ICA.ContCont.MultS.PC: Assess surrogacy in the causal-inference single-trial setting (Individual Causal Association, ICA) using a continuous univariate T and multiple continuous S, by simulating correlation matrices using an algorithm based on partial correlations

Description

The function ICA.ContCont.MultS quantifies surrogacy in the single-trial causal-inference framework where T is continuous and there are multiple continuous S. This function provides an alternative for ICA.ContCont.MultS.

Usage

ICA.ContCont.MultS.PC(M=1000,N,Sigma,Seed=123,Show.Progress=FALSE)

Value

An object of class ICA.ContCont.MultS.PC with components,

R2_H: The multiple-surrogate individual causal association value(s).
Corr.R2_H: The corrected multiple-surrogate individual causal association value(s).
Lower.Dig.Corrs.All: A data.frame that contains the matrix that contains the identifiable and unidentifiable correlations (lower diagonal elements) that were used to compute ($R^2_{H}$) in the run.

Arguments

M: The number of multivariate ICA values ($R^2_{H}$) that should be sampled. Default M=1000.
N: The sample size of the dataset.
Sigma: A matrix that specifies the variance-covariance matrix between $T_0$, $T_1$, $S_{10}$, $S_{11}$, $S_{20}$, $S_{21}$, ..., $S_{k0}$, and $S_{k1}$ (in this order, the $T_0$ and $T_1$ data should be in Sigma[c(1,2), c(1,2)], the $S_{10}$ and $S_{11}$ data should be in Sigma[c(3,4), c(3,4)], and so on). The unidentifiable covariances should be defined as NA (see example below).
Seed: The seed that is used. Default Seed=123.
Show.Progress: Should progress of runs be graphically shown? (i.e., 1% done..., 2% done..., etc). Mainly useful when a large number of S have to be considered (to follow progress and estimate total run time).

Author

Alvaro Florez

Details

The multivariate ICA ($R^2_{H}$) is not identifiable because the individual causal treatment effects on $T$, $S_1$, ..., $S_k$ cannot be observed. A simulation-based sensitivity analysis is therefore conducted in which the multivariate ICA ($R^2_{H}$) is estimated across a set of plausible values for the unidentifiable correlations. To this end, consider the variance covariance matrix of the potential outcomes $\boldsymbol{\Sigma}$ (0 and 1 subscripts refer to the control and experimental treatments, respectively): $$\boldsymbol{\Sigma} = \left(\begin{array}{ccccccccc} \sigma_{T_{0}T_{0}}\\ \sigma_{T_{0}T_{1}} & \sigma_{T_{1}T_{1}}\\ \sigma_{T_{0}S1_{0}} & \sigma_{T_{1}S1_{0}} & \sigma_{S1_{0}S1_{0}}\\ \sigma_{T_{0}S1_{1}} & \sigma_{T_{1}S1_{1}} & \sigma_{S1_{0}S1_{1}} & \sigma_{S1_{1}S1_{1}}\\ \sigma_{T_{0}S2_{0}} & \sigma_{T_{1}S2_{0}} & \sigma_{S1_{0}S2_{0}} & \sigma_{S1_{1}S2_{0}} & \sigma_{S2_{0}S2_{0}}\\ \sigma_{T_{0}S2_{1}} & \sigma_{T_{1}S2_{1}} & \sigma_{S1_{0}S2_{1}} & \sigma_{S1_{1}S2_{1}} & \sigma_{S2_{0}S2_{1}} & \sigma_{S2_{1}S2_{1}}\\ ... & ... & ... & ... & ... & ... & \ddots\\ \sigma_{T_{0}Sk_{0}} & \sigma_{T_{1}Sk_{0}} & \sigma_{S1_{0}Sk_{0}} & \sigma_{S1_{1}Sk_{0}} & \sigma_{S2_{0}Sk_{0}} & \sigma_{S2_{1}Sk_{0}} & ... & \sigma_{Sk_{0}Sk_{0}}\\ \sigma_{T_{0}Sk_{1}} & \sigma_{T_{1}Sk_{1}} & \sigma_{S1_{0}Sk_{1}} & \sigma_{S1_{1}Sk_{1}} & \sigma_{S2_{0}Sk_{1}} & \sigma_{S2_{1}Sk_{1}} & ... & \sigma_{Sk_{0}Sk_{1}} & \sigma_{Sk_{1}Sk_{1}}. \end{array}\right)$$

