A non‐linear model for censored and mismeasured time varying covariates in survival models, with applications in human immunodeficiency virus and acquired immune deficiency syndrome studies

In survival regression analysis, when the time‐dependent covariates are censored and measured with errors, a joint model is often considered for the longitudinal covariate data and the survival data. Typically, an empirical linear (mixed) model is assumed for the time‐dependent covariates. However, such an empirical linear covariate model may be inappropriate for the (unobserved) censored covariate values that may behave quite differently from the observed covariate process. In applications such as human immunodeficiency virus–acquired immune deficiency syndrome studies, a mechanistic non‐linear model can be derived for the covariate process on the basis of the underlying data generation mechanisms and such a non‐linear covariate model may provide better ‘predictions’ for the censored and mismeasured covariate values. We propose a joint Cox and non‐linear mixed effect model to model survival data with censored and mismeasured time varying covariates. We use likelihood methods for inference, implemented by the Monte Carlo EM algorithm. The models and methods are evaluated by simulations. An acquired immune deficiency syndrome data set is analysed in detail, where the time‐dependent covariate is a viral load which may be censored because of a lower detection limit and may also be measured with errors. The results based on linear and non‐linear covariate models are compared and new insights are gained.