  Typically, case-control studies to estimate odds-ratios associating risk
factors with disease incidence from logistic regression only include cases with
newly diagnosed disease. Recently proposed methods allow incorporating
information on prevalent cases, individuals who survived from disease diagnosis
to sampling, into cross-sectionally sampled case-control studies under
parametric assumptions for the survival time after diagnosis. Here we propose
and study methods to additionally use prospectively observed survival times
from prevalent and incident cases to adjust logistic models for the time
between disease diagnosis and sampling, the backward time, for prevalent cases.
This adjustment yields unbiased odds-ratio estimates from case-control studies
that include prevalent cases. We propose a computationally simple two-step
generalized method-of-moments estimation procedure. First, we estimate the
survival distribution based on a semi-parametric Cox model using an
expectation-maximization algorithm that yields fully efficient estimates and
accommodates left truncation for the prevalent cases and right censoring. Then,
we use the estimated survival distribution in an extension of the logistic
model to three groups (controls, incident and prevalent cases), to accommodate
the survival bias in prevalent cases. In simulations, when the amount of
censoring was modest, odds-ratios from the two-step procedure were equally
efficient as those estimated by jointly optimizing the logistic and survival
data likelihoods under parametric assumptions. Even with 90% censoring they
were as efficient as estimates obtained using only cross-sectionally available
information under parametric assumptions. This indicates that utilizing
prospective survival data from the cases lessens model dependency and improves
precision of association estimates for case-control studies with prevalent
cases.
