Abstract
Survival analysis deals with failure time data. The presence of censoring makes the application of the classical parametric and nonparametric methods of survival analysis inadequate and as such need’s modifications. Parametric mixture models are applied where a single classical model may not suffice. The parametric mixture needs to be made more robust to address the heterogeneity of survival data. This paper proposed a mixture of two distributions for the analysis of survival data, the models consist of Gamma-Gamma, and Loglogistic-Gamma distributions. Data was simulated to investigate the performance of the models, and used to estimate the maximum likelihood parameters of the models by employing Expectation Maximization (EM). Parameters of the models were estimated and were all close the postulated values. Simulations were repeated to test the consistency and stability of the models through mean square error (MSE) and root mean square error (RMSE), and were all found to be stable and consistent. Real data was applied to determine the best fit among the mixture models and classical distributions using information criteria. Mixture models were found to model the data and the mixture of two different distributions gives the best fit.
Keywords: Survival analysis, Log-logistic, Gamma, Expectation maximization, Algorithm, Akaike information criteria, Maximum likelihood estimate, Mean square error, Root mean square error.
Received: 8 December 2021 / Revised: 17 January 2022 / Accepted: 2 February 2022/ Published: 15 February 2022
Contribution/ Originality
The study is significant in the sense that, all existing studies of mixture models need to be extended to enrich the study of the analysis of heterogeneous survival data, because of its wide application, such as, in biomedical sciences, industrial reliability or reliability engineering, social sciences, and business.
1. INTRODUCTION
Survival or reliability study is an area with its unique characteristic; it deals with the statistical methods of analysing survival data obtained from clinical studies of humans, laboratory study of animals and investigation of the durability of manufactured items, among other appropriate applications. Survival time can broadly be defined as the time to the occurrence of the event of interest, the event of interest can be the time to failure of a manufactured item, the time to occurrence of a disease, time to relapse, response to treatment, death, etc. The study of survival data has paid attention on predicting the probability of response, survival, or mean lifetime, comparing the survival distributions of experimental animals or, of human patients and the identification of risk and/or prognostic factors related to response, survival, and the development of a disease, [1]. Parametric and nonparametric methods are usually employed [2, 3].
Mixture models are being explored in survival and reliability analysis in recent times. Mixture models can be used to analyze failure-time data in a variety of ways. As a flexible way of modelling data, mixture models have several applications in situations where a single model may not suffice. They are applied where the data is heterogenous. Some authors proposed a mixture of three classical distributions [4] to analyze survival data that has three different time overlapping phases [5].
Similarly, two component mixture models of different distributions were also studied [6] therefore, this paper wish to enrich the study of two component, different distributions, mixture models for the analysis of survival data.
2. SURVIVAL ANALYSIS
Let T denote the survival time, which is a non-negative absolutely continuous random variable that represents the life time of individuals. If F(t) is the cumulative distribution of T, the survival function is defined to be;
S(t) = P (an individual survives longer than t).
Equation 1 represent the survival function of an individual with survival time T.
2.1. The Probability Density Function
Similar to any other continuous random variable, the survival time T has a probability density function defined as the limit of the probability that an individual fails in the short interval (t, t + Δt) per unit width Δt.
Equation 2 is the probability density function as explained above in 2.1.
2.2. The Hazard Function
The hazard function h (t) of survival time T gives the conditional failure rate; it is defined as the probability of failure during small interval of time, given that the individual has survived to the beginning of the interval. It can be expressed as;
The hazard function is also known as the force of mortality, conditional mortality rate, and age-specific failure rate [1].
The survival function, S(t), probability density function, f(t), and hazard function, h(t), are mathematically equivalent.
The relationship is expressed thus:
Equations 5 through 9 presents the relationship between the hazard function, the probability density function and the survival function.
2.3. Gamma Distribution
The Gamma distribution has the pdf of the form:
2.4. Log-Logistic Distribution
The log-logistic distribution is related to the logistic distribution in an identical fashion to how the log-normal and normal distributions are related with each other.
A logarithmic transformation on the logistic distribution generates the log-logistic distribution. Because of its flexible shapes, the log-logistic distribution has been illustrated to provide useful fits to data from many different fields, including engineering, economics, hydrology, and medical sciences.
