The function ICA.BinBin.Grid.Full quantifies surrogacy in the single-trial causal-inference framework (individual causal association and causal concordance) when both the surrogate and the true endpoints are binary outcomes. This method provides an alternative for ICA.BinBin and ICA.BinBin.Grid.Sample. It uses an alternative strategy to identify plausible values for \(\pi\). See Details below.
ICA.BinBin.Grid.Full(pi1_1_, pi1_0_, pi_1_1, pi_1_0, pi0_1_, pi_0_1,
Monotonicity=c("General"), pi_1001=seq(0, 1, by=.02),
pi_1110=seq(0, 1, by=.02), pi_1101=seq(0, 1, by=.02),
pi_1011=seq(0, 1, by=.02), pi_1111=seq(0, 1, by=.02),
pi_0110=seq(0, 1, by=.02), pi_0011=seq(0, 1, by=.02),
pi_0111=seq(0, 1, by=.02), pi_1100=seq(0, 1, by=.02),
Seed=sample(1:100000, size=1))A scalar that contains \(P(T=1,S=1|Z=0)\), i.e., the proability that \(S=T=1\) when under treatment \(Z=0\).
A scalar that contains \(P(T=1,S=0|Z=0)\).
A scalar that contains \(P(T=1,S=1|Z=1)\).
A scalar that contains \(P(T=1,S=0|Z=1)\).
A scalar that contains \(P(T=0,S=1|Z=0)\).
A scalar that contains \(P(T=0,S=1|Z=1)\).
Specifies which assumptions regarding monotonicity should be made: Monotonicity=c("General"), Monotonicity=c("No"), Monotonicity=c("True.Endp"), Monotonicity=c("Surr.Endp"), or Monotonicity=c("Surr.True.Endp"). When a general analysis is requested (using Monotonicity=c("General") in the function call), all settings are considered (no monotonicity, monotonicity for \(S\) alone, for \(T\) alone, and for both for \(S\) and \(T\). Default Monotonicity=c("General").
A vector that specifies the grid of values that should be considered for \(\pi_{pi_1001}\). Default pi_1001=seq(0, 1, by=.02).
A vector that specifies the grid of values that should be considered for \(\pi_{pi_1110}\). Default pi_1110=seq(0, 1, by=.02).
A vector that specifies the grid of values that should be considered for \(\pi_{pi_1101}\). Default pi_1101=seq(0, 1, by=.02).
A vector that specifies the grid of values that should be considered for \(\pi_{pi_1011}\). Default pi_1011=seq(0, 1, by=.02).
A vector that specifies the grid of values that should be considered for \(\pi_{pi_1111}\). Default pi_1111=seq(0, 1, by=.02).
A vector that specifies the grid of values that should be considered for \(\pi_{pi_0110}\). Default pi_0110=seq(0, 1, by=.02).
A vector that specifies the grid of values that should be considered for \(\pi_{pi_0011}\). Default pi_0011=seq(0, 1, by=.02).
A vector that specifies the grid of values that should be considered for \(\pi_{pi_0111}\). Default pi_0111=seq(0, 1, by=.02).
A vector that specifies the grid of values that should be considered for \(\pi_{pi_1100}\). Default pi_1100=seq(0, 1, by=.02).
The seed to be used to generate \(\pi_r\). Default Seed=sample(1:100000, size=1).
An object of class ICA.BinBin with components,
An object of class data.frame that contains the valid \(\pi\) vectors.
The vector of the \(R_H^2\) values.
The vector of odds ratios for \(T\).
The vector of odds ratios for \(S\).
The vector of the entropies of \(\Delta_T\).
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. (2014) 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 ICA.BinBin.Grid.Full computes \(R_{H}^{2}\) using a grid-based approach where all possible combinations of the specified grids for the parameters that are allowed that are allowed to vary freely are considered. When it is not assumed that monotonicity holds for both \(S\) and \(T\), the computationally less demanding algorithm ICA.BinBin.Grid.Sample may be preferred.
Alonso, A., Van der Elst, W., & Molenberghs, G. (2014). Validation of surrogate endpoints: the binary-binary setting from a causal inference perspective.
Buyse, M., Burzykowski, T., Aloso, A., & Molenberghs, G. (2014). Direct estimation of joint counterfactual probabilities, with application to surrogate marker validation.
ICA.ContCont, MICA.ContCont, ICA.BinBin, ICA.BinBin.Grid.Sample
# NOT RUN {
# time consuming code part
# Compute R2_H given the marginals,
# assuming monotonicity for S and T and grids
# pi_0111=seq(0, 1, by=.001) and
# pi_1100=seq(0, 1, by=.001)
ICA <- ICA.BinBin.Grid.Full(pi1_1_=0.2619048, pi1_0_=0.2857143, pi_1_1=0.6372549,
pi_1_0=0.07843137, pi0_1_=0.1349206, pi_0_1=0.127451,
pi_0111=seq(0, 1, by=.01), pi_1100=seq(0, 1, by=.01), Seed=1)
# obtain plot of R2_H
plot(ICA, R2_H=TRUE)
# }
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