## 3 Results
### Working Examples: Fitting Finite Mixture Models to Lifespan Datasets
Based on the estimated parameters in Table 1 and using Equations (13) and (15), we obtain the probability density functions and survival functions of the Gompertz mixture distributions as illustrated in Figures 3 and 4. In our analysis, _tor1_, _sit2_, and _sod2_ mutants have RLS that can be fit into two subpopulations. Table 1 demonstrates _scg9_, _lap4_, and _pmr1_ mutants can even be fitted into three subpopulations. Estimated model parameters of single-mode Gompertz aging model for shape parameter \(\alpha\) ranged between 0.05 and 0.25, rate parameter \(\beta\) ranged between 0.0001 and 0.04, which supports that our mixture parameters are correctly estimated by the previous studies [9,14]. We can deduce that the Gompertz mixture model method provides the best fit for the RLS of budding yeast. This agrees with our assumption that a mixture model may be biologically meaningful. ## 4 Discussion

A common task in yeast aging studies is modeling the lifespan distribution. In this study, we have developed and introduced an R package called **fitmix** that provides several models and estimation techniques for modeling replicative lifespan distributions from both individual lifespan data and grouped lifespan data, as well as providing additional functionality for simulating finite mixture distributions and fitting curves to division-probability of the data. In our analysis, we show that RLS in many mutants, such as _tor1_, _sch9_, _sit2_, _hap4_, _sod2_, _pmr1_, can be fit into two subpopulations, some of them even three subpopulations. Our study suggests replicative aging in these probably have at least two subpopulations in the other mutants as well (Supplementary Materials), and we argue that these are promising biological insights or hypotheses that may be pursued by experimental biologists. Since experimental studies are expensive and time-consuming, our mixture-modeling approach provides a reasonable alternative research method to assess the aging process in many more yeast mutants. To help facilitate the comparison of the different model estimation methods provided by the **fitmix**, we include multiple goodness-of-fit measures that the user can determine the best model technique for their purposes. The **fitmix** package employs the EM algorithm that might have some other alternatives. However, from the previous section, we can see that simply evaluating Equation (22) to find such parameters would be very hard. An

Figure 4: Overlay of Gompertz mixture fitting curves with RLS histograms using experimental replicative lifespan datasets for the laboratory (**a**) _BY4741_ and (**b**) _BY4742_ wildtype reference strains. Red fitting curves represent the Gompertz mixture fit to the RLS data of (**a**) _BY4741_ and (**b**) _BY4742_, and the probability density functions of lifespan data are represented with the blue dashed fitting curves. See Table 1 for estimated parameters of the finite Gompertz mixture fits. analytical approach to the estimation of parameters requires an explicit formula for the maximum likelihood investigation [33]. Fortunately, there is an iterative method we can use to achieve this purpose. It is called the Expectation-Maximization, or simply EM algorithm [34; 35]. It is widely used for optimization problems where the objective function has complexities such as the one that has been encountered in the **fitmix** package. To avoid converging to a local maximum of the observed data likelihood function, we compare our estimated mixture parameters depending on starting values previously reported [9; 14; 31]. While this package was developed in the context of yeast research, the models we fit and simulate have numerous applications throughout other biological and medical fields, such as cancer survival data, disease datasets, and lifespan datasets of single units.