Penalized estimation of linear transformation models for interval-censored data with time-dependent covariates

We investigate efficient estimation strategies for partially linear transformation models with time-dependent covariates under interval censoring. The unknown monotone function is approximated using a monotone $B$-spline basis to enable flexible semiparametric modeling, and we develop a computationally efficient nested hybrid EM algorithm that integrates Newton's method with isotonic regression. To support large-sample inference, we propose a straightforward variance-covariance estimation procedure for the regression parameters and introduce a score test to assess the adequacy of the proportional hazards (PH) specification within the broader class of transformation models. The numerical performance of the penalized estimators is examined extensively and compared with both the time-invariant covariate model by Lu et al. and the semiparametric transformation model by Zeng et al. Finally, the proposed methodology is applied to data from the National Alzheimer's Coordinating Center (NACC) to demonstrate its practical utility in a real-world clinical setting.