An Extension of Log-Logistic Distribution for Analyzing Survival Data

In this paper, a new generalization of Log-Logistic distribution using Alpha Power transformation is proposed. The new distribution is named as Alpha Power Log-Logistic distribution. A comprehensive account of some of its statistical properties are derived. The maximum likelihood estimation procedure is used to estimate the parameters. The importance and utility of the proposed model are proved empirically using two real life data sets.


Introduction
Log-Logistic (LL) distribution also known as Fisk distribution in economics is one of the important continuous probability distribution having massive application in survival analysis. It also finds application in areas like finance and insurance. If logarithm of a random variable X follows Logistic distribution then X follows LL distribution. The properties of LL distribution made it an attractive alternative to distributions which were conventionally used in the analysis of survival data. Kleiber and Kotz (2003) discussed the application of LL distribution in economics, Collet (2003) showed its application in medical field, Ashkar and Mahdi (2006) used LL distribution to analyze stream flow data etc. Some other authors who studied the properties and utility of LL distribution are Singh et al. (1988), Nandram (1989), Diekmann (1992), Bacon (1993), Little et al. (1994) etc.
The cumulative distribution function (cdf) and probability density function (pdf) of LL distribution are given by Eq.
A number of authors extended this distribution to make it more flexible and increase its applicability in diverse fields. Map. Gui (2013) developed a new class of LL distribution using Marshall Olkin transformation. Lemonte (2014) proposed four parameter Beta Log-Logistic distribution and studied its properties. Lately, a prominent generalization technique Known as Alpha Power (AP) transformation was suggested by Mahadavi and Kundu (2015) which has been exploited by various authors to achieve flexibility. The cdf of AP transformation is given as The pdf of AP transformation is given as In this article, the LL distribution is generalized by using Alpha Power (AP) transformation and the new model so obtained named as Alpha Power Log-Logistic (APLL) distribution. It is more flexible and exhibits more complex shapes of density and hazard rate functions. Also, the proposed model outclasses some well-established models in terms of two real life data sets. The rest of the article is unfolded as: in Section 2, the pdf and cdf of the proposed model i.e., APLL distribution are defined. Section 3 deals with the reliability measures of the APLL distribution. The expansion of pdf and cdf is discussed in Section 4. Some of the important statistical properties are explored in Section 5. The parameter estimation is discussed in Section 6. The simulation study and applicability of the model is debated in Section 7 and 8 respectively. Finally, some conclusions are provided in Section 9.

APLL Distribution
A random variable X is said to follow two parameter APLL distributions with scale parameter θ > 0 and shape parameter α > 0 if its cdf takes the following form: The corresponding pdf is given as The plots of density function for different parameter combinations are presented in Figure 1.

Reliability Analysis
In this section, the reliability measures for APLL distribution are investigated.

Reliability function
The reliability function denoted by R AP LL (x) is the probability that an item does not fail before time say x and for APLL distribution, it is given as An Extension of Log-Logistic Distribution for Analyzing Survival Data

Hazard rate function
The Hazard rate Function denoted by h AP LL (t) is the probability of instantaneous rate of death and is given as

Reverse hazard rate function
The reverse hazard rate function for APLL distribution is denoted by φ AP LL (x) and is given as The behavior of reliability function and hazard rate function of APLL distribution for different values of the parameters is illustrated in Figure 2.

Mixture representation
Using the power series expansion, b a = ∞ j=0 , the pdf and cdf of APLL distribution can be expressed in terms of an an alternative representation given by Eq. (5) and (6) respectively.

Statistical properties
In this section, some important statistical properties of APLL distribution are discussed.

Simulation and quantiles
Upon inverting Eq. (3) , we get Where U ∼ unif orm(0, 1). Using Eq. (7), the APLL distribution can easily be simulated. Also the p th quantile of APLL distribution is given as

Moments
The r th moment about origin of APLL distribution can be obtained as Upon substituting Eq. (5) in the above given equation, we get An Extension of Log-Logistic Distribution for Analyzing Survival Data Substituting x θ = t and solving, we get Putting r=1 in Eq. (8), we get the mean of APLL distribution The variance of APLL distribution is given as The u th incomplete moment about origin is defined by ψ u = n 0 x u f AP LL (x)dx and for APLL distribution, it can be obtained as where B x (l, m) = x 0 y l−1 (1 − y) m−1 dy.

