The function MICA.Sample.ContCont quantifies surrogacy in the multiple-trial causal-inference framework. It provides a faster alternative for MICA.ContCont. See Details below.
MICA.Sample.ContCont(Trial.R, D.aa, D.bb, T0S0, T1S1, T0T0=1, T1T1=1, S0S0=1, S1S1=1,
T0T1=seq(-1, 1, by=.001), T0S1=seq(-1, 1, by=.001), T1S0=seq(-1, 1, by=.001),
S0S1=seq(-1, 1, by=.001), M=50000)A scalar that specifies the trial-level correlation coefficient (i.e., \(R_{trial}\)) that should be used in the computation of \(\rho_{M}\).
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 \(\rho_{M}\).
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 \(\rho_{M}\).
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 \(\rho_{M}\).
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 \(\rho_{M}\).
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.
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.
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.
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.
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=.001).
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=.001).
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=.001).
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=.001).
The number of runs that should be conducted. Default 50000.
An object of class MICA.ContCont with components,
An object of class numeric which contains the total number of matrices that can be formed as based on the user-specified correlations.
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.
A scalar or vector of the \(\rho_{\Delta}\) values.
A scalar or vector of the \(\rho_{M}\) values.
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).
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, and retains the positive definite ones for the computation of \(\rho_{M}\).
In contrast, the function MICA.Sample.ContCont samples random values for \(\rho_{S_{0}T_{1}}\), \(\rho_{S_{1}T_{0}}\), \(\rho_{T_{0}T_{1}}\), and \(\rho_{S_{0}S_{1}}\) based on a uniform distribution with user-specified minimum and maximum values, and retains the positive definite ones for the computation of \(\rho_{M}\).
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.
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.
ICA.ContCont, MICA.ContCont, plot Causal-Inference ContCont, UnifixedContCont, UnimixedContCont, BifixedContCont, BimixedContCont
# NOT RUN {
#Time consuming (>5 sec) code part
# 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 {-1, -0.999, ..., 1} is considered for the
# correlations between the counterfactuals:
SurMICA <- MICA.Sample.ContCont(Trial.R=.80, D.aa=5, D.bb=10, T0S0=.8, T1S1=.8,
T0T0=90, T1T1=100, S0S0=10, S1S1=15, T0T1=seq(-1, 1, by=.001),
T0S1=seq(-1, 1, by=.001), T1S0=seq(-1, 1, by=.001),
S0S1=seq(-1, 1, by=.001), M=10000)
# Examine and plot the vector of the generated MICA values:
summary(SurMICA)
plot(SurMICA, ICA=FALSE, MICA=TRUE)
# Same analysis, but now assume that D.aa=.5 and D.bb=.1:
SurMICA <- MICA.Sample.ContCont(Trial.R=.80, D.aa=.5, D.bb=.1, T0S0=.8, T1S1=.8,
T0T0=90, T1T1=100, S0S0=10, S1S1=15, T0T1=seq(-1, 1, by=.001),
T0S1=seq(-1, 1, by=.001), T1S0=seq(-1, 1, by=.001),
S0S1=seq(-1, 1, by=.001), M=10000)
# Examine and plot the vector of the generated MICA values:
summary(SurMICA)
plot(SurMICA)
# }
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