Bivariate Quantile Residual Life Functions for Estimating Immunological and Virological Recovery in HIV Patients

Monitoring both immunological and virological outcome measures provides the most accurate and meaningful assessment of the human immunodeficiency virus (HIV) treatment effectiveness. However, evaluating treatment efficacy based solely on either measure, using commonly employed survival and hazard functions, can introduce bias. In addition, heterogeneity in treatment-related recovery often results in skewed immunological and virological data. To address these challenges, we extend traditional survival analysis to the bivariate setting by introducing the bivariate median residual life function (MeRL) as a robust alternative for assessing treatment efficacy. In this framework, treatment efficacy is evaluated using the bivariate vector of time to immunological restoration and virological suppression. Unlike the mean residual life, the MeRL is less influenced by outliers and heavy-tailed distributions, which are frequently encountered in clinical studies. This paper proposes novel estimators for the bivariate MeRL function under order restrictions, and establishes their strong consistency and asymptotic distributions. Simulation studies were conducted to evaluate the performance of the proposed estimators. Finally, we demonstrated the practical utility of our approach through an application to real-world bivariate HIV data, offering valuable insights for both researchers and healthcare practitioners.