensure that the resulting conditional error function is non-increasing are also available.
Imports:	ggplot2, methods
License:	GPL (≥ 3)
Encoding:	UTF-8
RoxygenNote:	7.3.2
Suggests:	knitr, rmarkdown, testthat (≥ 3.0.0)
VignetteBuilder:	knitr
Config/testthat/edition:	3
URL:	https://github.com/morten-dreher/optconerrf
BugReports:	https://github.com/morten-dreher/optconerrf/issues
NeedsCompilation:	no
Packaged:	2025-09-03 19:51:20 UTC; Morten
Author:	Morten Dreher [aut, cre], Werner Brannath [aut, cph], Cornelia Ursula Kunz [ctb], Johanna zur Verth [aut]
Maintainer:	Morten Dreher <morten.dreher@outlook.de>
Repository:	CRAN
Date/Publication:	2025-09-09 13:40:07 UTC

Simple range check for numeric variables

Description

This function performs very basic range checks for numeric variables and throws an error if the range is violated. A custom hint may be added to the message.

Usage

.rangeCheck(variable, range, allowedEqual, hint = "")

Arguments

Power results for optimal conditional error design

Description

A class for power results of the optimal conditional error function.

Simulation results for optimal conditional error design

Description

A class for simulation results of the optimal conditional error function.

Optimal Conditional Error Design

Description

A class for a trial design object using the optimal conditional error function.

Details

This object should not be created directly; use getDesignOptimalConditionalErrorFunction() with suitable arguments to create a design.

Create a design object for the optimal conditional error function.

Description

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

getDesignOptimalConditionalErrorFunction(
  alpha,
  alpha1,
  alpha0,
  conditionalPower = NA_real_,
  delta1 = NA_real_,
  delta1Min = NA_real_,
  delta1Max = Inf,
  ncp1 = NA_real_,
  ncp1Min = NA_real_,
  ncp1Max = Inf,
  useInterimEstimate = TRUE,
  firstStageInformation,
  likelihoodRatioDistribution,
  minimumSecondStageInformation = 0,
  maximumSecondStageInformation = Inf,
  minimumConditionalError = 0,
  maximumConditionalError = 1,
  conditionalPowerFunction = NA,
  levelConstantMinimum = 0,
  levelConstantMaximum = 10,
  enforceMonotonicity = TRUE,
  ...
)

Arguments

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.

Value

An object of class TrialDesignOptimalConditionalError, which can be passed to other package functions.

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 deltaLR*sqrt(firstStageInformation) and the likelihood ratio is calculated as:

  l(p_1) = e^{\Phi^{-1}(1-p_1)\vartheta - \vartheta^2/2}.

  deltaLR 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 weightsDeltaLR, equal weights are assumed.

• likelihoodRatioDistribution="normal": calculates the likelihood ratio for a normally distributed prior of \vartheta with mean deltaLR*sqrt(firstStageInformation) (\mu) and standard deviation tauLR*sqrt(firstStageInformation) (\sigma). The parameters deltaLR and tauLR 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 mean kappaLR*sqrt(firstStageInformation) (\eta). The likelihood ratio is then calculated as:

  l(p_1) = \kappa \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 deltaMaxLR 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 usage of a fixed effect is indicated by setting useInterimEstimate=FALSE, in which case the fixed effect is provided by delta1 on the mean difference scale or by ncp1 on the non-centrality parameter scale (i.e., delta1*sqrt(firstStageInformation)). For an interim estimate, specified by useInterimEstimate=TRUE, a lower cut-off for the interim estimate must be provided, either by delta1Min on the mean difference scale, or ncp1Min on the non-centrality parameter scale. In addition, an upper limit of the estimate may be analogously provided by delta1Max or ncp1Max.

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 optimal conditional error function returned by getDesignOptimalConditionalErrorFunction() is transformed to be non-increasing in the first-stage p-value p_1 if found to be increasing on any interval. The necessary intervals and constants for the transformation are calculated by getMonotonisationConstants(). 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 minimumConditionalError and maximumConditionalError. By default, minimumConditionalError=0 and maximumConditionalError=1, i.e., no constraints are applied. The constraints may also be specified on the second-stage information via minimumSecondStageInformation and maximumSecondStageInformation. If both minimumConditionalError and maximumSecondStageInformation respectively maximumConditionalError and minimumSecondStageInformation are provided, both constraints will be applied.

Level constant

The level constant is determined by the helper function getLevelConstant(). 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 levelConstantMinimum and levelConstantMaximum.
If inappropriate constraints to the optimal conditional error function are provided via minimumConditionalError and maximumConditionalError or minimumSecondStageInformation and maximumSecondStageInformation, it may be impossible to find a level constant which exhausts the full alpha level.

