A parametric method for cumulative incidence modeling with a new four-parameter log-logistic distribution

Competing risks, which are particularly encountered in medical studies, are an important topic of concern, and appropriate analyses must be used for these data. One feature of competing risks is the cumulative incidence function, which is modeled in most studies using non- or semi-parametric methods. However, parametric models are required in some cases to ensure maximum efficiency, and to fit various shapes of hazard function.

We have used the stable distributions family of Hougaard to propose a new four-parameter distribution by extending a two-parameter log-logistic distribution, and carried out a simulation study to compare the cumulative incidence estimated with this distribution with the estimates obtained using a non-parametric method. To test our approach in a practical application, the model was applied to a set of real data on fertility history.

The results of simulation studies showed that the estimated cumulative incidence function was more accurate than non-parametric estimates in some settings. Analyses of real data indicated that the proposed distribution showed a much better fit to the data than the other distributions tested. Therefore, the new distribution is recommended for practical applications to parameterize the cumulative incidence function in competing risk settings.