A New Frequentist Implementation of the Daniels and Hughes Bivariate Meta-Analysis Model for Surrogate Endpoint Evaluation

Surrogate endpoints are used when the primary outcome is difficult to measure accurately. Determining if a measure is suitable to use as a surrogate endpoint is a challenging task and a variety of meta-analysis models have been proposed for this purpose. The Daniels and Hughes bivariate model for trial-level surrogate endpoint evaluation is gaining traction but presents difficulties for frequentist estimation and hitherto only Bayesian solutions have been available. This is because the marginal model is not a conventional linear model and the number of unknown parameters increases at the same rate as the number of studies. This second property raises immediate concerns that the maximum likelihood estimator of the model's unknown variance component may be downwardly biased. We derive maximum likelihood estimating equations to motivate a bias adjusted estimator of this parameter. The bias correction terms in our proposed estimating equation are easily computed and have an intuitively appealing algebraic form. A simulation study is performed to illustrate how this estimator overcomes the difficulties associated with maximum likelihood estimation. We illustrate our methods using two contrasting examples from oncology.