The log logistic distribution is characterized by two parameters, γ (positive shape parameter) and η (positive scale parameter). A log-logistic random variable X with parameters γ and η has probability density function and survival function as follows;
The log-logistic distribution was used in survival analysis [7] Similarly, it was used to model economic data [8].
3. METHODOLOGY
The Expectation maximization algorithm (EM) was used to achieve the maximum likelihood estimation of the parameters of the model, it is a way to find maximum-likelihood estimates for model parameters when your data is incomplete, has missing data points, or has unobserved variables. It is an iterative way to approximate the maximum likelihood function. A model selection criterion based on the Akaike Information (AIC) was employed to find the mixture model that gives the best fit.
Data is simulated from a two-component parametric mixture of Gamma-Gamma, and Loglogistic-Gamma distributions. The model is evaluated by simulated data set, before it’s applied to real dataset.
3.1. Parametric Mixture Model
Let Ti, . . . ,Tn be n independent random variables, where Tj is the survival time of the jth subject. We assume that the probability density function f(t) of Tj is a mixture.
3.2. Gamma-Gamma Mixture Model
The mixture model of distributions assumes the population consist of two distinct sub groups or classes, therefore, the gamma mixture can be written as:
fgm-gm (t) = πfgm(t; λ1, k1) + (1-π)fgm(t; λ2, k2) (20)
where π is the mixture weight of the distributions and λ1, λ2, k1, and k1 are shapes and scales of the two components respectively.
3.3. Log-logistic Gamma Mixture Model
The mixture of the densities of Log-logistic and Gamma can be represented as:
fll-gm (t) = πfll(t; γ1, η1) + (1-π)fgm(t; λ2, k2) (21)
where is the weights of the mixture and γ1, η1 are the shape and scale of the loglogistic component and λ2, k2 are the shape and scale of the gamma component of the distribution. The Expectation Maximization (EM) algorithm [9] is a general approach to maximum likelihood estimation for problems of finite mixture models [10]. Starting value initialisation is very important in the EM algorithm because the likelihood surface of mixture models tend to have multiple modes [11]. The EM algorithm typically produces improved result when started from reasonable initial values [12]. The EM algorithm is a broadly applicable algorithm that provides an iterative procedure for computing Maximum Likelihood Estimates (MLE), [10].
Suppose the density of a random variable Yhas an a component mixture form.
Maximum likelihood estimation of mixture model is cumbersome to solve using the traditional method of taking derivative with respect to each parameter. These, and some other difficulties made modelling heterogeneous data unattractive for a long time [13].
3.4. Mixture Model Selection
Regarding the mixture density estimation problem, the problem of determining the proper number of components and proper mathematical form of each component is faced. In other words, one need to determine which mixture model fits the data better. The question we try to answer here is a model selection problem.
The statistic Akaike Information Criterion (AIC) appears to be adequate for model selection in the mixture density estimation [13]. Here, we follow the model selection approach using AIC proposed by Akaike [14].
4. RESULTS
The simulated data contains n = 100 observations. The Maximum Likelihood Estimates (MLE), of the parameters of each component of a mixture, and the Akaike Information Criteria (AIC) are used for model selection in each case, mean square error (MSE) and root mean square error (RMSE) are also employed, the MSE and RMSE are among the many ways to quantify the difference between an estimator and the true value of the quantity being estimated.
4.1. Gamma-Gamma
Dataset is simulated from a two-component parametric mixture of Gamma model. The simulated data contains n = 100 observations and the proportion parameters for each component is λ1=0.5, λ2 =0.5. Table 1 list the MLEs of the parameters and the mixing proportions of the mixture model, it can be seen that, the parameters of the model were estimated successfully, because the estimated parameters are closed to the postulated values. Table 2 shows that, the EM is also consistent and stable in its estimation because of the relatively small values of MSE and RMSEs.
Table 1. MLEs of gamma-gamma model with no repetitions.
| Gamma-Gamma | ||||||
Parameter |
λ1 |
λ2 |
α1 |
α2 |
β1 |
β2 |
Postulate |
0.5 |
0.5 |
40 |
6 |
20 |
1 |
Estimate |
0.485 |
0.515 |
39.999 |
6.007 |
19.752 |
0.963 |
Table 2. MLEs of gamma-gamma model with 300 repetitions.
