Purpose of the package

The R package optconerrf can be used to derive the optimal conditional error function proposed by Brannath & Bauer (2004), which minimises the expected second-stage sample size of a two-stage adaptive clinical trial design while controlling for the conditional power. The package implements a more general approach by using the second-stage information rather than the second-stage sample-size and introduces extensions to the original methodology. In case the specification of the optimal conditional error function leads to a non-monotone function, the package can be used to implement the monotone transformation detailed in Brannath et al. (2024).

This document provides a brief introduction to the central functions of the package. These are:

• getDesignOptimalConditionalErrorFunction() to create a design which facilitates the use of the remaining functions
• getOptimalConditionalErrorFunction() to calculate the optimal conditional error for a given first-stage p-value
• getSecondStageInformation() to calculate the second-stage information required to reach the target conditional power for a given first-stage p-value
• getExpectedSecondStageInformation() to calculate the expected second-stage information for a specification of the optimal conditional error function under an effect assumption

Prerequisites to computing the optimal conditional error function

In optconerrf, the tested set of hypotheses is always assumed to be:

\[ H_0: \Delta \leq 0 \enspace \text{vs.} \enspace H_1: \Delta>0, \] where \(\Delta\) is the effect size of interest.

The optimal conditional error function is defined as:

\[ \alpha_2(p_1) = \psi \left( \frac{-e^{c_0}}{Q(p_1)}\right). \]

The individual parameters and functions are the following:

• \(p_1\) is the first-stage p-value.
• \(\psi\) is the inverse of the function \(\nu'(u) = -2\cdot(\Phi^{-1}(1-u)+\Phi^{-1}(CP))/\phi(\Phi^{-1}(1-u))\). For \(\psi\), no closed form is available and its solution is found through the uniroot() function in optconerrf.
• \(c_0\) is the level constant, required for the optimal conditional error function to preserve the overall type I error rate.
• \(Q(p_1)=l(p_1)/\Delta_1^2\) is the ratio of the likelihood ratio \(l(p_1)\) of \(p_1\) and an effect assumption \(\Delta_1\) at which the conditional power \(CP\) should be reached. The likelihood ratio requires an assumption about the true treatment effect, denoted \(\vartheta\). This is the effect assumption under which the second-stage information is minimised.

Note that \(\vartheta\) denotes a non-centrality parameter rather than the corresponding mean difference \(\Delta\). The non-centrality parameter \(\vartheta\) is proportional to \(\Delta\) and additionally takes into account the first-stage information \(I_1\) of the trial. It is calculated as:

\[ \vartheta = \Delta\cdot \sqrt{I_1}. \]

In optconerrf, the treatment effect for the conditional power can be specified either as the mean difference delta1 or the non-centrality parameter ncp1, corresponding to \(\vartheta_1 = \Delta_1 \cdot \sqrt{I_1}\).

The optconerrf package is based on a design object which contains the required specifications of the optimal conditional error functions. This object is created by the function getDesignOptimalConditionalErrorFunction() and can be simply passed to other functions instead of specifying each argument separately. The following section explains how to derive the design object.

Creating the design object

For the creation of a design object, the following parameters are required and discussed in more detail below:

• Overall type I error rate and early stopping boundaries
• Effect for likelihood ratio
• Conditional power
• Effect for conditional power
• First-stage information
• Optional: Constraints providing minimum and maximum conditional error (or maximum and minimum second-stage information, respectively)

In addition to these arguments, which must be specified by the user, the following fields are automatically calculated and added to the design object:

• Level constant
• Monotonisation constants to ensure a non-increasing conditional error function (if necessary and not explicitly inhibited by the user)

Using the design object

As an example, we consider an adaptive design with overall type I error rate alpha=0.025, early rejection and futility boundaries alpha1=0.0154 and alpha0=0.5 with a target conditional power conditionalPower=0.9 at a fixed alternative delta1=0.25 (note that useInterimEstimate=FALSE). We assume a fixed effect of deltaLR=0.25 for the calculation of the likelihood ratio (likelihoodRatioDistribution="fixed") and firstStageInformation=50. Note that this design approximates the original trial design used in Brannath & Bauer (2004) and achieves an overall power of 90%.

design <- getDesignOptimalConditionalErrorFunction(
  alpha = 0.025, alpha1 = 0.0154, alpha0 = 0.5, conditionalPower = 0.9,
  delta1 = 0.25, likelihoodRatioDistribution = "fixed", deltaLR = 0.25,
  firstStageInformation = 50, useInterimEstimate = FALSE
)

