MICA.ContCont: Assess surrogacy in the causal-inference multiple-trial setting (Meta-analytic Individual Causal Association; MICA) in the continuous-continuous case

Description

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

Usage

MICA.ContCont(Trial.R, D.aa, D.bb, 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))

Arguments

Trial.R

A scalar that specifies the trial-level correlation coefficient (i.e., $R_{trial}$) that should be used in the computation of $\rho_{M}$.

D.aa

A scalar that specifies the between-trial variance of the treatment effects on the surrogate endpoint (i.e., $d_{aa}$) that should be used in the computation of $\rho_{M}$.

D.bb

A scalar that specifies the between-trial variance of the treatment effects on the true endpoint (i.e., $d_{bb}$) that should be used in the computation of $\rho_{M}$.

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 $\rho_{M}$.

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 $\rho_{M}$.

T0T0

A scalar that specifies the variance of the true endpoint in the control treatment condition that should be considered in the computation of $\rho_{M}$. 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 $\rho_{M}$. 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 $\rho_{M}$. 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 $\rho_{M}$. 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 $\rho_{M}$. 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 $\rho_{M}$. 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 $\rho_{M}$. 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 $\rho_{M}$. Default seq(-1, 1, by=.1).

Value

An object of class MICA.ContCont with components,

Total.Num.Matrices

An object of class numeric which contains the total number of matrices that can be formed as based on the user-specified correlations.

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 $\rho_{M}$ values.

ICA

A scalar or vector of the $\rho_{\Delta}$ values.

MICA

A scalar or vector of the $\rho_{M}$ values.

Warning

The theory that relates the causal-inference and the meta-analytic frameworks in the multiple-trial setting (as developped in Alonso et al., submitted) assumes that a reduced or semi-reduced modelling approach is used in the meta-analytic framework. Thus $R_{trial}$, $d_{aa}$ and $d_{bb}$ should be estimated based on a reduced model (i.e., using the Model=c("Reduced") argument in the functions UnifixedContCont, UnimixedContCont, BifixedContCont, or BimixedContCont) or based on a semi-reduced model (i.e., using the Model=c("SemiReduced") argument in the functions UnifixedContCont, UnimixedContCont, or BifixedContCont).

Details

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

In the multiple-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):

$$\rho_{M}=\rho(\Delta_{Tij},\:\Delta_{Sij})=\frac{\sqrt{d_{bb}d_{aa}}R_{trial}+\sqrt{V(\varepsilon_{\Delta Tij})V(\varepsilon_{\Delta Sij})}\rho_{\Delta}}{\sqrt{V(\Delta_{Tij})V(\Delta_{Sij})}},$$

where

$$V(\varepsilon_{\Delta Tij})=\sigma_{T_{0}T_{0}}+\sigma_{T_{1}T_{1}}-2\sqrt{\sigma_{T_{0}T_{0}}\sigma_{T_{1}T_{1}}}\rho_{T_{0}T_{1}},$$ $$V(\varepsilon_{\Delta Sij})=\sigma_{S_{0}S_{0}}+\sigma_{S_{1}S_{1}}-2\sqrt{\sigma_{S_{0}S_{0}}\sigma_{S_{1}S_{1}}}\rho_{S_{0}S_{1}},$$ $$V(\Delta_{Tij})=d_{bb}+\sigma_{T_{0}T_{0}}+\sigma_{T_{1}T_{1}}-2\sqrt{\sigma_{T_{0}T_{0}}\sigma_{T_{1}T_{1}}}\rho_{T_{0}T_{1}},$$ $$V(\Delta_{Sij})=d_{aa}+\sigma_{S_{0}S_{0}}+\sigma_{S_{1}S_{1}}-2\sqrt{\sigma_{S_{0}S_{0}}\sigma_{S_{1}S_{1}}}\rho_{S_{0}S_{1}}.$$

The correlations between the counterfactuals (i.e., $\rho_{S_{0}T_{1}}$, $\rho_{S_{1}T_{0}}$, $\rho_{T_{0}T_{1}}$, and $\rho_{S_{0}S_{1}}$) are not identifiable from the data. 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 correlations that are involved in the computation of $\rho_{M}$, the function MICA.ContCont constructs all possible matrices that can be formed as based on the specified values, identifies the matrices that are positive definite (i.e., valid correlation matrices), and computes $\rho_{M}$ for each of these matrices. An examination of the vector of the obtained $\rho_{M}$ values allows for a straightforward examination of the impact of different assumptions regarding the correlations between the counterfactuals on the results (see also plot Causal-Inference ContCont), and 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.

Notes

A single $\rho_{M}$ 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

# NOT RUN {
 #time-consuming code parts
# Generate the vector of MICA values when R_trial=.8, rho_T0S0=rho_T1S1=.8,
# sigma_T0T0=90, sigma_T1T1=100,sigma_ S0S0=10, sigma_S1S1=15, D.aa=5, D.bb=10,  
# and when the grid of values {0, .2, ..., 1} is considered for the 
# correlations between the counterfactuals:
SurMICA <- MICA.ContCont(Trial.R=.80, D.aa=5, D.bb=10, T0S0=.8, T1S1=.8,
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 the generated MICA values:
summary(SurMICA)
plot(SurMICA)


# Same analysis, but now assume that D.aa=.5 and D.bb=.1:
SurMICA <- MICA.ContCont(Trial.R=.80, D.aa=.5, D.bb=.1, T0S0=.8, T1S1=.8,
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 the generated MICA values:
summary(SurMICA)
plot(SurMICA)


# Same as first analysis, but specify vectors for rho_T0S0 and rho_T1S1: 
# Sample from normal with mean .8 and SD=.1 (to account for uncertainty 
# in estimation) 
SurMICA <- MICA.ContCont(Trial.R=.80, D.aa=5, D.bb=10, 
T0S0=rnorm(n=10000000, mean=.8, sd=.1), 
T1S1=rnorm(n=10000000, mean=.8, sd=.1),
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))
# }

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