Computes the surrogate predictive function (SPF) based on sensitivity-analyis, i.e., \(r(i,j)=P(\Delta T=i|\Delta S=j)\), in the setting where both \(S\) and \(T\) are binary endpoints. For example, \(r(-1,1)\) quantifies the probability that the treatment has a negative effect on the true endpoint (\(\Delta T=-1\)) given that it has a positive effect on the surrogate (\(\Delta S=1\)). All quantities of interest are derived from the vectors of 'plausible values' for \(\pi\) (i.e., vectors \(\pi\) that are compatible with the observable data at hand). See Details below.
SPF.BinBin(x)A fitted object of class ICA.BinBin, ICA.BinBin.Grid.Full, or ICA.BinBin.Grid.Sample.
The vector of values for \(r(1, 1)\), i.e., \(P(\Delta T=1|\Delta S=1\)).
The vector of values for \(r(-1, 1)\).
The vector of values for \(r(0, 1)\).
The vector of values for \(r(1, 0)\).
The vector of values for \(r(-1, 0)\).
The vector of values for \(r(0, 0)\).
The vector of values for \(r(1, -1)\).
The vector of values for \(r(-1, -1)\).
The vector of values for \(r(0, -1)\).
The assumption regarding monotonicity under which the result was obtained.
All \(r(i,j)=P(\Delta T=i|\Delta S=j)\) are derived from \(\pi\) (vector of potential outcomes). Denote by \(\bold{Y}'=(T_0,T_1,S_0,S_1)\) the vector of potential outcomes. The vector \(\bold{Y}\) can take 16 values and the set of parameters \(\pi_{ijpq}=P(T_0=i,T_1=j,S_0=p,S_1=q)\) (with \(i,j,p,q=0/1\)) fully characterizes its distribution.
Based on the data and assuming SUTVA, the marginal probabilites \(\pi_{1 \cdot 1 \cdot}\), \(\pi_{1 \cdot 0 \cdot}\), \(\pi_{\cdot 1 \cdot 1}\), \(\pi_{\cdot 1 \cdot 0}\), \(\pi_{0 \cdot 1 \cdot}\), and \(\pi_{\cdot 0 \cdot 1}\) can be computed (by hand or using the function MarginalProbs). Define the vector
$$\bold{b}'=(1, \pi_{1 \cdot 1 \cdot}, \pi_{1 \cdot 0 \cdot}, \pi_{\cdot 1 \cdot 1}, \pi_{\cdot 1 \cdot 0}, \pi_{0 \cdot 1 \cdot}, \pi_{\cdot 0 \cdot 1})$$ and \(\bold{A}\) is a contrast matrix such that the identified restrictions can be written as a system of linear equation
$$\bold{A \pi} = \bold{b}.$$
The matrix \(\bold{A}\) has rank \(7\) and can be partitioned as \(\bold{A=(A_r | A_f)}\), and similarly the vector \(\bold{\pi}\) can be partitioned as \(\bold{\pi^{'}=(\pi_r^{'} | \pi_f^{'})}\) (where \(f\) refers to the submatrix/vector given by the \(9\) last columns/components of \(\bold{A/\pi}\)). Using these partitions the previous system of linear equations can be rewritten as $$\bold{A_r \pi_r + A_f \pi_f = b}.$$
The functions ICA.BinBin, ICA.BinBin.Grid.Sample, and ICA.BinBin.Grid.Full contain algorithms that generate plausible distributions for \(\bold{Y}\) (for details, see the documentation of these functions). Based on the output of these functions, SPF.BinBin computes the surrogate predictive function.
Alonso, A., Van der Elst, W., & Molenberghs, G. (2015). Assessing a surrogate effect predictive value in a causal inference framework.
ICA.BinBin, ICA.BinBin.Grid.Sample, ICA.BinBin.Grid.Full, plot.SPF.BinBin
# NOT RUN {
# Use ICA.BinBin.Grid.Sample to obtain plausible values for pi
ICA_BINBIN_Grid_Sample <- 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("General"), M=2500)
# Obtain SPF
SPF <- SPF.BinBin(ICA_BINBIN_Grid_Sample)
# examine results
summary(SPF)
plot(SPF)
# }
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