ICA_alpha_ContCont: Assess surrogacy using a Rényi divergence based family of...

In Surrogate: Evaluation of Surrogate Endpoints in Clinical Trials

Assess surrogacy using a Rényi divergence based family of metrics in the causal-inference single-trial setting in normal case

Description

This function is a set of metrics to evaluate surrogacy. ICA_{\alpha} have the similar mathematical properties with ICA.ContCont().

Usage

ICA_alpha_ContCont(
  alpha = numeric(),
  T0S0,
  T1S1,
  T0T0 = 1,
  T1T1 = 1,
  S0S0 = 1,
  S1S1 = 1,
  T0T1 = seq(-1, 1, by = 0.1),
  T0S1 = seq(-1, 1, by = 0.1),
  T1S0 = seq(-1, 1, by = 0.1),
  S0S1 = seq(-1, 1, by = 0.1)
)

Arguments

alpha	(numeric) is order ⁠alpha in [0, infinity]⁠
T0S0	A scalar or vector that specifies the correlation(s) between the surrogate and the true endpoint in the control treatment condition (\rho_{T0S0})
T1S1	A scalar or vector that specifies the correlation(s) between the surrogate and the true endpoint in the experimental treatment condition (\rho_{T1S1})
T0T0	A scalar that specifies the variance of the true endpoint in the control treatment condition
T1T1	A scalar that specifies the variance of the true endpoint in the experimental treatment condition
S0S0	A scalar that specifies the variance of the surrogate endpoint in the control treatment condition
S1S1	A scalar that specifies the variance of the surrogate endpoint in the experimental treatment condition
T0T1	A scalar or vector that contains the correlation(s) between the counterfactuals T0 and T1
T0S1	A scalar or vector that contains the correlation(s) between the counterfactuals T0 and S1
T1S0	A scalar or vector that contains the correlation(s) between the counterfactuals T1 and S0
S0S1	A scalar or vector that contains the correlation(s) between the counterfactuals S0 and S1

Value

• Total.Num.Matrices: An object of class numeric that contains the total number of matrices that can be formed as based on the user-specified correlations in the function call.

• Pos.Def: A data.frame that contains the positive definite matrices that can be formed based on the user-specified correlations. These matrices are used to compute the vector of the \rho_{\Delta} values.

• rho: A scalar or vector that contains the individual causal association \rho_{\Delta}

• ICA: A scalar or vector that contains the individual causal association \rho_{\Delta}^2=ICA

• ICA_alpha: A scalar or vector that contains the individual causal association ICA_{\alpha}

• Sigmas: A data.frame that contains the \sigma_{\Delta T} and \sigma_{\Delta S}

New family of surrogacy metrics:

ICA was extended as ICA_{\alpha} by proposing a new family of surrogacy metrics derived from Rényi divergence. Based on this family of divergences, one can propose a more general set of surrogacy metrics defined as:

ICA_{\alpha} = 1 - e^{-2 D_{\alpha}[f(\Delta T,\Delta S), f(\Delta T)f(\Delta S)]}

when \alpha=1 then

ICA_{1}=1-e^{-2D_1[f(\Delta T,\Delta S),f(\Delta T)f(\Delta S)]}=1-e^{-2I(\Delta T, \Delta S)}=\rho_{\Delta}^2=ICA

References

Deliorman, G., Alonso, A., & Pardo, M. C. (2025). A Rényi-Divergence-Based Family of Metrics for the Evaluation of Surrogate Endpoints in a Causal Inference Framework. Statistics in Biopharmaceutical Research.

Examples

ICA_alpha_ContCont(alpha=0.5,
                  T0S0 = 0.9597334, T1S1=0.9644139,
                  T0T0=544.3285, T1T1=550.6597,
                  S0S0=180.6831, S1S1=180.9433,
                  T0T1=seq(-1, 1, by = 0.5),  T0S1=seq(-1, 1, by = 1),
                  T1S0=seq(-1, 1, by = 0.5), S0S1=seq(-1, 1, by = 1))

#More than one alpha value can be calculated simultaneously:

ICA_alpha_ContCont(alpha=c(1.25,1.5,2),
                  T0S0 = 0.9597334, T1S1=0.9644139,
                  T0T0=544.3285, T1T1=550.6597,
                  S0S0=180.6831, S1S1=180.9433,
                  T0T1=-0.9, T0S1=-0.9,
                  T1S0=-0.9, S0S1=-0.9)

Surrogate documentation built on Oct. 16, 2026, 9:07 a.m.