PCA.ContCont: 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.

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 $-1$, $-0.99$, $-0.98$, …, $1$.

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 $\delta$.

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.

References

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

Examples

Run this code

# NOT RUN {
# 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
# }

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