  The Cox regression model is a popular model for analyzing the relationship
between a covariate and a survival endpoint. The standard Cox model assumes a
constant covariate effect across the entire covariate domain. However, in many
epidemiological and other applications, the covariate of main interest is
subject to a threshold effect: a change in the slope at a certain point within
the covariate domain. Often, the covariate of interest is subject to some
degree of measurement error. In this paper, we study measurement error
correction in the case where the threshold is known. Several bias correction
methods are examined: two versions of regression calibration (RC1 and RC2, the
latter of which is new), two methods based on the induced relative risk under a
rare event assumption (RR1 and RR2, the latter of which is new), a maximum
pseudo-partial likelihood estimator (MPPLE), and simulation-extrapolation
(SIMEX). We develop the theory, present simulations comparing the methods, and
illustrate their use on data concerning the relationship between chronic air
pollution exposure to particulate matter PM10 and fatal myocardial infarction
(Nurses Health Study (NHS)), and on data concerning the effect of a subject's
long-term underlying systolic blood pressure level on the risk of
cardiovascular disease death (Framingham Heart Study (FHS)). The simulations
indicate that the best methods are RR2 and MPPLE.