Once the design object is created using getDesignOptimalCondtionalErrorFunction(), it can be easily passed to other functions of optconerrf. The design object is an instance of the class TrialDesignOptimalConditionalError and offers implementations of the generic functions print(), plot() and summary(). The print() function provides a comprehensive overview of user-specified parameters (such as early stopping boundaries and effect sizes) as well as internally calculated variables (such as the level constant):

# Comprehensive design output
print(design)
#> Optimal Conditional Error Function Design: 
#>  
#> General design parameters: 
#>   Overall significance level: 0.025 
#>   First-stage efficacy boundary (p-value scale): 0.0154 
#>   Binding first-stage futility boundary (p-value scale): 0.5 
#> 
#> Conditional power specification: 
#>   Target conditional power: 0.9 
#>   Alternative: 0.25 
#>   First-stage non-centrality parameter: 1.767767 
#>   First-stage information: 50 
#> 
#> Likelihood ratio specification: 
#>   Fixed parameter(s) in likelihood ratio:  0.25 
#>   Parameter weights:  1 
#> 
#> Level constant: 
#>   Constant: 7.964514 
#>   Searched on interval: [0, 10]

The plot() function has 4 different modes of operation, which are specified via the argument type: Passing design and type = 1 to the plot() function produces a plot visualising the optimal conditional error function resulting from the specification in design in relation to the first-stage p-value. As type = 1 is also the default value, the argument can be omitted to achieve this function behaviour:

plot(design) # Plot of the optimal conditional error function

Using type = 2 produces a plot of the second-stage information resulting from the design specification in relation to the first-stage p-value. The second-stage information \(I_2(p_1)\) of a first-stage p-value \(p_1\) is calculated as:

\[ I_2(p_1) = \frac{\bigl( \Phi^{-1}(1-\alpha_2(p_1)) + \Phi^{-1}(CP) \bigr)^2}{\Delta_1^2}. \]

plot(design, type = 2) # Plot of the second-stage information

The remaining plot types can be used to visualise the likelihood ratio \(l(p_1)\) (type = 3) and the function \(Q(p_1)\) (type = 4) resulting from the design specification and will not be shown here.

To access individual fields of the object, e.g., the overall type I error rate (alpha) or the level constant (levelConstant) the following code can be used:

design$alpha
#> [1] 0.025
design$levelConstant
#> [1] 7.964514
print(design$levelConstant, digits = 12) # For more precision
#> [1] 7.96451445086

Computing the optimal conditional error function and second-stage information

Values for the optimal conditional error function are calculated using the function getOptimalConditionalError(). For this function, the p-value of the first stage (argument firstStagePValue) and a design object, created using getDesignOptimalConditionalErrorFunction(), must be provided.

For the specification above, the optimal conditional error for an exemplary first-stage p-value of 0.1 can be calculated as:

getOptimalConditionalError(
  firstStagePValue = 0.1, design = design
)
#> [1] 0.03136691

Multiple first-stage p-values may also be provided as a vector. For first-stage p-values with \(p_1 \leq \alpha_1\) or \(p_1 > \alpha_0\), a conditional error of 1 or 0 is returned, respectively.

getOptimalConditionalError(
  firstStagePValue = c(0.0005, 0.1, 0.05, 0.5, 0.8),
  design = design
)
#> [1] 1.000000000 0.031366907 0.061042561 0.003081757 0.000000000

The calculation of the second-stage information is achieved in a similar fashion, using getSecondStageInformation():

getSecondStageInformation(
  firstStagePValue = 0.1, design = design
)
#> [1] 158.0175

getSecondStageInformation(
  firstStagePValue = c(0.0005, 0.1, 0.05, 0.5, 0.8),
  design = design
)
#> [1]   0.0000 158.0175 127.9281 258.6313   0.0000

Note that the second-stage information for first-stage p-values with \(p_1 \leq \alpha_1\) or \(p_1 > \alpha_0\) is always 0, as the trial is stopped early for such results.

Calculation of expected second-stage information

With the function getExpectedSecondStageInformation(), a calculation of the expected second-stage information that a specification of the optimal conditional error function yields can be calculated. The calculation of the expected second-stage information \(\mathbb{E}(I_2)\) is performed as:

\[ \mathbb{E}(I_2) = \int_{\alpha_1}^{\alpha_0} \frac{ (\Phi^{-1}(1-\alpha_2(p_1)) + \Phi^{-1}(CP_{\Delta_1}))^2\cdot l(p_1)}{\Delta_1^2}dp_1. \] The effect \(\Delta_1\) provides the effect for which the conditional power \(CP_{\Delta_1}\) is targeted. \(\Delta_1\) is extracted from the design object and may depend on the interim data.

