Time‐to‐event data with time‐varying biomarkers measured only at study entry,with applications to Alzheimer's disease

Relating time‐varying biomarkers of Alzheimer's disease to time‐to‐event using a Cox model is complicated by the fact that Alzheimer's disease biomarkers are sparsely collected, typically only at study entry; this is problematic since Cox regression with time‐varying covariates requires observation of the covariate process at all failure times. The analysis might be simplified by using study entry as the time origin and treating the time‐varying covariate measured at study entry as a fixed baseline covariate. In this paper, we first derive conditions under which using an incorrect time origin of study entry results in consistent estimation of regression parameters when the time‐varying covariate is continuous and fully observed. We then derive conditions under which treating the time‐varying covariate as fixed at study entry results in consistent estimation. We provide methods for estimating the regression parameter when a functional form can be assumed for the time‐varying biomarker, which is measured only at study entry. We demonstrate our analytical results in a simulation study and apply our methods to data from the Rush Religious Orders Study and Memory and Aging Project and data from the Alzheimer's Disease Neuroimaging Initiative.