getLikelihoodRatio: Calculate Likelihood Ratio

In optconerrf: Optimal Monotone Conditional Error Functions

Calculate Likelihood Ratio

Description

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

Usage

getLikelihoodRatio(firstStagePValue, design)

Arguments

firstStagePValue	First-stage p-value or p-values. Must be a numeric vector between 0 and 1.
design	An object of class TrialDesignOptimalConditionalError created by getDesignOptimalConditionalErrorFunction(). Contains all necessary arguments to calculate the optimal conditional error function for the specified case.

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)

optconerrf documentation built on Sept. 9, 2025, 5:29 p.m.