Fano.BinBin: Evaluate the possibility of finding a good surrogate in the setting where both \(S\) and \(T\) are binary endpoints

Description

The function Fano.BinBin evaluates the existence of a good surrogate in the single-trial causal-inference framework when both the surrogate and the true endpoints are binary outcomes. See Details below.

Usage

Fano.BinBin(pi1_,  pi_1, rangepi10=c(0,min(pi1_,1-pi_1)), 
fano_delta=c(0.1), M=100, Seed=1)

Arguments

pi1_

A scalar or a vector of plausibel values that represents the proportion of responders under treatment.

pi_1

A scalar or a vector of plausibel values that represents the proportion of responders under control.

rangepi10

Represents the range from which \(\pi_{10}\) is sampled. By default, Monte Carlo simulation will be constrained to the interval \([0,\min(\pi_{1\cdot},\pi_{\cdot0})]\) but this allows the user to specify a more narrow range. rangepi10=c(0,0) is equivalent to the assumption of monotonicity for the true endpoint.

fano_delta

A scalar or a vector that specifies the values for the upper bound of the prediction error \(\delta\). Default fano_delta=c(0.2).

The number of random samples that have to be drawn for the freely varying parameter \(\pi_{10}\). Default M=1000. The number of random samples should be sufficiently large in relation to the length of the interval rangepi10. Typically M=1000 yields a sufficiently fine grid. In case rangepi10 is a single value: M=1

Seed

The seed to be used to sample the freely varying parameter \(\pi_{10}\). Default Seed=1.

Value

An object of class Fano.BinBin with components,

R2_HL

The sampled values for \(R^2_{HL}\).

H_Delta_T

The sampled values for \(H{\Delta T}\).

PPE_T

The sampled values for \(PPE_{T}\).

minpi10

The minimum value for \(\pi_{10}\).

maxpi10

The maximum value for \(\pi_{10}\).

samplepi10

The sampled value for \(\pi_{10}\).

delta

The specified vector of upper bounds for the prediction errors.

uncertainty

Indexes the sampling of \(pi1\_\).

pi_00

The sampled values for \(\pi_{00}\).

pi_11

The sampled values for \(\pi_{11}\).

pi_01

The sampled values for \(\pi_{01}\).

pi_10

The sampled values for \(\pi_{10}\).

Details

Values for \(\pi_{10}\) have to be uniformly sampled from the interval \([0,\min(\pi_{1\cdot},\pi_{\cdot0})]\). Any sampled value for \(\pi_{10}\) will fully determine the bivariate distribution of potential outcomes for the true endpoint. The treatment effect should be positive.

The vector \(\bold{\pi_{km}}\) fully determines \(R^2_{HL}\).

References

Alonso, A., Van der Elst, W., & Molenberghs, G. (2014). Validation of surrogate endpoints: the binary-binary setting from a causal inference perspective.

Examples

Run this code

# NOT RUN {
# Conduct the analysis assuming no montonicity
# for the true endpoint, using a range of
# upper bounds for prediction errors 
Fano.BinBin(pi1_ = 0.5951 ,  pi_1 = 0.7745, 
fano_delta=c(0.05, 0.1, 0.2), M=1000)


# Conduct the same analysis now sampling from
# a range of values to allow for uncertainty

Fano.BinBin(pi1_ = runif(n=20,min=0.504,max=0.681), 
pi_1 = runif(n=20,min=0.679,max=0.849), 
fano_delta=c(0.05, 0.1, 0.2), M=10, Seed=2)
# }

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