Moment Generating Function
The moment generating function of APLL distribution can be obtained using the relation using Eq. (8) in Eq. (11), we get the required expression as given by Eq. (12).

Mean Deviation about Mean and Median
The mean deviation about mean is defined as An Extension of Log-Logistic Distribution for Analyzing Survival Data using Eq. (5), we get Also the expression for Mean deviation about medain for APLL distribution takes the following form: ; θ > 1.

Mean residual life (MRL) and Mean Waiting Time (MWT)
The MRL is defined as The MWT is defined as Upon substituting Eq. (10) for u = 1 and Eq. (5) in Eq. (15) and Eq. (16), we get the required expressions for MRL and MWT for APLL distribution.

Renyi Entropy
The Renyi Entropy given by Renyi (1961) as a measure of uncertainty is defined as using Eq. (3), we get Using the expansion α s = ∞ j=0 (logα) j s j j! in (17), I ν reduces to An Extension of Log-Logistic Distribution for Analyzing Survival Data

L-moments
The L-moments of APLL distribution can be obtained as We have substituting the value of f i:n (x) in Eq. (19) and solving, we get

Parameter Estimation
Let η = (α, θ) T be the vector of parameters of APLL distribution. The log-likelihood function denoted by l, computed from a sample of size n drawn from APLL distribution is given as: logα.
An Extension of Log-Logistic Distribution for Analyzing Survival Data The elements of score matrix U (η) = ∂l ∂η = ∂l ∂α , ∂l ∂θ T are Upon equating Eqs. (22) and (23) to zero and solving them simultaneously, we obtain the ML estimates of α and θ. Methods such as Newton Raphson can be used to solve such non-linear Equations. For fixed θ in (23) we obtainα(θ) as followsα and then we can obtainθ from (22) by solving the following equation: Onceθ has been obtained, we can evaluate the value ofα asα =α(θ).
The second order partial derivatives of APLL distribution for l exist. Thus, the asymptotic sampling distribution ofη is is the observed Fisher information matrix given by: where The 100(1 − φ) confidence interval for α and θ cab be determined asα ± Z φ is the upper φ − th percentile of standard Normal distribution.

Simulation
In this section, we present the simulation study to illustrate the behavior of MLEs for different sample sizes. Samples of sizes (n) 50, 100, 300, 500 and 1000 were drawn from APLL distribution for the two parameter combination (α, θ) i.e., (3,3) and (0.5, 0.8). The results are reported in Table 1. It can be clearly seen that the estimates are consistent as the standard deviations of the MLEs show a decreasing trend with increase in the sample size.
An Extension of Log-Logistic Distribution for Analyzing Survival Data

Application
To access the flexibility and establish the superiority of the APLL distribution, we compare the fits of APLL distribution with three well-established models for two real life data sets. The three models that are used for comparison are • Transmuted Log-Logistic (TLL) distribution with pdf where α is the transmuted parameter.
• Exponentiated Log-Logistic (ELL) distribution with pdf The first data set consists of the time between failures for 30 repairable items and was also analyzed by Murthy et al. (2004). The Ml estimates and values of comparison criterions for APLL distribution and competitive models are reported in Table 2 and Table 3 respectively.The second data set represents the survival times (in years) after diagnosis of 43 patients with a certain kind of leukemia extracted from Kleiber and Kotz (2003). The Ml estimates and values of comparison criterions for APLL distribution and competitive models are reported in Table 4 and Table 5 respectively. The criterion such as log l, AIC, SIC, AICc and HQIC are used as performance comparing tools. Also, the values of K-S statistic and associated p-value is computed.
The results obtained in Table 3 and Table 4 reveal that APLL distribution has the least value of all the comparison criterions, hence APLL distribution can be considered a strong competitor to other distributions compared here for An Extension of Log-Logistic Distribution for Analyzing Survival Data    fitting data. The relative histogram and fitted APLL distribution for first and second data set are presented in Figure  3(a) and3(b) respectively. Also, to compare the empirical distribution of the data with APLL distribution graphically, the QQ-plot for both the data sets is displayed in Figure 4.

Conclusion
In this paper, a new lifetime distribution namely APLL distribution is proposed and studied. The new distribution is more flexible and its hazard rate function exhibits complex shapes. The new distribution is compared with three well-established models using two real life data sets. The results showed that APLL distribution provides better fit than the competitive models. We hope that this distribution attracts wider application in diverse fields.