In medical research with time-to-event data, there may be more than one final outcome of interest, and this circumstance can complicate the statistical analysis. In such cases, events other than the desired one(s) are considered as competing risks when their occurrence prevents the event of interest [1, 2]. An important quantity in competing risk settings is the cumulative incidence function (CIF), which makes it possible to calculate the probability of a particular event. In contrast, the cause-specific hazard function (CSHF) calculates the instantaneous rate of the event. For example, in fertility studies in women, researchers are interested in calculating the cumulative live birth rate in the presence of competing risks over time. Competing events, such as the probability of stillborn fetuses or abortions, can be calculated.
Most competing risk analyses of CIF are estimated non- or semi-parametrically [3, 4]. However, the parametric model is another available approach for modeling CIF. The advantage of parametric methods compared to non- and semi-parametric ones is that if a parametric model is selected correctly, it can predict the probability of the occurrence of events in the long term and provide additional insights about the time to failure and hazard functions . Also, when the survival pattern follows a particular parametric model, the estimates from true model fit are usually more accurate than the non-parametric estimates.
The best known distributions for modeling CIF are the Weibull and Gompertz distributions. However, these are suitable only for hazard functions that increase or decrease monotonically; they are inadequate when the hazard function shape is unimodal. In such cases, simple distributions such as the two-parameter log-logistic or log-normal distributions are likely to be better choices. One approach to the construction of flexible parametric models is to add a shape parameter to provide a wide range of hazard shapes and improve the models in survival data. In 1996, Mudholkar et al. proposed a generalized Weibull family with a range of hazard shapes  and Foucher et al. in 2005 applied this distribution in semi-Markov models . In 2006, Sparling et al. presented a three-parameter family of survival distributions that included the Weibull, negative binomial, and log-logistic distributions as special cases . These distributions can fit U-shapes or unimodal shapes for the hazard function, and therefore can be appropriate for survival data.
In light of the issues summarized above, a more efficient parametric distribution with various shapes of hazard patterns would appear to be useful for estimating CIF in competing risk situations. In recent years, various parametric distributions have been developed specifically for analyzing competing risk data that offer more flexibility. For example, in 2006 Jeong introduced a new parametric distribution for modeling CIF . In 2009, Wahed et al. developed Weibull's distribution, resulting in a beta-Weibull four-parameter distribution for use in competing risks . Here, we propose a new four-parameter log-logistic distribution by extension of a two-parameter log-logistic distribution that contains different kinds of hazard shapes in survival data and increases the efficiency of the CIF over the non-parametric approaches. Also, this is an improper distribution which enjoys more flexibility for modeling of CIF. Therefore, it would be suitable for competing risk models. We have performed a simulation study to compare CIF estimates obtained with the four-parameter distribution and a non-parametric method. After using simulated data to assess the method, we analyzed a real data set to examine the efficiency of our proposed distribution.