  The standard Cox model in survival analysis assumes that the covariate effect
is constant across the entire covariate domain. However, in many applications,
there is interest in considering the possibility that 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 value of this threshold is
unknown and need to be estimated. In addition, often, the covariate of interest
is not measured exactly, but rather is subject to some degree of measurement
error. In this paper, we discuss estimation of the model parameters under an
independent additive error model where the covariate of interesting is measured
with error and the potential threshold value in this covariate is unknown. As
in earlier work which discussed the case of konwn threshold, we study the
performance of several bias correction methods: two versions of regression
calibration (RC1 and RC2), two versions of the fitting a model for the induced
relative risk (RR1 and RR2), maximum pseudo-partial likelihood estimator
(MPPLE) and simulation-extrapolation (SIMEX). These correction methods are
compared with the naive estimator. We develop the relevant theory, present a
simulation study comparing the several correction methods, and illustrate the
use of the bias correction methods in data from the Nurses Health Study (NHS)
concerning the relationship between chronic air pollution exposure to
particulate matter of diameter 10 $\mu$m or less (PM$_{10}$). The simulation
results suggest that the best overall choice of bias correction method is
either the RR2 method or the MPPLE method.

Applications