Survival Analysis of CABG Patients by Parametric Estimations In Modifiable Risk Factors-Hypertension and Diabetes

In this paper, modifiable risk factors (hypertension and diabetes) of Coronary Artery Bypass Graft Surgery (CABG) patients are considered. The objective is to analyse survivor’s proportions of CABG patients in the considered risk factors, in complete and incomplete Populations, using suitable survival models. A new approach of complete population from its incomplete population of CABG patients of 12 years observations is used for the survival analysis. In the complete population, censored patients are proportionally included into the known survived and died patients respectively. The availability of a complete population may represent better behaviour of lifetimes / survival proportions for medical research. Survival p roportions of the CABG patients of complete and incomplete populations, with respect to the risk factors (hypertension and diabetes) are obtained from suitable, lifet ime representing models (Weibull and Exponential). Maximum likelihood method, in -conjunction with Davidon-Fletcher-Powell (DFP) optimization method and Cubic Interpolation method is used in estimation of survivor’s proportions from the parametric models.


Introduction
Th e Co ro n a ry A rte ry D is eas e ( CA D) is d u e to accumulat ion of cholesterol and other material, called plaque, within inner walls of the coronary arteries (the arteries that supply blood with o xygen and nutrients to heart muscles). As this build -up grows (Arteriosclerosis), co mparatively less blood can flow through the arteries. Over the time (which differs fro m individual to indiv idual) heart weakens. This leads to chest pain (An g in a) wh ich is a sy mp to m o f Myocard ial In farct ion (M I). When th e clot (thro mbus) completely cuts off the hearts blood supply, this may leads to permanent heart damage, called as heart attack (M I). Heart failure means the heart cannot pump required blood to the rest of the body [10]. CAD is the lead ing cause o f death wo rld wid e (see W illiam, St eph en , Van -Th o mas and Robert ) [36], John [18], Hansson [13], A xel, Yiwen, Dalit, Veena, Elaine, Cat ia, Matthew, Jonathan, Edward & Len [3] and Sun & Hoong [33] ). Th e sy mp to ms and s igns o f CABG patients comprising 2011 patients using Kaplan Meier method [19]. The patients were grouped with respect to Male, Female, Age, Hypertension, Diabetes, and Ejection Fraction, Vessels, Congestive Heart Failure, Elective and Emergency Surgery. The patients were undergone through a first re-operation at Emory University hospitals from 1975 to 1993. This study also comprises the same data set of 2011 patients. The details of patients are given in the article [35].
In this paper we present survival analysis of the CABG patients with respect to some modifiab le risk factors, Hypertension & Diabetes, in co mplete and inco mplete Populations. Khan, Saleem and Mahmud in the article [21] proposed a procedure, to make an inco mplete population (IP) a co mplete population (CP). The differences between the means of survival proportions of the CABG patients, obtained by using survival models (Weibull, Exponential etc) are statistically insignificant at 5% level of significance for details see the article [21].
The importance of paramet ric models for analysis of lifetime date has been indicated by Mann, Schefer and Singpurwala [28], Nelson [29], Cy rus [8], Lawless [25], Klein & Moeschberger [22] and Sridhar and Mun Choon Chan [32]. The Exponential distribution model has been used by Lee, Kim and Jung [27] in med ical research for survival data of patients. The Weibull distribution model has been used for survival analysis by Cohen [6], Gross and Clark [12], Bunday [5], Crow [7], Klein & Moeschberger [22], Lawrencce [26], Abrenthy [1], Hisada & Arizino [16], Lawless [25], David & M itchel [9] and Lang [23]. In part icular, the survival study of chronic d iseases, such as AIDS and Cancer, has been carried out by Bain and Englehardt [4], Khan & Mah mud [20], Klein & Moeschberger [22], Lawless [25] and Swaminathan and Brenner [34] using Exponential and Weibull distributions. Lanju & William [24] used Weibull distribution to hu man survival data of patients with plas ma cell and in response-adaptive randomization for survival trials respectively. Lee, Kim and Jung [27] used the exponential in medical research for survival data of the patients. Khan, Saleem and Mohmud [21] concluded that the survival data of the CABG patients has been best modeled by the Weibull and Exponential distributions. In this paper, the survivor proportions of the CABG patients are obtained for incomp lete and complete populations of the CABG patients by parametric models (Weibull and Exponential), using data of CA BG patients Hypertension and Diabetes. Maximu m likelihood method, in-conjunction with DFP optimization method and Cubic Interpolation method is used. A subroutine for maximizing log-likelihood function of each model is developed in FORTRAN p rogram to obtain the estimates of the parameters of the model. The survival proportions of IP and CP of the CA BG patients with respect to modifiab le risk factors are presented in term of statistics and graphs (survival curves), discussed and concluded.

Methodology
Khan, Saleem and Mah mud mentioned that the method proposed by Kaplan Meier [19] and latter discussed by William [35] in 1995 and Lawless [25] is: Saleem, Mah mud and Khan [31] mentioned the form of likelihood function proposed by Klein & Moeschberger [22] and Lawless [25], for a survival model, in the presence of censored data. The maximu m likelihood method works by developing a likelihood function based on the available data and finding the estimates of parameters of a probability distribution that maximizes the likelihood function. This may be achieved by using iterative method: see Bunday & Al-Mutwali [5] and Khan & Mah mud [20] The likelihood function for all observed died and censored individuals is of the form: Where, the first sum is for failure and the second sum is for all censored individuals. Setting of indiv iduals at time i t we get: In this study time is partitioned into intervals, which are of unit length t starting from zero. Moreover, failures and censoring of the patients occur in each interval i of equal length of time t, 1, 2,...,12. i = For co mplete population the term for censored observations is dropped from the likelihood function.

Application
Khan, Saleem and Mah mud [21] presented detail application of above methodology for parametric model (Weibull distribution). Same procedure is fo llowed for second parametric model (Exponential distribution with b =1) considered in this article. The methodology is reproduced here. The probability density function (pdf) of Weibull d istribution is: where q is vector of parameters a and b ; a is a scale parameter and b is a shape parameter; a ,b and t > 0. where, By using (2), (3) and (4) in the DFP optimization method, we find the parameters estimates for wh ich value of the likelihood function is maximu m. For co mplete population we drop the term for censored observations from likelihood function. Same procedure is followed for Exponential model.

Discussion
The graphs in fig 5& 6

Conclusion 2
The differences between the means of survival proportions (obtained by using Weibull and exponential distributions) of CP and IP of