Although longitudinal and survival data are collected in the same study, they are usually analyzed separately. Measurement errors and missing data problems arise because of separate analysis of these two data. Therefore, joint model should be used instead of separate analysis. The standard joint model frequently used in the literature is obtained by combining the linear mixed effect model of longitudinal data and Cox regression model with survival data. Nevertheless, to use the Cox regression model for survival data, the assumption of proportional hazards must be provided. Parametric survival sub-models should be used instead of the Cox regression model for the survival sub-model of the JM where the assumption is not provided. In this article, parametric joint modeling of longitudinal data and survival data that do not provide the assumption of proportional hazards are investigated. For the survival data, the model with Exponential, Weibull, Log-normal, Log-logistic, and Gamma accelerated failure time models and the linear mixed effect model are combined with random effects and the models were applied in primary biliary cirrhosis data set obtained from Mayo Clinic. After determining the best parametric joint model according to Akaike and Bayesian information criterions, the best available model was compared with standard joint model and of separate analysis of survival data and longitudinal data. As a results, in the studies where longitudinal and survival data are obtained together, it is seen that the parametric joint model gives more better results than the standard joint model when the proportional hazard assumption is not provided.

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In order to obtain a better fit for the observational data, various cosmological models have been developed in the literature. For this purpose, we propose an adiabatic expansion of the Universe with a polytropic fluid matter-radiation content and a varying cosmological constant. Basic motivation of this study basically differs from the previous models because our model does not neglect the pressure of the matter content of the Universe. We present the details of our new derivations by obtaining the polytropic matter density and luminosity distances. After constructing the model, two large collections of observational data of supernovae type Ia (SNe Ia) are conducted to diagnose the model. Examining the fitness of our new models of a polytropic universe with the observational data, we find a significant influence of polytropic matter on the cosmological parameters of the expanding universe. We find that our flat constant Λ model and varying Λ model describes the SNe Ia data significantly better than the other models including the standard model for Gold Data and Union2.1 Data, respectively.

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