The function ICA.BinBin.Grid.Sample 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.Full. It uses an alternative strategy to identify plausible values for \(\pi\). See Details below.
ICA.BinBin.Grid.Sample(pi1_1_, pi1_0_, pi_1_1, pi_1_0, pi0_1_,
pi_0_1, Monotonicity=c("General"), M=100000,
Volume.Perc=0, Seed=sample(1:100000, size=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 \(Z=0\).
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 \(P(T=0,S=1|Z=1)\).
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").
The number of random samples that have to be drawn for the freely varying parameters. Default M=100000. This argument is not used when Volume.Perc=0. Default M=10000.
Note that the marginals that are observable in the data set a number of restrictions on the unidentified correlations. For example, under montonicity for \(S\) and \(T\), it holds that \(\pi_{0111}<=min(\pi_{0\cdot1\cdot}, \pi_{\cdot1\cdot1})\) and \(\pi_{1100}<=min(\pi_{1\cdot0\cdot}, \pi_{\cdot1\cdot0})\). For example, when \(min(\pi_{0\cdot1\cdot}, \pi_{\cdot1\cdot1})=0.10\) and \(min(\pi_{1\cdot0\cdot}, \pi_{\cdot1\cdot0})=0.08\), then all valid \(\pi_{0111}<=0.10\) and all valid \(\pi_{1100}<=0.08\). The argument Volume.Perc specifies the fraction of the 'volume' of the paramater space that is explored. This volume is computed based on the grids G=0, 0.01, ..., maximum possible value for the counterfactual probability at hand. E.g., in the previous example, the 'volume' of the parameter space would be \(11*9=99\), and when e.g., the argument Volume.Perc=1 is used a total of \(99\) runs will be conducted. Notice that when monotonicity is not assumed, relatively high values of Volume.Perc will lead to a large number of runs and consequently a long analysis time.
The seed to be used to generate \(\pi_r\). Default M=100000.
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 \(\Delta_T\).
The 'volume' of the parameter space when monotonicity is not assumed.
The 'volume' of the parameter space when monotonicity for \(T\) is assumed.
The 'volume' of the parameter space when monotonicity for \(S\) is assumed.
The 'volume' of the parameter space when monotonicity for \(S\) and \(T\) is assumed.
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 number of possible combinations become very high. The function ICA.BinBin.Grid.Sample considers a random sample of all possible combinations.
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 parts
# 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.Sample(pi1_1_=0.261, pi1_0_=0.285,
pi_1_1=0.637, pi_1_0=0.078, pi0_1_=0.134, pi_0_1=0.127,
Monotonicity=c("Surr.True.Endp"), M=2500, Seed=1)
# obtain plot of R2_H
plot(ICA, R2_H=TRUE)
# }
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