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In this paper, we present a new approach (Kalman Filter Smoothing) to estimate and forecast survival of Diabetic and Non Diabetic Coronary Artery Bypass Graft Surgery (CABG) patients. Survival proportions of the patients are obtained from a lifetime representing parametric model (Weibull distribution with Kalman Filter approach). Moreover, an approach of complete population (<i>CP</i>) from its incomplete population (<i>IP</i>) of the patients with 12 years observations/follow-up is used for their survival analysis [1]. The survival proportions of the <i>CP</i> obtained from Kaplan Meier method are used as observed values <i>y<sub>t</sub></i> at time <i>t</i> (input) for Kalman Filter Smoothing process to update time varying parameters. In case of <i>CP</i>, the term representing censored observations may be dropped from likelihood function of the distribution. Maximum likelihood method, in-conjunction with Davidon-Fletcher-Powell (DFP) optimization method [2] and Cubic Interpolation method is used in estimation of the survivor’s proportions. The estimated and forecasted survival proportions of <i>CP</i> of the Diabetic and Non Diabetic CABG patients from the Kalman Filter Smoothing approach are presented in terms of statistics, survival curves, discussion and conclusion.

The Coronary Artery Disease (CAD) is a chronic disease, which progresses with age at different rates. CAD is a result of built-up of fats on the inner walls of the coronary arteries. Thus, the sizes of coronary arteries become narrow and as a result the blood flow to the heart muscles is reduced/blocked. Therefore, the heart muscles do not receive required oxygenated blood, which leads to the heart attack. CAD is a leading cause of death worldwide (see Hansson [

William, Ellis, Josef, Ralph and Robert [

The Weibull distribution model has been used for survival analysis by Abrenthy [

The dynamic linear model (DLM) and Kalman Filter (KF) equations have been described by Harrison and Steven [

In our study, Kalman filter technique is applied to estimate parameters of Weibull probability distribution using Diabetic and Non Diabetic CABG patient’s data sets. For construction of KF equations, survivor function of the probability distribution is linearized by transformation of double-log. The procedure to construct linear form of the survivor function, as advocated by researchers (see Gross and Clark [

For the estimation of survival proportions Kaplan Meier [

[

viduals/patients) and number of individuals at risk at time

dividuals,

dividual’s

Observation Equation:

System Equation:

where

and

The KF equations of Weibull probability distribution models are constructed by linearizing survival function of the distribution with transformation; double-log. The parameters of the probability distributions are estimated at each time t, by maximizing log-likelihood function of lognormal distribution (which is transformed form of Weibull distribution), through the Davidon-Fletcher-Powel method of optimization. For the entire system, the parametric space at each time point t is

Since the values of survival proportions

searchers, Meinhold and Singpurwala [

nonlinear, monotonically decreasing function of t and is survival function,

form is a requirement for filtering techniques. Thus to implement KF a random quantity

defined, which require that

Now,

setting

The corresponding system equation is:

Comparing Equations (3) and (4) with (1) and (2), we find that:

To find maximum likelihood estimates we consider negative log-likelihood function say

where,

be obtained as:

For derivation of

A subroutine for maximizing log-likelihood function of the double-lognormal distribution along with KF process (subroutine) is developed in FORTRAN program.

The subroutine in-conjunction with DFP optimization method is used to find the optimal initial estimates of the parameters included in the model

The optimal initial estimates of parameters obtained by maximizing the log-likelihood function are presented in

The results (survival proportions obtained by using Weibull distribution and KF approach

time point t as explained earlier) of

Parameters | ||||
---|---|---|---|---|

Estimates | Gradients | Estimates | Gradients | |

1.2999 | 1.0546 | |||

1.2197 | 1.9704 | |||

Value of Log-Likelihood | 161.4857 | 84.236929169 |

Years (t) | ||
---|---|---|

0 | 1 | 1 |

1 | 0.90 | 0.979 |

2 | 0.87 | 0.949 |

3 | 0.85 | 0.916 |

4 | 0.83 | 0.880 |

5 | 0.79 | 0.843 |

6 | 0.77 | 0.805 |

7 | 0.73 | 0.767 |

8 | 0.69 | 0.729 |

9 | 0.65 | 0.692 |

10 | 0.61 | 0.656 |

11 | 0.56 | 0.620 |

12 | 0.53 | 0.586 |

13 | 0.552 | |

14 | 0.520 | |

15 | 0.489 |

The graphs of observed survival proportions from the complete population

Years (t) | ||
---|---|---|

0 | 1 | 1 |

1 | 0.85 | 0.9200 |

2 | 0.82 | 0.8410 |

3 | 0.79 | 0.7667 |

4 | 0.73 | 0.6978 |

5 | 0.66 | 0.6343 |

6 | 0.60 | 0.5760 |

7 | 0.53 | 0.5225 |

8 | 0.51 | 0.4737 |

9 | 0.45 | 0.4291 |

10 | 0.36 | 0.3885 |

11 | 0.34 | 0.3515 |

12 | 0.23 | 0.3179 |

13 | 0.2873 | |

14 | 0.2596 | |

15 | 0.2345 |

We are thankful to our reviewers, whose constructive criticism has resulted in a clearer presentation of our work and inclusion of additional useful reference material.

M.Saleem,K. H.Khan,NusratYasmin, (2015) Estimation and Forecasting Survival of Diabetic CABG Patients (Kalman Filter Smoothing Approach). American Journal of Computational Mathematics,05,405-413. doi: 10.4236/ajcm.2015.54035

Consider p.d.f of log-normal distribution:

Let

then,

Let

To find maximum likelihood estimates, we consider negative log-likelihood function say

where

tion,

For partial derivatives, differentiating above equation with respect to

and