Regression analysis with multiplicative and time-varying additive regression coefficients with examples from breast and colon cancer.

Abstract

Regression analysis may be used to simplify the representation of mortality rates when there are many significant prognostic covariates or to adjust for confounding effects. The principal request of the regression model in this range of use is to have unbiased parameter estimates. A model with constant multiplicative and time-varying additive regression coefficients is discussed. The model allows some covariate effects to be multiplicative while allowing others to have a time-varying additive effect. Thus, it is a mix of classical Cox regression and Aalen's additive risk model. A major characteristic of cancer mortality rates, in contrast to general mortality rates, is that hazard rates, after a potentially initial increase, decrease, although not always tending to zero. Cancer diseases, like breast and colon cancer, have significantly increased cause-specific mortality rates even 20 years after diagnosis. Another major feature in cancer survival analysis is that many covariate effects are time-varying. Some covariate effects, like age at diagnosis, may only be significant for a limited time after diagnosis. Furthermore, some treatment procedures may initially decrease the mortality, while the long-term effect may be opposite. A third issue is that average covariate effects are very often not multiplicative. Estimation is carried out iteratively; the cumulative additive regression functions are estimated non-parametrically using a least-squares method and the multiplicative parameters are estimated from the partial likelihood. The method is applied on 3201 female breast cancer and 1372 male colon cancer patients.