Multivar.PCA.ContCont: Compute the multivariate predictive causal association (PCA) in the Continuous-continuous case

Description

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.

Usage

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 \(-1\), \(-0.99\), \(-0.98\), …, \(1\).

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.

Value

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.

Examples

Run this code

# NOT RUN {
# 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)
# }

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