SPF.BinBin {Surrogate}

Evaluate the surrogate predictive function (SPF) in the binary-binary setting (sensitivity-analysis based approach)

Description

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.

Usage

SPF.BinBin(x)

Arguments

x

A fitted object of class ICA.BinBin, ICA.BinBin.Grid.Full, or ICA.BinBin.Grid.Sample.

Details

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.

Value

r_1_1

The vector of values for r(1, 1), i.e., P(\Delta T=1|\Delta S=1).

r_min1_1

The vector of values for r(-1, 1).

r_0_1

The vector of values for r(0, 1).

r_1_0

The vector of values for r(1, 0).

r_min1_0

The vector of values for r(-1, 0).

r_0_0

The vector of values for r(0, 0).

r_1_min1

The vector of values for r(1, -1).

r_min1_min1

The vector of values for r(-1, -1).

r_0_min1

The vector of values for r(0, -1).

Monotonicity

The assumption regarding monotonicity under which the result was obtained.

Author(s)

Wim Van der Elst, Paul Meyvisch, Ariel Alonso, & Geert Molenberghs

References

Alonso, A., Van der Elst, W., & Molenberghs, G. (2015). Assessing a surrogate effect predictive value in a causal inference framework.

See Also

ICA.BinBin, ICA.BinBin.Grid.Sample, ICA.BinBin.Grid.Full, plot.SPF.BinBin

Examples

# 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)