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)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.
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.
Wim Van der Elst, Ariel Alonso, & Geert Molenberghs
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
if (FALSE) #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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