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. https://www.jstor.org/stable/3695393

Brannath, W., Dreher, M., zur Verth, J., Scharpenberg, M. (2024). Optimal monotone conditional error functions. https://arxiv.org/abs/2402.00814

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.
getDesignOptimalConditionalErrorFunction(
  alpha = 0.025, alpha1 = 0.001, alpha0 = 0.5, conditionalPower = 0.9,
  delta1 = 0.25, likelihoodRatioDistribution = "fixed", deltaLR = 0.25,
  firstStageInformation = 80, useInterimEstimate = FALSE,
  minimumSecondStageInformation = 40, maximumSecondStageInformation = 160
)

# 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).
getDesignOptimalConditionalErrorFunction(
  alpha = 0.025, alpha1 = 0.001, alpha0 = 0.5, conditionalPower = 0.9,
  delta1Min = 0.25, likelihoodRatioDistribution = "maxlr",
  firstStageInformation = 80, useInterimEstimate = TRUE
)

Calculate Expected Second-stage Information

Description

Calculate the expected second-stage information using the optimal conditional error function with specific assumptions.

Usage

getExpectedSecondStageInformation(
  design,
  likelihoodRatioDistribution = NULL,
  ...
)

Arguments

Details

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.

Value

Expected second-stage information.

References

Brannath, W. & Bauer, P. (2004). Optimal conditional error functions for the control of conditional power. Biometrics. https://www.jstor.org/stable/3695393

See Also

getDesignOptimalConditionalErrorFunction(), getSecondStageInformation()

Examples

# Get a design
design <- getDesignOptimalConditionalErrorFunction(
alpha = 0.025, alpha1 = 0.001, alpha0 = 0.5, conditionalPower = 0.9,
delta1 = 0.25, likelihoodRatioDistribution = "fixed", deltaLR = 0.25,
firstStageInformation = 80, useInterimEstimate = FALSE,
)
# Calculate expected information under correct specification
getExpectedSecondStageInformation(design)

# Calculate expected information under the null hypothesis
getExpectedSecondStageInformation(
 design = design, likelihoodRatioDistribution = "fixed", deltaLR = 0
)

Internal variation of getPsi() used by findLevelConstant()

Description

Calculate psi for the given scenario, incl. constant

Usage

getInnerPsi(firstStagePValue, constant, design)

Arguments

Details

Internal function called by getIntegral() that calculates a point value of getPsi().

Value

Point value of getPsi().

Internal calculation of the integral over psi

Description

Helper function that integrates over the optimal conditional error function. Used to find appropriate level constant.

Usage

getIntegral(constant, design)

Arguments

Details

Internal function called by findLevelConstant() that should be solved (i.e., root should be found).

Value

Distance of integral over psi to (alpha-alpha1).

Calculate the integral over a partially constant function

Description

This function aims to provide a more accurate integration routine than the integrate() function for a partially constant function.

Usage

getIntegralWithConstants(constant, design)

Arguments

Value

Integral over the partially constant function.

Get Level Constant for Optimal Conditional Error Function

Description

Find the constant required such that the conditional error function meets the overall level condition.

Usage

getLevelConstant(design)

Arguments

Details

The level condition is defined as:

\alpha = \alpha_1 + \int_{\alpha_1}^{\alpha_0} \alpha_2(p_1)dp_1.

The constant c_0 of the optimal conditional error function is calibrated such that it meets the level condition. For a valid design, the additional following condition must be met to be able to exhaust the level \alpha:

\alpha_1 + CP(\alpha_0-\alpha_1)>\alpha.

This condition is checked by getLevelConstant() and the execution is terminated if it is not met.
In case a conditional power function is used, the condition is instead:

\alpha_1 + \int_{\alpha_1}^{\alpha_0} CP(p_1)dp_1>\alpha.

Value

A list that contains the constant (element $root) and other components provided by uniroot(). The level constant is calculated corresponding to the mean difference scale.

References

Brannath, W. & Bauer, P. (2004). Optimal conditional error functions for the control of conditional power. Biometrics. https://www.jstor.org/stable/3695393

Brannath, W., Dreher, M., zur Verth, J., Scharpenberg, M. (2024). Optimal monotone conditional error functions. https://arxiv.org/abs/2402.00814

Calculate Likelihood Ratio

Description

Calculate the likelihood ratio of a p-value for a given distribution.

