Evaluating principal surrogate markers in vaccine trials in the presence of multiphase sampling

This article focuses on the evaluation of vaccine-induced immune responses as principal surrogate markers for predicting a given vaccine's effect on the clinical endpoint of interest. To address the problem of missing potential outcomes under the principal surrogate framework, we can utilize baseline predictors of the immune biomarker(s) or vaccinate uninfected placebo recipients at the end of the trial and measure their immune biomarkers. Examples of good baseline predictors are baseline immune responses when subjects enrolled in the trial have been previously exposed to the same antigen, as in our motivating application of the Zostavax Efficacy and Safety Trial (ZEST). However, laboratory assays of these baseline predictors are expensive and therefore their subsampling among participants is commonly performed. In this article, we develop a methodology for estimating principal surrogate values in the presence of baseline predictor subsampling. Under a multiphase sampling framework, we propose a semiparametric pseudo-score estimator based on conditional likelihood and also develop several alternative semiparametric pseudo-score or estimated likelihood estimators. We derive corresponding asymptotic theories and analytic variance formulas for these estimators. Through extensive numeric studies, we demonstrate good finite sample performance of these estimators and the efficiency advantage of the proposed pseudo-score estimator in various sampling schemes. We illustrate the application of our proposed estimators using data from an immune biomarker study nested within the ZEST trial.