Surrogate (version 1.7)

PPE.BinBin: Evaluate a surrogate predictive value based on the minimum probability of a prediction error in the setting where both \(S\) and \(T\) are binary endpoints

Description

The function PPE.BinBin assesses a surrogate predictive value using the probability of a prediction error in the single-trial causal-inference framework when both the surrogate and the true endpoints are binary outcomes. It additionally assesses the indivdiual causal association (ICA). See Details below.

Usage

PPE.BinBin(pi1_1_, pi1_0_, pi_1_1, pi_1_0, 
pi0_1_, pi_0_1, M=10000, Seed=1)

Arguments

pi1_1_

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

pi_1_1

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

pi_1_0

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

pi0_1_

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

pi_0_1

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

M

The number of valid vectors that have to be obtained. Default M=10000.

Seed

The seed to be used to generate \(\pi_r\). Default Seed=1.

Value

An object of class PPE.BinBin with components,

index

count variable

PPE

The vector of the PPE values.

RPE

The vector of the RPE values.

PPE_T

The vector of the \(PPE_T\) values indicating the probability on a prediction error without using information on \(S\).

R2_H

The vector of the \(R_H^2\) values.

H_Delta_T

The vector of the entropies of \(\Delta_T\).

H_Delta_S

The vector of the entropies of \(\Delta_S\).

I_Delta_T_Delta_S

The vector of the mutual information of \(\Delta_S\) and \(\Delta_T\).

Details

In the continuous normal setting, surroagacy can be assessed by studying the association between the individual causal effects on \(S\) and \(T\) (see ICA.ContCont). In that setting, the Pearson correlation is the obvious measure of association.

When \(S\) and \(T\) are binary endpoints, multiple alternatives exist. Alonso et al. (2016) proposed the individual causal association (ICA; \(R_{H}^{2}\)), which captures the association between the individual causal effects of the treatment on \(S\) (\(\Delta_S\)) and \(T\) (\(\Delta_T\)) using information-theoretic principles.

The function PPE.BinBin computes \(R_{H}^{2}\) using a grid-based approach where all possible combinations of the specified grids for the parameters that are allowed to vary freely are considered. It additionally computes the minimal probability of a prediction error (PPE) and the reduction on the PPE using information that \(S\) conveys on \(T\). Both measures provide complementary information over the \(R_{H}^{2}\) and facilitate more straightforward clinical interpretation. No assumption about monotonicity can be made.

References

Alonso A, Van der Elst W, Molenberghs G, Buyse M and Burzykowski T. (2016). An information-theoretic approach for the evaluation of surrogate endpoints based on causal inference.

Meyvisch P., Alonso A.,Van der Elst W, Molenberghs G. (2018). Assessing the predictive value of a binary surrogate for a binary true endpoint, based on the minimum probability of a prediction error.

See Also

ICA.BinBin.Grid.Sample

Examples

# NOT RUN {
# Conduct the analysis 
 
# }
# NOT RUN {
 # time consuming code part
PPE.BinBin(pi1_1_=0.4215, pi0_1_=0.0538, pi1_0_=0.0538,
           pi_1_1=0.5088, pi_1_0=0.0307,pi_0_1=0.0482, 
           Seed=1, M=10000) 
# }
# NOT RUN {
# }