Regression models for survival data: The generalized time-dependent logistic family

A three-parameter family of survival models is introduced. The base-line density is derived and the main properties of the model, including a frailty interpretation, are discussed. Several different regression models are considered, and one of these, which is non-proportional hazards and which has other interesting properties-the generalized time-dependent logistic model-is developed in some detail. The method of maximum likelihood is used to estimate the parameters and to generate the observed information in the presence of right censoring. Zn a reduced form the family is characterized by a hazard function given explicitly by the time-dependent multiple-logistic function. Consequently the natural logarithm of the 'conditional odds on instantaneous failure' is a linear function of time. This model is shown to describe, closely, the survival pattern of 855 incident cases of lung cancer studied in Northern Ireland. The data are analysed in relation to sex and age at diagnosis and the results are compared with those obtained by using the proportional hazards model of Cox