Likelihood Function-based Estimation for Clustered Censored Data with Embedded AFT Models

The accelerated failure time (AFT) model offers a directly interpretable, time-ratio alternative to Cox regression and is attractive for clustered right-censored data. We develop an embedded-likelihood AFT framework that (i) transforms observed times so that, at the truth, the transformed failure times are i.i.d. and independent of covariates, and (ii) uses these transformed data to derive score-type estimating equations under two common semiparametric embeddings: proportional hazards (PH) and proportional odds (PO). Estimation proceeds via a two-level algorithm that separates outer updates for the regression vector from inner profiling of the shared frailty, scale, and frailty-variance parameters. We provide sandwich standard errors and optional cluster bootstrap inference including the consistency and asymptotic normality for the PH/PO embeddings. Comprehensive simulations spanning multiple cluster sizes, censoring levels, and error/frailty distributions indicate that the embedded-likelihood estimators match, and occasionally improve upon, penalized partial likelihood (PPL) for regression effects, with near-nominal coverage using either sandwich or bootstrap standard errors. Heavy-tailed errors mainly inflate variance, while frailty-law misspecification primarily impacts the frailty variance but leaves regression estimates well centered with robust inference. In a Diabetic Retinopathy Study reanalysis, PH and PO-embedded AFT fits yield effect estimates consistent with PPL, slightly better AIC/BIC for the Cox-AFT embedding, and a clinically meaningful between-patient heterogeneity captured by the frailty. The embedded-likelihood AFT formulation provides a unified, interpretable, and computationally practical route to likelihood-based inference for clustered censored data, delivering performance competitive with established approaches while retaining the advantages of the AFT perspective.