ICA.ContCont: Assess surrogacy in the causal-inference single-trial setting (Individual Causal Association, ICA) in the Continuous-continuous case

Description

The function ICA.ContCont quantifies surrogacy in the single-trial causal-inference framework. See Details below.

Usage

ICA.ContCont(T0S0, T1S1, T0T0=1, T1T1=1, S0S0=1, S1S1=1, T0T1=seq(-1, 1, by=.1), 
T0S1=seq(-1, 1, by=.1), T1S0=seq(-1, 1, by=.1), S0S1=seq(-1, 1, by=.1))

Value

An object of class ICA.ContCont with components,

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.
ICA: A scalar or vector that contains the individual causal association (ICA; $\rho_{\Delta}$) value(s).
GoodSurr: A data.frame that contains the ICA ($\rho_{\Delta}$), $\sigma_{\Delta_{T}}$, and $\delta$.

Arguments

T0S0: A scalar or vector that specifies the correlation(s) between the surrogate and the true endpoint in the control treatment condition that should be considered in the computation of $\rho_{\Delta}$.
T1S1: A scalar or vector that specifies the correlation(s) between the surrogate and the true endpoint in the experimental treatment condition that should be considered in the computation of $\rho_{\Delta}$.
T0T0: A scalar that specifies the variance of the true endpoint in the control treatment condition that should be considered in the computation of $\rho_{\Delta}$. Default 1.
T1T1: A scalar that specifies the variance of the true endpoint in the experimental treatment condition that should be considered in the computation of $\rho_{\Delta}$. Default 1.
S0S0: A scalar that specifies the variance of the surrogate endpoint in the control treatment condition that should be considered in the computation of $\rho_{\Delta}$. Default 1.
S1S1: A scalar that specifies the variance of the surrogate endpoint in the experimental treatment condition that should be considered in the computation of $\rho_{\Delta}$. Default 1.
T0T1: A scalar or vector that contains the correlation(s) between the counterfactuals T0 and T1 that should be considered in the computation of $\rho_{\Delta}$. Default seq(-1, 1, by=.1), i.e., the values $-1$, $-0.9$, $-0.8$, ..., $1$.
T0S1: A scalar or vector that contains the correlation(s) between the counterfactuals T0 and S1 that should be considered in the computation of $\rho_{\Delta}$. Default seq(-1, 1, by=.1).
T1S0: A scalar or vector that contains the correlation(s) between the counterfactuals T1 and S0 that should be considered in the computation of $\rho_{\Delta}$. Default seq(-1, 1, by=.1).
S0S1: A scalar or vector that contains the correlation(s) between the counterfactuals S0 and S1 that should be considered in the computation of $\rho_{\Delta}$. Default seq(-1, 1, by=.1).

Author

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

Details

Based on the causal-inference framework, it is assumed that each subject j has four counterfactuals (or potential outcomes), i.e., $T_{0j}$, $T_{1j}$, $S_{0j}$, and $S_{1j}$. Let $T_{0j}$ and $T_{1j}$ denote the counterfactuals for the true endpoint ($T$) under the control ($Z=0$) and the experimental ($Z=1$) treatments of subject j, respectively. Similarly, $S_{0j}$ and $S_{1j}$ denote the corresponding counterfactuals for the surrogate endpoint ($S$) under the control and experimental treatments, respectively. The individual causal effects of $Z$ on $T$ and $S$ for a given subject j are then defined as $\Delta_{T_{j}}=T_{1j}-T_{0j}$ and $\Delta_{S_{j}}=S_{1j}-S_{0j}$, respectively.

