PCAContCont function

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

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.

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):

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.

Returns

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 $\delta$.

References

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

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

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