  We develop a maximum likelihood estimating approach for time-to-event Weibull
regression models with outcome-dependent sampling, where sampling of subjects
is dependent on the residual fraction of the time left to developing the event
of interest. Additionally, we propose a two-stage approach which proceeds by
iteratively estimating, through a pseudo score, the Weibull parameters of
interest (i.e., the regression parameters) conditional on the inverse
probability of sampling weights; and then re-estimating these weights (given
the updated Weibull parameter estimates) through the profiled full likelihood.
With these two new methods, both the estimated sampling mechanism parameters
and the Weibull parameters are consistently estimated under correct
specification of the conditional referral distribution. Standard errors for the
regression parameters are obtained directly from inverting the observed
information matrix in the full likelihood specification and by either
calculating bootstrap or robust standard errors for the hybrid pseudo
score/profiled likelihood approach. Loss of efficiency with the latter approach
is considered. Robustness of the proposed methods to misspecification of the
referral mechanism and the time-to-event distribution is also briefly examined.
Further, we show how to extend our methods to the family of parametric
time-to-event distributions characterized by the generalized gamma
distribution. The motivation for these two approaches came from data on time to
cirrhosis from hepatitis C viral infection in patients referred to the
Edinburgh liver clinic. We analyze these data here.
