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.
PPE.BinBin(pi1_1_, pi1_0_, pi_1_1, pi_1_0,
pi0_1_, pi_0_1, M=10000, Seed=1)An object of class PPE.BinBin with components,
count variable
The vector of the PPE values.
The vector of the RPE values.
The vector of the \(PPE_T\) values indicating the probability on a prediction error without using information on \(S\).
The vector of the \(R_H^2\) values.
The vector of the entropies of \(\Delta_T\).
The vector of the entropies of \(\Delta_S\).
The vector of the mutual information of \(\Delta_S\) and \(\Delta_T\).
A scalar that contains values for \(P(T=1,S=1|Z=0)\), i.e., the probability that \(S=T=1\) when under treatment \(Z=0\).
A scalar that contains values for \(P(T=1,S=0|Z=0)\).
A scalar that contains values for \(P(T=1,S=1|Z=1)\).
A scalar that contains values for \(P(T=1,S=0|Z=1)\).
A scalar that contains values for \(P(T=0,S=1|Z=0)\).
A scalar that contains values for \(P(T=0,S=1|Z=1)\).
The number of valid vectors that have to be obtained. Default M=10000.
The seed to be used to generate \(\pi_r\). Default Seed=1.
Paul Meyvisch, Wim Van der Elst, Ariel Alonso, Geert Molenberghs
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.
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.
ICA.BinBin.Grid.Sample
# Conduct the analysis
if (FALSE) # 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)
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