  There is a surge in medical follow-up studies that include longitudinal
covariates in the modeling of survival data. So far, the focus has been largely
on right-censored survival data. We consider survival data that are subject to
both left truncation and right censoring. Left truncation is well known to
produce biased sample. The sampling bias issue has been resolved in the
literature for the case which involves baseline or time-varying covariates that
are observable. The problem remains open, however, for the important case where
longitudinal covariates are present in survival models. A joint likelihood
approach has been shown in the literature to provide an effective way to
overcome those difficulties for right-censored data, but this approach faces
substantial additional challenges in the presence of left truncation. Here we
thus propose an alternative likelihood to overcome these difficulties and show
that the regression coefficient in the survival component can be estimated
unbiasedly and efficiently. Issues about the bias for the longitudinal
component are discussed. The new approach is illustrated numerically through
simulations and data from a multi-center AIDS cohort study.
