The Use of Split Exponential and Split Weibull Analyse Survival Data With Long Term Survivors

The split population model is a flexible way of extending the standard survival analytical methods to failure time data in which susceptibles and long-term survivors coexist. Susceptibles would develop the event with certainty if complete follow-up were possible, but the long-term survivors would never experience the event. A study was conducted to allow the effects of covariates on the probability that an individual is immune, and the immune probability vary from individual to individual. In effect, we are associating with each individual a distinct probability of being immune, which depends on the covariate information specific to that individual. And then fitted a few models using the maximum likelihood estimation to determine whether the covariates are significant or not. Several popular distributions on the survival data analysis as endorsed by graphical techniques were used. We applied the split exponential and the split Weibull models together with deviance test, a parametric test for the presence of immunes, and a test for outlier, to test for sufficient follow-up in the samples where there may or may not be immunes presences. We presented the probability of eventual immune for the ith individual as the logit model and logistic model. We will work with two data sets, firstly a Clinical Trial in the Treatment of Carcinoma of the Oropharynx and secondly Stanford Heart Transplant data. The results from the data analyses for a Clinical Trial in the Treatment of Carcinoma of the Oropharynx data show that the simple exponential model produces a fit not significantly worse than the simple Weibull model and the simple split Weibull model no better than the simple split exponential model, also shown that no evidence of immune population and all covariates are not significant. The results from the data analyses for Stanford Heart Transplant data show that the simple Weibull model is significantly better than the simple exponential model, and the simple split Weibull model is better than the simple split exponential model. We have calculated the maximum log-likelihood function value for both the logit exponential and logistic exponential models. They are exactly similar for both the Clinical Trial in the Treatment of Carcinoma of the Oropharynx and Stanford Heart Transplant data. So, we suggest that both the logit exponential and logistic exponential models are equally superior.

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