  We study inference for censored survival data where some covariates are
distorted by some unknown functions of an observable confounding variable in a
multiplicative form. Example of this kind of data in medical studies is the
common practice to normalizing some important observed exposure variables by
patients' body mass index (BMI), weight or age. Such phenomenon also appears
frequently in environmental studies where ambient measure is used for
normalization, and in genomic studies where library size needs to be normalized
for next generation sequencing data. We propose a new covariate-adjusted Cox
proportional hazards regression model and utilize the kernel smoothing method
to estimate the distorting function, then employ an estimated maximum
likelihood method to derive estimator for the regression parameters. We
establish the large sample properties of the proposed estimator. Extensive
simulation studies demonstrate that the proposed estimator performs well in
correcting the bias arising from distortion. A real data set from the National
Wilms' Tumor Study (NWTS) is used to illustrate the proposed approach.
