Penalised likelihood estimation of joint models for partly-interval censored survival and longitudinal data

<p dir="ltr">Studies of long-term or chronic diseases frequently give rise to data where the primary outcome is a time-to-event and where important predictors may be time-varying. For example, in oncology studies, changing treatment regimens or longitudinally-measured biomarkers may be important predictors of clinical outcomes such as death, recurrence, or disease progression. Survival analysis methods accounting for time-varying covariates, including the extended Cox model and joint models, have been widely studied for right censored survival outcomes. A particularly important application of these models has been dynamic prediction, where individual-level survival probabilities are updated as new information becomes available. However, the process of data collection in these studies frequently produces interval censored event times, possibly alongside exact event times, right censoring times, or left censoring times. Interval censoring occurs when the exact timing of the event of interest is unobserved, but it is known to have occurred between two time points, such as two sequential medical assessments. Survival analysis methods that can accommodate both a more complex partly-interval censoring scheme and time-varying covariates have only been given limited consideration in existing statistical literature. In this thesis we address this gap by investigating novel methodologies for fitting Cox models with time-varying covariates and partly-interval censored survival data. Building on the penalised likelihood approach of Ma et al. (2021), we first develop a method to fit an extended Cox model with time-varying covariates observed without measurement error. We then further develop this methodology to the more complex case of joint models, where longitudinal covariates may be subject to measurement error. Our penalised likelihood methodology includes a spline approximation to the baseline hazard function, which is smoothed via the penalty term in the likelihood. We employ a Newton multiplicative-iterative algorithm to estimate model parameters while constraining the spline coefficients (and therefore the baseline hazard estimate) to be non-negative, and automatic smoothing parameter selection is conducted using a restricted maximum likelihood approach. The penalised likelihood is used to derive asymptotic covariance estimates for all model parameters, including the spline coefficients, allowing for statistical inference on both regression coefficients and estimated survival quantities such as the survival function. After developing a penalised likelihood method for joint models under partly-interval censoring, we subsequently investigate methods for the dynamic prediction of individual conditional survival probabilities. We derive a novel likelihood-based method for estimating individual conditional survival probabilities and their variances from a joint model. Additionally, we extend established estimators for assessing the accuracy of dynamic predictions to account for partly-interval censoring. The performance of our proposed penalised likelihood approach to the extended Cox model, the joint model, and dynamic prediction is investigated via extensive simulation studies, including comparisons to existing methods for both right censored and interval censored data. We illustrate the applicability of our proposed methods by analysing data from studies of HIV, breast cancer and metastatic melanoma, highlighting the direct relevance of our methods to clinical research and decision-making.</p>