A Fractional Survival Model

A survival model is derived from the exponential function using the concept of fractional differentiation. The hazard function of the proposed model generates various shapes of curves including increasing, increasing-constant-increasing, increasing-decreasing-increasing, and so-called bathtub hazard curve. The model also contains a parameter that is the maximum of the survival time.


Introduction
The one-parameter exponential distribution has been applied on many fields. The common form of the probability density function (PDF) is ( ) μ . The hazard function is equal to the first derivative of the CDF divided by the survival function which is equal to 1 minus the CDF also known as the mortality function in survival analysis. Therefore, the hazard function of the one-parameter exponential distribution is µ, a constant. A constant hazard rate does not describe all the observed phenomena in many fields. It is desirable to have a model with a non-constant hazard function. Based on the one-parameter exponential distribution model, we derive a fractional survival model by taking an arbitrary order of the CDF. We then apply the model to 3 cases of real world data.

The Model
Let DZ(t) denote the first derivative of the CDF of the one-parameter exponential function, then the hazard function is =µ. Many works have been done to derive models with various hazard rates. Instead of varying the hazard rates, we here employ the idea of Stiassnie (1979) who used a model with an arbitrary order of differentiation to explain the dynamics of viscoelastic materials. Moreover, the arbitrary order is not necessarily an integer. Therefore, we may choose to take the derivative to some arbitrary order of the CDF to be given by ; ; F a b c is the confluent hypergeometric function (Gurland, 1958;Muller, 2001) with 3 arguments.
However, ( ) Z t is just the incomplete Gamma distribution, a special case of the confluent hypergeometric function (Luke, 1959). To make the model more general, the first argument of ( ) Z t is substituted by a. Then, ( ) The fractional survival function S(t) is, therefore, 1-( ) The negative sign in the third argument in equation (2) can be eliminated using Kummer's formula (Gurland, 1958) Taking the first derivative of ( ) F t in equation (1) and using the differential formula of the confluent hypergeometric function (Abramowitz & Stegun, 1972), the probability density function is Or, using Kummer's formula, the PDF is The confluent hypergeometric function can be represented by a series and an integral expression (Gurlan, 1958;Muller 2001). Either expression has its own restrictions on the values of the first and the second arguments. In this article, the confluent hypergeometric function is numerically evaluated by Muller's (20001) algorithm based on the series expression in which the second argument cannot be zero or a negative integer. Applying the restrictions to our fractional survival model, λ cannot be zero or a negative integer, and 0 ≤ t ≤ T.

Application
Case The  Figure 5 shows the full-scaled hazard curve with increasing-decreasing-increasing rates.
Case 2 We use the same data in case 1 to fit the proposed model on time to first marriage. The sample was censored in 2003 with 17338 married individuals and 6029 unmarried. The average year to the first marriage was 24.01.
The maximum likelihood estimates and the standard errors based on the second derivatives valued at the maximized log-likelihood function are in Table 2.

Case 3
The data in this application contains survival times in month from 5880 patients after they received coronary artery bypass grafting (CABG). Among these patients, 545 died during the study. The average survival time to death after receiving the procedure was 47 months. The data is available at www.clevelandclinic.org/heartcenter/hazard/default.htm. The maximum likelihood estimates and the standard errors based on the second derivatives valued at the maximized log-likelihood function are in Table 3. The plots of the cumulative hazard curves of the proposed model and the Nelson-Aalen estimates are in figure 9. The survival curves of the proposed model and the Nelson-Aalen estimates are in figure 10. The decreasing-constant-increasing hazard curve of the fitted model is in figure 11. The hazard is known as the bathtub curve which is also the three phase hazard function described by Sergeant, Blackstone and Meyns (1997) who analyzed CABG data as well.

Discussion
In this study, we proposed a survival function with flexible hazard curves with the parameter T that can be regarded as the maximum survival time. However, in case1, the estimated T is not significantly different from 0 at 0.05 significance level. It suggests that the model may be overparametized for the data. A model with an increasing-decreasing hazard curve may be a good candidate. In case 2, the estimated T is 94.512 which suggests that the maximum age to the first marriage be 94.512. In case 3, the estimated T suggests that the maximum survival month after CABG be 225.265.

Acknowledgement
This work is part of the first author's master thesis finished at the University of Alabama at Birmingham in 1995 under the supervision of Professor Emeritus Malcolm E. Turner. It converges for all real values of a, c, and z and c cannot be a negative integer or zero. The confluent hypergeoemtric function can also be presented as an intergral form (Gurland, 1958) ( )   When the confluent hypergeometric is evaluated under the integral expression, a -λ must be less than 1 (Gurland, 1958) and, therefore, the first and the second derivative of the moment generating function are undefined. The confluent hypergeometric can also be evaluated under the series expression and both derivatives of the moment generating function are 0. It means that the expectation and the variance are both 0 which implies that t is not a random variable. Therefore, the mean and the variance of the proposed distribution do not exit.