Varying coefficient frailty models with applications in single molecular experiments

Motivated by an analysis of single molecular experiments in the study of T‐cell signaling, a new model called varying coefficient frailty model with local linear estimation is proposed. Frailty models have been extensively studied, but extensions to nonconstant coefficients are limited to spline‐based methods that tend to produce estimation bias near the boundary. To address this problem, we introduce a local polynomial kernel smoothing technique with a modified expectation‐maximization algorithm to estimate the unknown parameters. Theoretical properties of the estimators, including their unbiased property near the boundary, are derived along with discussions on the asymptotic bias‐variance trade‐off. The finite sample performance is examined by simulation studies, and comparisons with existing spline‐based approaches are conducted to show the potential advantages of the proposed approach. The proposed method is implemented for the analysis of T‐cell signaling. The fitted varying coefficient model provides a rigorous quantification of an early and rapid impact on T‐cell signaling from the accumulation of bond lifetime, which can shed new light on the fundamental understanding of how T cells initiate immune responses.