PCA.ContCont {EffectTreat}

Compute the predictive causal association (PCA) in the Continuous-continuous case

Description

The function PCA.ContCont computes the predictive causal association (PCA) when S=pretreatment predictor and T=True endpoint are continuous normally distributed endpoints. See Details below.

Usage

PCA.ContCont(T0S, T1S, T0T0=1, T1T1=1, SS=1, T0T1=seq(-1, 1, by=.01))

Arguments

T0S

A scalar or vector that specifies the correlation(s) between the pretreatment predictor and the true endpoint in the control treatment condition that should be considered in the computation of \rho_{\psi}.

T1S

A scalar or vector that specifies the correlation(s) between the pretreatment predictor and the true endpoint in the experimental treatment condition that should be considered in the computation of \rho_{\psi}.

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_{\psi}. 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_{\psi}. Default 1.

SS

A scalar that specifies the variance of the pretreatment predictor endpoint. Default 1.

T0T1

A scalar or vector that contains the correlation(s) between the counterfactuals T_0 and T_1 that should be considered in the computation of \rho_{\psi}. Default seq(-1, 1, by=.01), i.e., the values -1, -0.99, -0.98, ..., 1.

Details

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

The correlation between the individual causal effect of Z on T and S_{j} (the pretreatment predictor) equals (for details, see Alonso et al., submitted):

\rho_{\psi}=\frac{\sqrt{\sigma_{T1T1}}\rho_{T1S}-\sqrt{\sigma_{T0T0}}\rho_{T0S}}{\sqrt{\sigma_{T0T0}+\sigma_{T1T1}-2\sqrt{\sigma_{T0T0}\sigma_{T1T1}}}\rho_{T0T1}},

where the correlation \rho_{T_{0}T_{1}} is 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 \rho_{T_{0}T_{1}} in the above expression, the function PCA.ContCont constructs all possible matrices that can be formed as based on these values and the estimable quantities \rho_{T_{0}S}, \rho_{T_{1}S}, identifies the matrices that are positive definite (i.e., valid correlation matrices), and computes \rho_{\psi} for each of these matrices. The obtained vector of \rho_{\psi} values can subsequently be used to e.g., conduct a sensitivity analysis.

Notes

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

Value

An object of class PCA.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_{\psi} values.

PCA

A scalar or vector that contains the PCA (\rho_{\psi}) value(s).

GoodSurr

A data.frame that contains the PCA (\rho_{\psi}), \sigma_{\psi_{T}}, and \delta.

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

References

Alonso, A., Van der Elst, W., & Molenberghs, G. (submitted). Validating predictors of therapeutic success: a causal inference approach.

Examples

# Based on the example dataset
    # load data in memory
data(Example.Data)
    # compute corr(S, T) in control treatment, gives .77
cor(Example.Data$S[Example.Data$Treat==-1], 
Example.Data$T[Example.Data$Treat==-1])
   # compute corr(S, T) in experimental treatment, gives .71
cor(Example.Data$S[Example.Data$Treat==1], 
Example.Data$T[Example.Data$Treat==1])
   # compute var T in control treatment, gives 263.99 
var(Example.Data$T[Example.Data$Treat==-1])
   # compute var T in experimental treatment, gives 230.64  
var(Example.Data$T[Example.Data$Treat==1])
   # compute var S, gives 163.65   
var(Example.Data$S)

# Generate the vector of PCA.ContCont values using these estimates 
# and the grid of values {-1, -.99, ..., 1} for the correlations
# between T0 and T1:
PCA <- PCA.ContCont(T0S=.77, T1S=.71, T0T0=263.99, T1T1=230.65, 
                    SS=163.65, T0T1=seq(-1, 1, by=.01))

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


# Other example

# Generate the vector of PCA.ContCont values when rho_T0S=.3, rho_T1S=.9, 
# sigma_T0T0=2, sigma_T1T1=2,sigma_SS=2, and  
# the grid of values {-1, -.99, ..., 1} is considered for the correlations
# between T0 and T1:
PCA <- PCA.ContCont(T0S=.3, T1S=.9, T0T0=2, T1T1=2, SS=2, 
T0T1=seq(-1, 1, by=.01))

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

# 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 PCA values)
PCA$Pos.Def