The survival analysis methods are usually conducted based on assumption that the population is homogeneity. However, generally, this assumption in most cases is unrealistic, because of unobserved risk factors or subject specific random effect. Disregarding the heterogeneity leads to unbiased results. So frailty model as a mixed model was used to adjust for uncertainty that cannot be explained by observed factors in survival analysis. In this paper, family of power variance function distributions that includes gamma and inverse Gaussian distribution were introduced and evaluated for frailty effects. Finally the proportional hazard frailty models with Weibull baseline hazard as a parametric model used for analyzing survival data of the colorectal cancer patients.

In survival studies‎, ‎frailty models are used to explain the unobserved heterogeneity hazards‎. ‎In most cases‎, ‎they are usually considered as the product of the function of the frailty random variable and baseline hazard rate‎. ‎Which is useful for right censored data‎. ‎In this paper‎, ‎the frailty model is explained as the product of the frailty random variable and baseline reversed hazard rate‎, ‎which can be used for left censored data‎. ‎The general reversed hazard rate frailty model is introduced and the distributional properties of the proposed model and lifetime random variables are studied‎. ‎Some dependency properties between lifetime random variable and frailty random variable are investigated‎. ‎It is shown that some stochastic orderings preserved from frailty random variables to lifetime variables‎. ‎Some theorems are used to obtain numerical results‎. ‎The application of the proposed model is discussed in the analysis of left censored data‎. ‎The results are used to model lung cancer data‎.