Estimation and Forecasting Survival of Diabetic CABG Patients (Kalman Filter Smoothing Approach)


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 (CP) from its incomplete population (IP) of the patients with 12 years observations/follow-up is used for their survival analysis [1]. The survival proportions of the CP obtained from Kaplan Meier method are used as observed values yt at time t (input) for Kalman Filter Smoothing process to update time varying parameters. In case of CP, 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 CP 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.

Share and Cite:

Saleem, M. , Khan, K. and Yasmin, N. (2015) Estimation and Forecasting Survival of Diabetic CABG Patients (Kalman Filter Smoothing Approach). American Journal of Computational Mathematics, 5, 405-413. doi: 10.4236/ajcm.2015.54035.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Khan, K.H., Saleem, M. and Mahmud, Z. (2011) Survival Proportions of CABG Patients: A New Approach. International Journal of Computational Science and Mathematics (IJCSM), 3, 293-302.
[2] Sorenson, H. (1985) Kalman Filtering: Theory and Application. IEEE Press, Los Alamitos.
[3] Hansson, G.K. (2005) Inflammation, Atherosclerosis, and Coronary Artery Disease. The New England Journal of Medicine, 352, 1685-1695.
[4] John, H.L. (2003) Hand Book of Patient Care in Cardiology Surgery. Lippincott Williams & Wilkins.
[5] Sun, Z. and Hong, N. (2011) Coronary Computed Tomography Angiography in Coronary Artery Disease. World Journal of Cardiology, 3, 303-310.
[6] Weintraub, W.S., Jones, E.L., Craver, J.M., et al. (1995) In-Hospital and Long-Term Outcome after Reoperative Coronary Artery Bypass Graft Surgery. Circulation, 92, II50-II57.
[7] Goldstein, L.B., Adams, R., Alberts, M.J., Appel, L.J., Brass, C., Bushnell, A., Culebras, T., De Graba, P. and Guyton, J.R. (2006) American Heart Association; American Stroke Association Stroke Council. Primary Prevention of Ischemic Stroke. American Journal of Ophthalmology: American Heart Association, 142, 716.
[8] Jennifer, H.R. (2008) After Coronary Artery Bypass Graft Surgery-Recovering from Open Heart Surgery.
[9] Heart and Stroke Foundation Canada (1997) Heart Disease and Stroke Statistics. Tipping the Scales of Progress: Heart Disease and Stroke in Canada.
[10] American Heart Association Dallas, Texas (2007) Heart Disease and Stroke Statistics. Circulation, 115, 169-171.
[11] Kaplan, E.L. and Meier, P. (1958) Nonparametric Estimations from Incomplete Observations. Journal of the American Statistical Association, 53, 457-481.
[12] William, S., Weintraub, M., Stephen, D., Clements, J.M., Van Thomas, C., Robert, A. and Guyton, N. (2003) Twenty Years Survival after Coronary Artery Surgery. American Heart Association, Dallas.
[13] Saleem, M., Khan, K.H. and Mahmud, Z. (2014) Long Term Survival of CABG Patients in Age Groups Using Complete and Incomplete Populations: (A New Approach). International Journal of Scientific & Engineering Research, 5, 21-28.
[14] Saleem, M., Khan, K.H. and Mahmud, Z. (2012) Survival Analysis of CABG Patients by Parametric Estimations in Modifiable Risk Factors—Hypertension and Diabetes. American Journal of Mathematics and Statistics, 2, 120-128.
[15] Abernathy, R.B. (1998) The New Weibull Handbook. 3rd Edition, SAE Publications, Warren dale.
[16] Bunday, B.D. and Al Mutwali, I.A. (1981) Direct Optimization for Calculation of Maximum Likelihood Estimates of Parameters of the Weibull Distribution. IEEE Transactions on Reliability, R-30, 367-369.
[17] Cohen, A.C. (1965) Maximum Likelihood Estimation in the Weibull Distribution Based on Complete and on Censored Samples. Technometrics, 7, 579-588.
[18] Crow, L.H. (1982) Confidence Interval Procedures for the Weibull Process with Applications to Reliability Growth. Technometrics, 24, 67-72.
[19] Gross, A.J. and Clark, V. (1975) Survival Distribution: Reliability Applications in the Biomedical Sciences. Wiley, Hoboken.
[20] Klein, P.J. and Moeschberger, L.M. (1997, 2003) Survival Analysis: Techniques for Censored and Truncated Data. Series: Statistics for Biology and Health, 2nd Edition, Springer, Berlin.
[21] Lang, W. (2010) Mixed Effects Models for Complex Data. Math & Statistics Library, Stanford.
[22] Lawless, J.F. (1982, 2003) Statistical Models and Methods for Lifetime Data. John Wiley and Sons, Inc., New York.
[23] Kurlansky, P., Herbert, M., Prince, S. and Mack, M.J. (2015) Improved Long-Term Survival for Diabetic Patients with Surgical versus Interventional Revascularization. The Annals of Thoracic Surgery, 99, 1298-1305.
[24] Bain, L.J. and Englehardt, M. (1991) Statistical Analysis of Reliability and Life-Testing Models: Theory and Methods. 2nd Edition, Marcel Dekker, New York.
[25] Khan, K.H. and Mahmud, Z. (1999) Weibull Distribution Model for the Breast Cancer Survival Data Using Maximum Likelihood Method. Journal of Research (Science), 10, 45-49.
[26] Swaminathan, R. and Brenner, H. (1998, 2011) Statistical Methods for Cancer Survival Analysis.
[27] Zhang, L.J. and Rosenberger, W.F. (2007) Response-Adaptive Randomization for Survival Trials: The Parametric Approach. Journal of the Royal Statistical Society: Series C (Applied Statistics), 56, 153-165.
[28] Harrison, P.J. and Stevens, C.F. (1976) Bayesian Forecasting. Journal of the Royal Statistical Society, 3, 205-228.
[29] Greg, W. and Gary, B. (2004) An Introduction to the Kalman Filter. University of North Carolina, Chapel Hill.
[30] Frank, S.S. (2006) Autonomous Mobile Robots: Sensing, Control, Decision-Making and Application. CRC/Taylor & Francis, Boca Raton.
[31] Meinhold, R.J. and Singpurwalla, N.D. (1987) A Kalman-Filter Smoothing Approach for Extrapolations in Certain Dose-Response, Damage-Assessment, and Accelerated-Life-Testing Studies. The American Statistician, 41, 101-106.
[32] Anatoli, I., Kenneth, G.M. and James, W.V. (1983) Mortality and Aging in a Heterogeneous Population: A Stochastic Process Model with Observed and Unobserved Variables. Theoretical Population Biology, 27, 154-175.
[33] Ludwig, F. (1994) Dynamic Modeling and Penalized Likelihood Estimation for Discrete Time Survival Data. Biometrika, 81, 317-330.
[34] Kalbfleisch, J.D. and Prentice, R.L. (1980) The Statistical Analysis of Failure Time Data. Wiley. John Wiley & Sons, Inc., Hoboken.
[35] Meinhold, R.J. and Singpurwalla, N.D. (1983) Understanding the Kalman Filter. The American Statistician, 37, 123-127.
[36] Fletcher, R. and Powell, M.J.D. (1963) A Rapid Convergent Decent Method for Minimization. The Computer Journal, 6, 163-168.

Copyright © 2022 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.