The identifiable correlations are fixed at their estimated values and the unidentifiable correlations are independently and randomly sampled using an algorithm based on partial correlations (PC). In the function call, the unidentifiable correlations are marked by specifying NA in the Sigma matrix (see example section below). The PC algorithm generate each correlation matrix progressively based on parameterization of terms of the correlations $\rho_{i,i+1}$, for $i=1,\ldots,d-1$, and the partial correlations $\rho_{i,j|i+1,\ldots,j-1}$, for $j-i>2$ (for details, see Joe, 2006 and Florez et al., 2018). Based on the identifiable variances, these correlation matrices are converted to covariance matrices $\boldsymbol{\Sigma}$ and the multiple-surrogate ICA are estimated (for details, see Van der Elst et al., 2017).

This approach to simulate the unidentifiable parameters of $\boldsymbol{\Sigma}$ is computationally more efficient than the one used in the function ICA.ContCont.MultS.

References

Florez, A., Alonso, A. A., Molenberghs, G. & Van der Elst, W. (2018). Simulation of random correlation matrices with fixed values: comparison of algorithms and application on multiple surrogates assessment.

Joe, H. (2006). Generating random correlation matrices based on partial correlations. Journal of Multivariate Analysis, 97(10):2177-2189.

Van der Elst, W., Alonso, A. A., & Molenberghs, G. (2017). Univariate versus multivariate surrogate endpoints.

Examples

Run this code

if (FALSE) { 
# Specify matrix Sigma (var-cavar matrix T_0, T_1, S1_0, S1_1, ...)
# here for 1 true endpoint and 3 surrogates

s<-matrix(rep(NA, times=64),8)
s[1,1] <- 450; s[2,2] <- 413.5; s[3,3] <- 174.2; s[4,4] <- 157.5; 
s[5,5] <- 244.0; s[6,6] <- 229.99; s[7,7] <- 294.2; s[8,8] <- 302.5
s[3,1] <- 160.8; s[5,1] <- 208.5; s[7,1] <- 268.4 
s[4,2] <- 124.6; s[6,2] <- 212.3; s[8,2] <- 287.1
s[5,3] <- 160.3; s[7,3] <- 142.8 
s[6,4] <- 134.3; s[8,4] <- 130.4 
s[7,5] <- 209.3; 
s[8,6] <- 214.7 
s[upper.tri(s)] = t(s)[upper.tri(s)]

# Marix looks like (NA indicates unidentified covariances):
#            T_0    T_1  S1_0  S1_1  S2_0   S2_1  S2_0  S2_1
#            [,1]  [,2]  [,3]  [,4]  [,5]   [,6]  [,7]  [,8]
# T_0  [1,] 450.0    NA 160.8    NA 208.5     NA 268.4    NA
# T_1  [2,]    NA 413.5    NA 124.6    NA 212.30    NA 287.1
# S1_0 [3,] 160.8    NA 174.2    NA 160.3     NA 142.8    NA
# S1_1 [4,]    NA 124.6    NA 157.5    NA 134.30    NA 130.4
# S2_0 [5,] 208.5    NA 160.3    NA 244.0     NA 209.3    NA
# S2_1 [6,]    NA 212.3    NA 134.3    NA 229.99    NA 214.7
# S3_0 [7,] 268.4    NA 142.8    NA 209.3     NA 294.2    NA
# S3_1 [8,]    NA 287.1    NA 130.4    NA 214.70    NA 302.5

# Conduct analysis
ICA <- ICA.ContCont.MultS.PC(M=1000, N=200, Show.Progress = TRUE,
Sigma=s, Seed=c(123))

# Explore results
summary(ICA)
plot(ICA)
}

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