The likelihood ratio \(l(p_1)\) in this case represents the scenario under which the performance of a specific design is to be evaluated. For example, one may have specified a design assuming \(\Delta=0.25\) as a fixed effect for the likelihood ratio (as above), but now wishes to investigate the performance of this design in case \(\Delta=0\), i.e., if the null hypothesis is true. This can be achieved by the following code:

getExpectedSecondStageInformation(
  design = design, likelihoodRatioDistribution = "fixed", deltaLR = 0
)
#> [1] 97.14428

The specification of the likelihood ratio under which to calculate the expected second-stage information is given by likelihoodRatioDistribution = "fixed" and the corresponding treatment effect is specified by deltaLR = 0.

Alternatively, one may calculate the expected second-stage information of the design if it was correctly specified, i.e., for \(\Delta=0.25\):

getExpectedSecondStageInformation(
  design = design, likelihoodRatioDistribution = "fixed", deltaLR = 0.25
)
#> [1] 95.38173

A shortcut for the calculation of the expected second-stage information under correct specification is also available by omitting the arguments likelihoodRatioDistribution and deltaLR, in which case the likelihood ratio specification given in the design is extracted and used for the calculation.

getExpectedSecondStageInformation(
  design = design
)
#> [1] 95.38173

Multiplying the respective information values by a factor of 2 (approximately) yields the original results of Brannath & Bauer (2004).

In addition to calculating the expected second-stage information for a single fixed effect size (by specifying likelihoodRatioDistribution = "fixed" and a single value for deltaLR), it is also possible to provide multiple weighted fixed effect sizes (by specifying likelihoodRatioDistribution = "fixed", multiple values for deltaLR and possibly respective weights with weightsDeltaLR) or to use a distributional assumption for the effect size in the likelihood ratio (by specifying one of "normal", "exp" or "unif" for likelihoodRatioDistribution and the respective parameters).

Complementary functions

In addition to the core functions described above, the optconerrf package provides additional functions which relate to the operating characteristics of the design. An exact calculation of the overall power of a given design can be achieved by using the function getOverallPower():

getOverallPower(
  design, alternative = c(0, 0.25))
#> Power calculation results 
#>  
#>  Alternative:  0.00, 0.25 
#>  First-stage futility:  0.50000000, 0.03854994 
#>  First-stage efficacy:  0.0154000, 0.3475734 
#>  Overall power:  0.0250000, 0.9000624

The argument alternative may consist of multiple elements and provides the effect sizes for which the overall power should be calculated.

In addition to an exact calculation via getOverallPower(), the same operating characteristics may be simulated using the function getSimulationResults().

The summary() function is also implemented for objects of class TrialDesignOptimalConditionalError and combines selected results from getExpectedSecondStageInformation() and getOverallPower() to provide a general overview of the design performance:

summary(design)
#> Summary of the Optimal Conditional Error Function Design: 
#>  
#> General design parameters: 
#>   Overall significance level: 0.025 
#>   First-stage efficacy boundary (p-value scale): 0.0154 
#>   Binding first-stage futility boundary (p-value scale): 0.5 
#> 
#> Second-stage information: 
#>   Expected second-stage information (delta=0): 97.14428 
#>   Expected second-stage information (delta=delta1=0.25): 95.38173
#>   Expected second-stage information (Given likelihood ratio distr.): 95.38173 
#>   Expected second-stage information (delta=Mean of given likelihood ratio distr.=0.25): 95.38173
#>   Maximum second-stage information: 258.6313
#>  
#> Power and stopping probabilities: 
#>   Conditional power (fixed): 0.9 
#>   Overall power (delta1=0.25): 0.9000624
#>   Efficacy stopping probability (delta1=0.25): 0.3475734
#>   Futility stopping probability (delta1=0.25): 0.03854994
#>   Overall power (delta1=Mean of given likelihood ratio distr.= 0.25): 0.9000624
#>   Efficacy stopping probability (delta1=Mean of given likelihood ratio distr.= 0.25): 0.3475734
#>   Futility stopping probability (delta1=Mean of given likelihood ratio distr.= 0.25): 0.03854994

References

Brannath, W. & Bauer, P. (2004). Optimal conditional error functions for the control of conditional power. Biometrics, 60 (3), 715–723. 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

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