Multivar.PCAContCont function

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

The function Multivar.PCA.ContCont computes the predictive causal association (PCA) when $S$ = the vector of pretreatment predictors and $T$ = the True endpoint. All $S$ and $T$ should be continuous normally distributed endpoints. See Details below.

Multivar.PCA.ContCont(Sigma_TT, Sigma_TS, Sigma_SS, T0T1=seq(-1, 1, by=.01), M=NA)

Arguments

• Sigma_TT: The variance-covariance matrix $\bold{\Sigma}_{TT}=\left(\begin{array}{cc}\sigma_{T0T0} & \sigma_{T0T1} \\ \sigma_{T0T1} & \sigma_{T1T1}\end{array}\right)$.
• Sigma_TS: The matrix that contains the covariances $\sigma_{T0Sr}$, $\sigma_{T1Sr}$. For example, when there are $2$ pretreatment predictors $\bold{\Sigma}_{TS}=\left(\begin{array}{cc}\sigma_{T0S1} & \sigma_{T0S2} \\ \sigma_{T1S1} & \sigma_{T1S2}\end{array}\right)$.
• Sigma_SS: The variance-covariance matrix of the pretreatment predictors. For example, when there are $2$ pretreatment predictors $\bold{\Sigma}_{SS}=\left(\begin{array}{cc}\sigma_{S1S1} & \sigma_{S1S2} \\ \sigma_{S1S2} & \sigma_{S2S2}\end{array}\right)$.
• 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 $R^2_{\psi}$. Default seq(-1, 1, by=.01), i.e., the values $-1$, $-0.99$, $-0.98$, ..., $1$.
• M: If M=NA, all correlation(s) between the counterfactuals $T_0$ and $T_1$ specified in the argument T0T1 are used to compute $R^2_{\psi}$. If M=m, random draws are taken from T0T1 until m $R^2_{\psi}$ are found. Default M=NA.

Returns

An object of class Multivar.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 $R^2_{\psi}$ values.

• PCA: A scalar or vector that contains the PCA ($R^2_{\psi}$) value(s).

• R2_psi_g: A Data.frame that contains $R^2_{\psi g}$.

References

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

Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

Examples

# First specify the covariance matrices to be used 
Sigma_TT = matrix(c(177.870, NA, NA, 162.374), byrow=TRUE, nrow=2)
Sigma_TS = matrix(data = c(-45.140, -109.599, 11.290, -56.542,
-106.897, 20.490), byrow = TRUE, nrow = 2)
Sigma_SS = matrix(data=c(840.564, 73.936, -3.333, 73.936, 357.719,
-30.564, -3.333, -30.564, 95.063), byrow = TRUE, nrow = 3)

# Compute PCA
Results <- Multivar.PCA.ContCont(Sigma_TT = Sigma_TT,
Sigma_TS = Sigma_TS, Sigma_SS = Sigma_SS)

# Evaluate results
summary(Results)
plot(Results)