| Gamma-Gamma | |||||||
Parameter |
λ1 |
λ2 |
α1 |
α2 |
β1 |
β2 |
|
Postulate |
0.5 |
0.5 |
40 |
6 |
20 |
1 |
|
Estimate |
0.495 |
0.515 |
41.412 |
5.787 |
20.661 |
0.946 |
|
MSE |
1.46e-06 |
1.45e-06 |
1.70e-01 |
2.78e-03 |
4.00e-02 |
1.14e-04 |
|
RMSE |
0.001 |
0.001 |
0.412 |
0.053 |
0.200 |
0.011 |
|
To assess the consistency of the model, simulation was repeated 300 times, the Mean Square Error (MSE), and the Root Mean Square Error were obtained. It can be seen that, both MSE and the RMSE values are very small which shows the model is consistent. Figure 1 compares the density function of classical parametric model and the density of the mixture model, it can be seen that, the mixture model fits the data better.
Figure 1. Density of survival mixture model of gamma-gamma.
Table 3 shows that, the model estimates are very close to the postulated values, hence the model can be said to be efficient.
Table 3. MLEs of log-logistic-gamma model with no repetitions.
| Log-logistic-Gamma | ||||||
| Parameter | λ1 |
λ2 |
α1 |
α2 |
β1 |
β2 |
| Postulate | 0.5 |
0.5 |
10 |
2 |
4 |
0.1 |
| Estimate | 0.521 |
0.480 |
10.523 |
1.887 |
3.956 |
0.111 |
Simulations were repeated 300 times to test for consistency of the model and it is found to be consistent as can be seen from the MSE and RMSE in Table 4, which are very close to zero.
Table 4. MLEs of log-logistic - gamma model with 300 repetitions.
| Log-logistic-Gamma | |||||||
Parameter |
λ1 |
λ2 |
α1 |
α2 |
β1 |
β2 |
|
Postulate |
0.5 |
0.5 |
10 |
2 |
4 |
0.1 |
|
Estimate |
0.440 |
0.561 |
10.692 |
1.556 |
4.341 |
0.111 |
|
MSE |
4.19e-06 |
4.19e-06 |
1.34e-02 |
2.24e-04 |
2.26e-03 |
1.43e-06 |
|
RMSE |
0.002 |
0.002 |
0.116 |
0.015 |
0.0478 |
0.001 |
|
The densities of the two classical distributions were also compared to the model as shown in Figure 2 it can be seen that, the model fits the data better.
Figure 2. Density of survival mixture model of Gamma – log-logistic model.
4.2. Real Data Application
The data used is the Acute Myelogenous Leukemia (aml) dataset in R statistical software, it was first used by Rupert [15]. The question at the time was whether the standard course of chemotherapy should be extended (“maintained”) by additional cycles.
The MLEs of aml dataset using Gamma-Gamma model is in Table 5 and Figure 3, it shows the survival function alongside the K-M of the model.
Table 5. MLEs and AIC of the AML dataset using gamma-gamma model.
Figure 3. Kaplan meier (K-M) survival curve of gamma-gamma.
Figure 3 shows that, the mixture model fits the K-M curve better than the classical model as it is closer to the K-M curve.
Table 6. MLEs and AIC of loglogistic-gamma model.
Figure 4. Kaplan meier (K-M) survival curve of loglogistic-gamma model.
Figure 4 shows the Kaplan meier survival curve of the loglogistic-gamma distribution.
It is observed that, the Gamma-Gamma fits both the simulated and real data well, however Loglogistic-Gamma model gives the best fit to both datasets, because it has the lowest Akaike Information Criteria (AIC), and comparing the Kaplan Meier (K-M) survival curves it can also be seen that, the K-M survival curve of the loglogistic-Gamma in gives a better fit than the single classical distributions. The estimation of the parameters of the model were successful in both the simulated and the real data, as estimated values were relatively close to postulated values. As can be seen the mixture model of Loglogistic-Gamma gives the best fit for the real data among the proposed models.
5. CONCLUSION
The paper proposed a two-component mixture model of classical distributions, namely, Gamma-Gamma and Loglogistic-Gamma distributions to analyze heterogenous survival data, simulated and real data were employed to assess the performance of the models, the models were found to estimate the parameters successfully as the estimates were close to the postulated values. It is found that, the mixture of two different distributions i.e. Loglogistic-Gamma gives a best fit to the real data applied.
Funding: The study did not receive any financial support. |
Competing Interests: The authors declare no competing interest. |
Authors’ Contributions: All authors contributed to the conception and design of the study. |
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