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Parametric bootstrap for testing model fitting in the proportional hazards framework: An application to the survival analysis of Bruna dels Pirineus beef calves¹

J. Casellas^*,2, J. Tarrés^*,3, J. Piedrafita^* and L. Varona

^* Grup de Recerca en Remugants, Departament de Ciència Animal i dels Aliments, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain; and Àrea de Producció Animal, Centre UdL-IRTA, 25198 Lleida, Spain

² Corresponding author: joaquim.casellas{at}uab.es

Given that correct assumptions on the baseline survival function are determinant for the validity of further inferences, specific tools to test the fit of a model to real data become essential in proportional hazards models. In this sense, we have proposed a parametric bootstrap to test the fit of survival models. Monte Carlo simulations are used to generate new data sets from the estimates obtained through the assumed models, and then bootstrap intervals can be established for the survival function along the time space studied. Significant fitting deficiencies are revealed when the real survival function is not included within the bootstrap interval. We tested this procedure in a survival data set of Bruna dels Pirineus beef calves, assuming 4 parametric models (exponential, Weibull, exponential time-dependent, Weibull time-dependent) and the Cox’s semiparametric model. Fitting deficiencies were not observed for the Cox’s model and the exponential time-dependent model, whereas the Weibull time-dependent model suffered from moderate overestimation at different ages. Thus, the exponential time-dependent model appears to be preferable because of its correct fit for survival data of beef calves and its smaller computational and time requirements. Exponential and Weibull models were completely rejected due to the continuous over- and underestimation of the survival probability reported. Results here highlighted the flexibility of parametric models with time-dependent effects, achieving a fit comparable to nonparametric models.

Key Words: model fitting • parametric bootstrap • proportional hazard • survival analysis

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