Monte Carlo Computation of the Parameter \(\theta\) from Carlotti and Parast (2026)

This function implements a Monte Carlo approach to estimate the parameter \(\theta\) from Carlotti and Parast (2026) . This parameter represents the difference in treatment effects between the primary and surrogate outcomes, both measured using the probability that the treated outcome is larger than the control outcome.

Usage

compute_theta(MC_data)

Arguments

MC_data

A list containing:

• P: A matrix or data frame of potential outcomes with columns "Y1", "Y0", "S1", and "S0".

Value

A list containing the true values:

• V_Y: The Monte Carlo estimate of \(P(Y_{1i} > Y_{0i})\) computed on P.

• V_S: The Monte Carlo estimate of \(P(S_{1i} > S_{0i})\) computed on P.

• theta: The difference V_Y - V_S.

Details

The function processes data from a chosen data generating process, computing the sample probabilities for both the primary outcome \(Y\) and the surrogate \(S\): $$\hat{V}_Y = \frac{1}{n} \sum\limits^{n}_{i=1} I(Y_{1i} > Y_{0i}),$$ $$\hat{V}_S = \frac{1}{n} \sum\limits^{n}_{i=1} I(S_{1i} > S_{0i}).$$ Then, it calculates $$\hat{\theta} = \hat{V}_Y - \hat{V}_S.$$ This function is generally not intended to be called directly by the user and is instead used internally within BSET_no_X and BSET_X.

References

Carlotti P, Parast L (2026). “A Bayesian Critique of Rank-Based Methods for Surrogate Marker Evaluation.” arXiv preprint arXiv:2603.14381.