Usage

getLikelihoodRatio(firstStagePValue, design)

Arguments

Details

The calculation of the likelihood ratio for a first-stage p-value p_1 is done based on a distributional assumption, specified in the design object. The different options require different parameters, elaborated in the following.

• likelihoodRatioDistribution="fixed": calculates the likelihood ratio for a fixed \Delta. The non-centrality parameter of the likelihood ratio \vartheta is then computed as deltaLR*sqrt(firstStageInformation) and the likelihood ratio is calculated as:

  l(p_1) = e^{\Phi^{-1}(1-p_1)\vartheta - \vartheta^2/2}.

  deltaLR 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 weightsDeltaLR, equal weights are assumed.

• likelihoodRatioDistribution="normal": calculates the likelihood ratio for a normally distributed prior of \vartheta with mean deltaLR*sqrt(firstStageInformation) (\mu) and standard deviation tauLR*sqrt(firstStageInformation) (\sigma). The parameters deltaLR and tauLR 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 mean 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 deltaMaxLR 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).

Value

The value of the likelihood ratio for the given specification.

References

Brannath, W. & Bauer, P. (2004). Optimal conditional error functions for the control of conditional power. Biometrics. https://www.jstor.org/stable/3695393

Hung, H. M. J., O’Neill, R. T., Bauer, P. & Kohne, K. (1997). The behavior of the p-value when the alternative hypothesis is true. Biometrics. https://www.jstor.org/stable/2533093

Examples

# Get a design
design <- getDesignOptimalConditionalErrorFunction(
alpha = 0.025, alpha1 = 0.001, alpha0 = 0.5, conditionalPower = 0.9,
delta1 = 0.25, likelihoodRatioDistribution = "fixed", deltaLR = 0.25,
firstStageInformation = 80, useInterimEstimate = FALSE,
)

getLikelihoodRatio(firstStagePValue = c(0.05, 0.1, 0.2), design = design)

Return Monotone Function Values

Description

Applies the provided monotonisation constants to a specified, possibly non-monotone function. The returned function values are non-increasing.

Usage

getMonotoneFunction(
  x,
  fun,
  lower = NULL,
  upper = NULL,
  argument = NULL,
  nSteps = 10^4,
  epsilon = 10^(-5),
  numberOfIterationsQ = 10^4,
  design
)

Arguments

Details

The exact monotonisation process is outlined in Brannath et al. (2024), but specified in terms of the first-stage test statistic z_1 rather than the first-stage p-value p_1.
The algorithm can easily be translated to the use of p-values by switching the maximum and minimum functions, i.e., replacing \min\{q, Q(z_1)\} by \max\{q, Q(p_1)\} and \min\{q, Q(z_1)\} by \max\{q, Q(p_1\}.

Value

Monotone function values.

References

Brannath, W., Dreher, M., zur Verth, J., Scharpenberg, M. (2024). Optimal monotone conditional error functions. https://arxiv.org/abs/2402.00814

Calculate the Constants for Monotonisation

Description

Computes the constants required to make a function non-increasing on the specified interval. The output of this function is necessary to calculate the monotone optimal conditional error function. The output object is a list that contains the intervals on which constant values are required, specified by the minimum dls and maximum dus of the interval and the respective constants, qs.

Usage

getMonotonisationConstants(
  fun,
  lower = 0,
  upper = 1,
  argument,
  nSteps = 10^4,
  epsilon = 10^(-5),
  numberOfIterationsQ = 10^4,
  design
)

Arguments

Value

A list containing the monotonisation constants (element $qs) and the intervals on which they must be applied, specified via minimum (element qls) and maximum (element qus).

References

Brannath, W., Dreher, M., zur Verth, J., Scharpenberg, M. (2024). Optimal monotone conditional error functions. https://arxiv.org/abs/2402.00814

Calculate Nu

Description

Calculate the factor which relates \alpha_2 to the second-stage information for given conditional power.

Usage

getNu(alpha, conditionalPower)

Arguments

Details

Note that this function uses factor 1 instead of factor 2 (Brannath & Bauer 2004). This has no impact on the optimal conditional error function, as constant factors are absorbed by the level constant c_0.
The calculation is:

\nu(\alpha_2(p_1)) = (\Phi^{-1}(1-\alpha_2(p_1)) + \Phi^{-1}(CP))^2.

Value

Factor linking information and \alpha_2.

References

Brannath, W. & Bauer, P. (2004). Optimal conditional error functions for the control of conditional power. Biometrics. https://www.jstor.org/stable/3695393

Examples

getNu(alpha = 0.05, conditionalPower = 0.9)

# Returns 0 if alpha exceeds conditionalPower
getNu(alpha = 0.8, conditionalPower = 0.7)

Calculate the Derivate of Nu

Description

Calculates the derivative of nu for a given conditional error and conditional power.

