A General Model for the Hazard Rate with Covariables

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Abstract
A parametric model is proposed for the hazard function incorporating covariables. The model is flexible enough so that it does not unduly restrict the shape of the hazard function. This model does not require that the proportional hazards assumption be met, and it provides for testing whether the assumption is reasonable. The hazard rate is given as a polynomial in time with the coefficients of the various powers of time being possibly different functions of the vector of covariables. Methods for fitting the model to data and testing hypotheses about model parameters are presented. The degree of the polynomial for the hazard is chosen in a step-wise manner as part of the fitting of the model. Specification of the survival curve with covariables is straightforward as a result of the parametric nature of the model. Use of the model and methods for fitting and hypothesis testing are illustrated by application to 2 different cohort studies. For each analysis, a single covariate indicates in which of 2 treatment groups an individual belongs. A time-constant hazard and the proportional hazards assumption are adequate for the 1st cohort examined. For the 2nd cohort a time-varying hazard is required and the proportional hazards assumption is not suitable. Results using other methods are compared with those of the proposed method for both of the cohorts. Good agreement among the different approaches is observed.