Multivar.PCAContCont: Compute the multivariate predictive causal association (PCA)...
In EffectTreat: Prediction of Therapeutic Success

Description Usage Arguments Value Author(s) References Examples

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.

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

`Sigma_TT`	The variance-covariance matrix \bold{Σ}_{TT}=≤ft(\begin{array}{cc}σ_{T0T0} & σ_{T0T1} \\ σ_{T0T1} & σ_{T1T1}\end{array}\right).
`Sigma_TS`	The matrix that contains the covariances σ_{T0Sr}, σ_{T1Sr}. For example, when there are 2 pretreatment predictors \bold{Σ}_{TS}=≤ft(\begin{array}{cc}σ_{T0S1} & σ_{T0S2} \\ σ_{T1S1} & σ_{T1S2}\end{array}\right).
`Sigma_SS`	The variance-covariance matrix of the pretreatment predictors. For example, when there are 2 pretreatment predictors \bold{Σ}_{SS}=≤ft(\begin{array}{cc}σ_{S1S1} & σ_{S1S2} \\ σ_{S1S2} & σ_{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_{ψ}. 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_{ψ}. If `M=m`, random draws are taken from `T0T1` until `m` R^2_{ψ} are found. Default `M=NA`.

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_{ψ} values.
`PCA`	A scalar or vector that contains the PCA (R^2_{ψ}) value(s).
`R2_psi_g`	A `Data.frame` that contains R^2_{ψ g}.

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

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

# 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)