Usage

getNuPrime(alpha, conditionalPower)

Arguments

Details

The function \nu' is defined as

\nu'(p_1) = -2 \cdot (\Phi^{-1}(1-\alpha_2(p_1)) + \Phi^{-1}(CP))/\phi(\Phi^{-1}(1-\alpha_2(p_1))).

Note that in this implementation, the the factor -2 is used instead of -4, which is used in by Brannath & Bauer (2004), who explicitly investigate the setting of a balanced two-group trial. The argument conditionalPower is either the fixed target conditional power or the value of the conditional power function at the corresponding first-stage p-value.

Value

Value for nu prime.

References

Brannath, W. & Bauer, P. (2004). Optimal conditional error functions for the control of conditional power. Biometrics. https://www.jstor.org/stable/3695393

Examples

getNuPrime(alpha = 0.05, conditionalPower = 0.9)

Calculate the Optimal Conditional Error

Description

Calculate the Optimal Conditional Error

Usage

getOptimalConditionalError(firstStagePValue, design)

Arguments

Details

The optimal conditional error \alpha_2 given a first-stage p-value p_1 is calculated as:

\alpha_2(p_1)=\psi(-e^{c_0} \cdot \frac{\Delta_1^2}{l(p_1)}).

The level constant c_0 as well as the specification of the effect size \Delta_1 and the likelihood ratio l(p_1) must be contained in the design object (see ?getDesignOptimalConditionalErrorFunction). Early stopping rules are supported, i.e., for p_1 \leq \alpha_1, the returned conditional error is 1 and for p_1 > \alpha_0, the returned conditional error is 0.

Value

Value of the optimal conditional error function.

References

Brannath, W. & Bauer, P. (2004). Optimal conditional error functions for the control of conditional power. Biometrics. https://www.jstor.org/stable/3695393

See Also

getDesignOptimalConditionalErrorFunction()

Examples

# Create a design
design <- getDesignOptimalConditionalErrorFunction(
alpha = 0.025, alpha1 = 0.001, alpha0 = 0.5, conditionalPower = 0.9,
delta1 = 0.5, firstStageInformation = 40, useInterimEstimate = FALSE,
likelihoodRatioDistribution = "fixed", deltaLR = 0.5)

# Calculate optimal conditional error
getOptimalConditionalError(
firstStagePValue = c(0.1, 0.2, 0.3), design = design
)

Calculate the overall power

Description

Calculate the overall power and other operating characteristics of a design.

Usage

getOverallPower(design, alternative)

Arguments

Details

This function is used to evaluate the overall performance of a design. The probabilities for first-stage futility, first-stage efficacy and overall efficacy (i.e., overall power) are saved in an object of class PowerResultsOptimalConditionalError.

Value

The overall power of the design at the provided effect size.

See Also

getDesignOptimalConditionalErrorFunction(), getSimulationResults()

Examples

# Get a design
design <- getDesignOptimalConditionalErrorFunction(
alpha = 0.025, alpha1 = 0.001, alpha0 = 0.5, conditionalPower = 0.9,
delta1 = 0.25, likelihoodRatioDistribution = "fixed", deltaLR = 0.25,
firstStageInformation = 80, useInterimEstimate = FALSE,
)

getOverallPower(design, alternative = 0.25)

Calculate Psi, the Inverse of Nu Prime

Description

Get point-wise values of psi (inverse of nu prime)

Usage

getPsi(nuPrime, conditionalPower)

Arguments

Details

The function \psi is the inverse of:

\nu'(\alpha) = -2 \cdot(\Phi^{-1}(1-\alpha) + \Phi^{-1}(1-CP)) / \phi(\Phi^{-1}(1-\alpha))

. If the conditional power CP lies outside of the range 1-\Phi(2) \leq CP \leq \Phi(2), the calculation is slightly more complicated. The argument conditionalPower is either the fixed target conditional power or the value of the conditional power function at the corresponding first-stage p-value.

Value

The value of alpha which corresponds to nuPrime and lies between 0 and conditionalPower.

Examples

# Returns 0.05
getPsi(getNuPrime(alpha = 0.05, conditionalPower = 0.9), conditionalPower = 0.9)

Calculate Q

Description

Calculate the ratio of likelihood ratio and squared effect size.

