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

Description Usage Arguments Details Value Author(s) References Examples

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.

1	PCA.ContCont(T0S, T1S, T0T0=1, T1T1=1, SS=1, T0T1=seq(-1, 1, by=.01))

`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 ρ_{ψ}.
`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 ρ_{ψ}.
`T0T0`	A scalar that specifies the variance of the true endpoint in the control treatment condition that should be considered in the computation of ρ_{ψ}. 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 ρ_{ψ}. 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 ρ_{ψ}. Default `seq(-1, 1, by=.01)`, i.e., the values -1, -0.99, -0.98, ..., 1.

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 Δ_{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):

ρ_{ψ}=\frac{√{σ_{T1T1}}ρ_{T1S}-√{σ_{T0T0}}ρ_{T0S}}{√{σ_{T0T0}+σ_{T1T1}-2√{σ_{T0T0}σ_{T1T1}}}ρ_{T0T1}},

where the correlation ρ_{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 ρ_{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 ρ_{T_{0}S}, ρ_{T_{1}S}, identifies the matrices that are positive definite (i.e., valid correlation matrices), and computes ρ_{ψ} for each of these matrices. The obtained vector of ρ_{ψ} values can subsequently be used to e.g., conduct a sensitivity analysis.

Notes

A single ρ_{ψ} value is obtained when all correlations in the function call are scalars.

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 ρ_{ψ} values.
`PCA`	A scalar or vector that contains the PCA (ρ_{ψ}) value(s).
`GoodSurr`	A `data.frame` that contains the PCA (ρ_{ψ}), σ_{ψ_{T}}, and δ.

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

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

# 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