Surrogate (version 1.1)

MaxEntICABinBin: Use the maximum-entropy approach to compute ICA in the binary-binary setting

Description

In a surrogate evaluation setting where both \(S\) and \(T\) are binary endpoints, a sensitivity-based approach where multiple 'plausible values' for ICA are retained can be used (see functions ICA.BinBin, ICA.BinBin.Grid.Full, or ICA.BinBin.Grid.Sample). Alternatively, the maximum entropy distribution of the vector of potential outcomes can be considered, based upon which ICA is subsequently computed. The use of the distribution that maximizes the entropy can be justified based on the fact that any other distribution would necessarily (i) assume information that we do not have, or (ii) contradict information that we do have. The function MaxEntICABinBin implements the latter approach.

Usage

MaxEntICABinBin(pi1_1_, pi1_0_, pi_1_1,
pi_1_0, pi0_1_, pi_0_1, Method="BFGS", 
Fitted.ICA=NULL)

Arguments

pi1_1_

A scalar that contains the estimated value for \(P(T=1,S=1|Z=0)\), i.e., the probability that \(S=T=1\) when under treatment \(Z=0\).

pi1_0_

A scalar that contains the estimated value for \(P(T=1,S=0|Z=0)\).

pi_1_1

A scalar that contains the estimated value for \(P(T=1,S=1|Z=1)\).

pi_1_0

A scalar that contains the estimated value for \(P(T=1,S=0|Z=1)\).

pi0_1_

A scalar that contains the estimated value for \(P(T=0,S=1|Z=0)\).

pi_0_1

A scalar that contains the estimated value for \(P(T=0,S=1|Z=1)\).

Method

The maximum entropy frequency vector \(p^{*}\) is calculated based on the optimal solution to an unconstrained dual convex programming problem (for details, see Alonso et al., 2015). Two different optimization methods can be specified, i.e., Method="BFGS" and Method="CG", which implement the quasi-Newton BFGS (Broyden, Fletcher, Goldfarb, and Shanno) and the conjugent gradient (CG) methods (for details on these methods, see the help files of the optim() function and the references theirin). Alternatively, the \(\pi\) vector (obtained when the functions ICA.BinBin, ICA.BinBin.Grid.Full, or ICA.BinBin.Grid.Sample are executed) that is 'closest' to the vector \(\pi\) can be retained. Here, the 'closest' vector is defined as the vector where the sum of the squared differences between the components in the vectors \(\pi\) and \(\pi\) is smallest. The latter 'Minimum Difference' method can re requested by specifying the argument Method="MD" in the function call. Default Method="BFGS".

Fitted.ICA

A fitted object of class ICA.BinBin, ICA.BinBin.Grid.Full, or ICA.BinBin.Grid.Sample. Only required when Method="MD" is used.

Value

R2_H

The R2_H value.

Vector_p

The maximum entropy frequency vector \(p^{*}\)

H_max

The entropy of \(p^{*}\)

References

Alonso, A., & Van der Elst, W. (2015). A maximum-entropy approach for the evluation of surrogate endpoints based on causal inference.

See Also

ICA.BinBin, ICA.BinBin.Grid.Sample, ICA.BinBin.Grid.Full, plot MaxEntICA BinBin

Examples

# NOT RUN {
# Sensitivity-based ICA results using ICA.BinBin.Grid.Sample
ICA <- ICA.BinBin.Grid.Sample(pi1_1_=0.341, pi0_1_=0.119, pi1_0_=0.254,
pi_1_1=0.686, pi_1_0=0.088, pi_0_1=0.078, Seed=1, 
Monotonicity=c("No"), M=5000)

# Maximum-entropy based ICA
MaxEnt <- MaxEntICABinBin(pi1_1_=0.341, pi0_1_=0.119, pi1_0_=0.254,
pi_1_1=0.686, pi_1_0=0.088, pi_0_1=0.078)

# Explore maximum-entropy results
summary(MaxEnt)

# Plot results
plot(x=MaxEnt, ICA.Fit=ICA)
# }