In the single-trial causal-inference framework, surrogacy can be quantified as the correlation between the individual causal effects of $Z$ on $S$ and $T$ (for details, see Alonso et al., submitted):

$$\rho_{\Delta}=\rho(\Delta_{T_{j}},\:\Delta_{S_{j}})=\frac{\sqrt{\sigma_{S_{0}S_{0}}\sigma_{T_{0}T_{0}}}\rho_{S_{0}T_{0}}+\sqrt{\sigma_{S_{1}S_{1}}\sigma_{T_{1}T_{1}}}\rho_{S_{1}T_{1}}-\sqrt{\sigma_{S_{0}S_{0}}\sigma_{T_{1}T_{1}}}\rho_{S_{0}T_{1}}-\sqrt{\sigma_{S_{1}S_{1}}\sigma_{T_{0}T_{0}}}\rho_{S_{1}T_{0}}}{\sqrt{(\sigma_{T_{0}T_{0}}+\sigma_{T_{1}T_{1}}-2\sqrt{\sigma_{T_{0}T_{0}}\sigma_{T_{1}T_{1}}}\rho_{T_{0}T_{1}})(\sigma_{S_{0}S_{0}}+\sigma_{S_{1}S_{1}}-2\sqrt{\sigma_{S_{0}S_{0}}\sigma_{S_{1}S_{1}}}\rho_{S_{0}S_{1}})}},$$

where the correlations $\rho_{S_{0}T_{1}}$, $\rho_{S_{1}T_{0}}$, $\rho_{T_{0}T_{1}}$, and $\rho_{S_{0}S_{1}}$ are not estimable. It is thus warranted to conduct a sensitivity analysis (by considering vectors of possible values for the correlations between the counterfactuals -- rather than point estimates).

When the user specifies a vector of values that should be considered for one or more of the counterfactual correlations in the above expression, the function ICA.ContCont constructs all possible matrices that can be formed as based on these values, identifies the matrices that are positive definite (i.e., valid correlation matrices), and computes $\rho_{\Delta}$ for each of these matrices. The obtained vector of $\rho_{\Delta}$ values can subsequently be used to examine (i) the impact of different assumptions regarding the correlations between the counterfactuals on the results (see also plot Causal-Inference ContCont), and (ii) the extent to which proponents of the causal-inference and meta-analytic frameworks will reach the same conclusion with respect to the appropriateness of the candidate surrogate at hand.

The function ICA.ContCont also generates output that is useful to examine the plausibility of finding a good surrogate endpoint (see GoodSurr in the Value section below). For details, see Alonso et al. (submitted).

Notes

A single $\rho_{\Delta}$ value is obtained when all correlations in the function call are scalars.

References

Alonso, A., Van der Elst, W., Molenberghs, G., Buyse, M., & Burzykowski, T. (submitted). On the relationship between the causal-inference and meta-analytic paradigms for the validation of surrogate markers.

Examples

Run this code

if (FALSE)  #time-consuming code parts
# Generate the vector of ICA.ContCont values when rho_T0S0=rho_T1S1=.95, 
# sigma_T0T0=90, sigma_T1T1=100,sigma_ S0S0=10, sigma_S1S1=15, and  
# the grid of values {0, .2, ..., 1} is considered for the correlations
# between the counterfactuals:
SurICA <- ICA.ContCont(T0S0=.95, T1S1=.95, T0T0=90, T1T1=100, S0S0=10, S1S1=15,
T0T1=seq(0, 1, by=.2), T0S1=seq(0, 1, by=.2), T1S0=seq(0, 1, by=.2), 
S0S1=seq(0, 1, by=.2))

# Examine and plot the vector of generated ICA values:
summary(SurICA)
plot(SurICA)

# Obtain the positive definite matrices than can be formed as based on the 
# specified (vectors) of the correlations (these matrices are used to 
# compute the ICA values)
SurICA$Pos.Def

# Same, but specify vectors for rho_T0S0 and rho_T1S1: Sample from
# normal with mean .95 and SD=.05 (to account for uncertainty 
# in estimation) 
SurICA2 <- ICA.ContCont(T0S0=rnorm(n=10000000, mean=.95, sd=.05), 
T1S1=rnorm(n=10000000, mean=.95, sd=.05), 
T0T0=90, T1T1=100, S0S0=10, S1S1=15,
T0T1=seq(0, 1, by=.2), T0S1=seq(0, 1, by=.2), T1S0=seq(0, 1, by=.2), 
S0S1=seq(0, 1, by=.2))

# Examine results
summary(SurICA2)
plot(SurICA2)

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