Usage

getQ(firstStagePValue, design)

Arguments

Details

For more information on how to specify the likelihood ratio, see ?getLikelihoodRatio(). In case the optimal conditional error function is ever increasing in the first-stage p-value p_1, a monotone transformation of getQ() is needed for logical consistency and type I error rate control.
The formula for Q(p_1) is:

Q(p_1) = l(p_1) / \Delta_1^2,

where l(p_1) is the likelihood ratio and \Delta_1 is the effect size at which the conditional power should be achieved. The effect size may also depend on the interim data (i.e., on p_1) in case useInterimEstimate = TRUE was specified for the design object.

Value

Ratio of likelihood ratio and squared effect size.

References

Brannath, W., Dreher, M., zur Verth, J., Scharpenberg, M. (2024). Optimal monotone conditional error functions. https://arxiv.org/abs/2402.00814

Examples

# Get a design
design <- getDesignOptimalConditionalErrorFunction(
alpha = 0.025, alpha1 = 0.001, alpha0 = 0.5, conditionalPower = 0.9,
delta1 = 0.25, likelihoodRatioDistribution = "fixed", deltaLR = 0.25,
firstStageInformation = 80, useInterimEstimate = FALSE,
)

getQ(firstStagePValue = c(0.05, 0.1, 0.2), design = design)

Calculate the Second-stage Information

Description

Calculate second-stage information for given first-stage p-value and design.

Usage

getSecondStageInformation(firstStagePValue, design)

Arguments

Details

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).

Value

The second-stage information.

References

Brannath, W. & Bauer, P. (2004). Optimal conditional error functions for the control of conditional power. Biometrics. https://www.jstor.org/stable/3695393

See Also

getDesignOptimalConditionalErrorFunction(), getExpectedSecondStageInformation(), getOptimalConditionalError()

Examples

design <- getDesignOptimalConditionalErrorFunction(
  alpha = 0.025, alpha1 = 0.001, alpha0 = 0.5,
  conditionalPower = 0.9, delta1 = 0.25, useInterimEstimate = FALSE,
  firstStageInformation = 40, likelihoodRatioDistribution = "maxlr"
)

getSecondStageInformation(
  firstStagePValue = c(0.05, 0.1, 0.2), design = design
)

Simulate trials

Description

Simulate the rejection probability for a given design and alternative.

Usage

getSimulationResults(
  design,
  maxNumberOfIterations = 10000,
  alternative,
  seed = NULL
)

Arguments

Details

Simulates the probabilities of overall rejection as well as early futility and early efficacy for the provided scenario and design. This is done by generating random normally distributed test statistics and calculating their p-values.

Value

An object of class SimulationResultsOptimalConditionalError containing the simulation results.

See Also

getDesignOptimalConditionalErrorFunction(), getOverallPower()

Examples

design <- getDesignOptimalConditionalErrorFunction(
 alpha = 0.025, alpha1 = 0.001, alpha0 = 0.5, delta1 = 0.25,
 useInterimEstimate = FALSE,
 conditionalPower = 0.9, likelihoodRatioDistribution = "maxlr",
 firstStageInformation = 10
)

# Simulate under the null hypothesis and for a mean difference of 0.5
getSimulationResults(
 design = design, alternative = c(0, 0.5)
)

Integrate over information

Description

Internal function used by getExpectedSecondStageInformation() to calculate the integral over the information.

Usage

integrateExpectedInformation(
  firstStagePValue,
  design,
  likelihoodRatioDistribution,
  ...
)

Arguments

Value

Integral over the information of the second stage

Plot the optimal conditional error function

Description

The returned plot is a ggplot2 object and can be supplemented with additional layers using ggplot2 commands.

Usage

## S3 method for class 'TrialDesignOptimalConditionalError'
plot(x, range = c(0, 1), type = 1, plotNonMonotoneFunction = FALSE, ...)

Arguments

Value

No return value, plots the design.

Print power results

Description

Print an overview of exact power results.

Usage

## S3 method for class 'PowerResultsOptimalConditionalError'
print(x, ...)

Arguments

Value

No return value, prints the power calculation results.

Print simulation results

Description

Print an overview of simulation results.

Usage

## S3 method for class 'SimulationResultsOptimalConditionalError'
print(x, ...)

Arguments

Value

No return value, prints the simulation results.

Print optimal conditional error trial design

Description

Print an overview of the specified design parameters.

Usage

## S3 method for class 'TrialDesignOptimalConditionalError'
print(x, ...)

Arguments

Value

No return value, prints the design.

Summary of the optimal conditional error trial design

Description

Provide an overview of the operating characteristics of the optimal conditional error trial design.

Usage

## S3 method for class 'TrialDesignOptimalConditionalError'
summary(object, ...)

Arguments

Value

No return value, prints a summary of design performance.