ICA.ContCont.MultS: Assess surrogacy in the causal-inference single-trial setting (Individual Causal Association, ICA) using a continuous univariate T and multiple continuous S

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.

Usage

ICA.ContCont.MultS(M = 500, N, Sigma, 
G = seq(from=-1, to=1, by = .00001),
Seed=c(123), Show.Progress=FALSE)

Value

An object of class ICA.ContCont.MultS 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=500.
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).
G: A vector of the values that should be considered for the unidentified correlations. Default G=seq(-1, 1, by=.00001), i.e., values with range $-1$ to $1$.
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

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

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 ICA.ContCont.MultS function requires the user to specify a distribution $G$ for the unidentified correlations. Next, the identifiable correlations are fixed at their estimated values and the unidentifiable correlations are independently and randomly sampled from $G$. In the function call, the unidentifiable correlations are marked by specifying NA in the Sigma matrix (see example section below). The algorithm generates a large number of 'completed' matrices, and only those that are positive definite are retained (the number of positive definite matrices that should be obtained is specified by the M= argument in the function call). Based on the identifiable variances, these positive definite correlation matrices are converted to covariance matrices $\boldsymbol{\Sigma}$ and the multiple-surrogate ICA are estimated.

An issue with this approach (i.e., substituting unidentified correlations by random and independent samples from $G$) is that the probability of obtaining a positive definite matrix is very low when the dimensionality of the matrix increases. One approach to increase the efficiency of the algorithm is to build-up the correlation matrix in a gradual way. In particular, the property that a $\left(k \times k\right)$ matrix is positive definite if and only if all principal minors are positive (i.e., Sylvester's criterion) can be used. In other words, a $\left(k \times k\right)$ matrix is positive definite when the determinants of the upper-left $\left(2 \times 2\right)$, $\left(3 \times 3\right)$, ..., $\left(k \times k\right)$ submatrices all have a positive determinant. Thus, when a positive definite $\left(k \times k\right)$ matrix has to be generated, one can start with the upper-left $\left(2 \times 2\right)$ submatrix and randomly sample a value from the unidentified correlation (here: $\rho_{T_0T_0}$) from $G$. When the determinant is positive (which will always be the case for a $\left(2 \times 2\right)$ matrix), the same procedure is used for the upper-left $\left(3 \times 3\right)$ submatrix, and so on. When a particular draw from $G$ for a particular submatrix does not give a positive determinant, new values are sampled for the unidentified correlations until a positive determinant is obtained. In this way, it can be guaranteed that the final $\left(k \times k\right)$ submatrix will be positive definite. The latter approach is used in the current function. This procedure is used to generate many positive definite matrices. Based on these matrices, $\boldsymbol{\Sigma_{\Delta}}$ is generated and the multivariate ICA ($R^2_{H}$) is computed (for details, see Van der Elst et al., 2017).

References

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

Examples

Run this code

if (FALSE)  #time-consuming code parts
# 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(M=100, N=200, Show.Progress = TRUE,
  Sigma=s, G = seq(from=-1, to=1, by = .00001), Seed=c(123))

# Explore results
summary(ICA)
plot(ICA)

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