  The goals in clinical and cohort studies often include evaluation of the
association of a time-dependent binary treatment or exposure with a survival
outcome. Recently, several impactful studies targeted the association between
aspirin-taking and survival following colorectal cancer diagnosis. Due to
surgery, aspirin-taking value is zero at baseline and may change its value to
one at some time point. Estimating this association is complicated by having
only intermittent measurements on aspirin-taking. Naive, commonly-used, methods
can lead to substantial bias. We present a class of calibration models for the
distribution of the time of status change of the binary covariate. Estimates
obtained from these models are then incorporated into the proportional hazard
partial likelihood in a natural way. We develop nonparametric, semiparametric
and parametric calibration models, and derive asymptotic theory for the methods
that we implement in the aspirin and colorectal cancer study. Our methodology
allows to include additional baseline variables in the calibration models for
the status change time of the binary covariate. We further develop a risk-set
calibration approach that is more useful in settings in which the association
between the binary covariate and survival is strong.
