The estimation of progression to liver cirrhosis and identifying its risk factors are often of epidemiological interest in hepatitis C natural history study. In most hepatitis C cohort studies, patients were usually recruited to the cohort with a referral bias because clinically the patients with more rapid disease progression were preferentially referred to liver clinics. A pair of correlated event times may be observed for each patient, time to development of cirrhosis and time to referral to a cohort. This paper considers accelerated failure time models to study the effects of covariates on progression to cirrhosis. A new non-parametric estimator is proposed to handle a flexible bivariate distribution of the cirrhosis and referral times and to take the referral bias into account. The asymptotic normality of the proposed estimator is also provided. Numerical studies show that the coefficient estimator and its covariance function estimator perform well.