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.
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