Abstract
In this chapter a methodology is developed to measure the quantitative relationship between the probability that a policy will terminate by death in the ensuing year and the policyholder’s characteristics at the time of application for life insurance. Due to the dichotomous nature of the response variable (death or survival for the following year). multiple regression analysis is inappropriate.1 Consequently, this study utilizes a maximum likelihood algorithm to estimate the coefficients in a multiple logistic model. The Newton-Raphson technique provides for an iterative solution of the estimated betas after initial estimates are calculated by the application of Fisher’s linear discriminant function.
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Notes
P. Armitage and Edmund A. Gehan, “Statistical Methods for the Identification and Use of Prognostic Factors,” International Journal of Cancer 13 (1974) 21.
Lars Wilhelmsen, Hans Wedel, and Gosta Tibblin, “Multivariate Analysis of Risk Factors for Coronary Heart Disease,” Circulation 48 (1973): 952.
R.A. Fisher, “The Use of Multiple Measurements in Taxanomic Problems,” Annals of Eugenics 7 (1936): 179–88.
Michael L. Feldstein, Edwin D. Savlov, and Russel Hilf, “A Statistical Model for Predicting Response of Breast Cancer Patients to Cytotoxic Chemotherapy,” Cancer Research 38 (1978): 2545.
Ibid.
Jerome Cornfield, Tavia Gordon, and Willie W. Smith, “Quantal Response Curves for Experimentally Uncontrolled Variables,” Bulletin of the International Statistical Institute 38 (1961) 3: 97–115; Jerome Cornfield, “Joint Dependence of Risk of Coronary Heart Disease on Serum Cholesterol and Systolic Blood Pressure: A Discriminant Function Analysis,” Federation Proceedings 2 (1962): 58–61; and Jerome Cornfield, “Discriminant Functions,” Review of the International Statistical Institute 35 (1967): 142–53.
Jeanne Truett, Jerome Cornfield, and William Kannel, “A Multivariate Analysis of the Risk of Coronary Heart Disease in Framingham,” Journal of Chronic Diseases 20 (1967): 511–24.
Jerome Cornfield, “Bayes Theorem,” Review of the International Statistical Institute 35 (1967): 34–48.
David G. Kleinbaum and Lawrence L. Kupper, Applied Regression Analysis and other Multivariable Methods ( North Scituate, Mass.: Duxbury Press, 1978 ), p. 433.
For a complete description of the study, see Thomas A. Dawber, William B. Kannel, and Lorna P. Lyell, “An Approach to Longitudinal Studies in the Community: The Framingham Study,” Annals of the New York Academy of Sciences 107 (1963): 539–56.
Truett, Cornfield, and Kannell, “Multivariate Analysis,” p. 512.
James E. Grizzle, C. Frank Starmer, and Gary G. Koch, “Analysis of Categorical Data by Linear Models,” Biometrics 25-0969): 489–505.
J. Neyli n, “Contributions to the Theory of the X2 Test,” Proceedings of the First Berkeley Symposium on Mathematical Statistics and Probability,ed. J. Neyman (Berkeley: University of California Press, 1949). pp. 230–73.
Yvonne M.M. Bishop, Stephen E. Fienberg, and Paul W. Holland, Discrete Multivariate Analysis: Theory and Practice (Cambridge, Mass.: MIT Press, 1975), p. 354. Essentially, the GSK approach utilizes a weighted least-squares analysis. If several of the zero values are replaced by a positive value, the quadratic form is not properly defined.
Truett, Cornfield, and Karmen, “Multivariate Analysis,” p. 521.
The problem of assessing goodness of fit is currently under study. See David W. Hosmer and Stanley Lemeshow, “A Goodness of Fit Test for the Multiple Logistic Regression Model,” Biostatistics Technical Reports 79–1, Division of Public Health, University of Massachusetts.
Max Halperin, William C. Blackwelder. and Joel I. Verter, “Estimation of the Multivariate Logistic Risk Function: A Comparison of the Discriminant Function and Maximum Likelihood Approaches,” Journal of Chronic Diseases 24 (1971): 132.
Ibid., p. 126.
William B. Kannel, William P. Castelli, Tavia Gordon, and Patricia M. McNamara, “Serum Cholesterol, Lipoproteins, and the Risk of Coronary Heart Disease,” Annals of Internal Medicine 74 (1971): 1–12.
Precedent for the propriety of this approach has also been set in William B. Kannel, J.M. Seidman, W. Fercho, and William P. Castelli, “Vital Capacity and Congestive Heart Failure: The Framingham Study,” Circulation 49 (1974): 1160–66; Uri Goldbourt, Jack H. Medalie, and Henry N. Neufeld, “Clinical Myocardial Infarction-III. A Multivariate Analysis of Incidence, the Israel Ischemic Heart Disease Study,” Journal of Chronic Diseases 28 (1975): 217–37; William B. Kannel, Joseph T. Doyle, Patricia M. McNamara, Phillip Quickenton, and Tavia Gordon, “Precursors of Sudden Coronary Death,- Circulation 51 (1975): 606–13; ”Relationship of Blood Pressure, Serum Cholesterol, Smoking Habit, Relative Weight, and ECG Abnormalities to Incidence of Major Coronary Events: Final Report of the Pooling Project,“ Journal of Chronic Diseases 31 (1978): 201–306.
Strother H. Walker and David B. Duncan, “Estimation of the Probability of an Event as a Function of Several Independent Variables,” Biometrika 54 (1967): 167–79.
Halperin, Blackwelder, and Verter, “Estimation of the Logistic Function,” p. 152.
Stuart C. Hartz, “A Statistical Model for Assessing the Need for Medical Care in a Health Screening Program,” Clinical Chemistry 19 (1973): 114.
Final Report of the Pooling Project,“ p. 265.
Ping-Hwa Hsu, Francis A.L. Mathewson, Hassan A.H. Abu-Zeid, and Simon W. Rabkin, “Change in Risk Factors and the Development of Chronic Disease: A Methodological Illustration,” Journal of Chronic Diseases 30 (1977): 567–84.
Jerome Cornfield and Sheela Mitchell, “Selected Risk Factors in Coronary Disease,” Archives of Environmental Health 19 (1969): 385.
Michael J. Cowell and Brian L. Hirst, “Mortality Differences Between Smokers and Nonsmokers,” Transactions of the Society of Actuaries 32 (1980): 185–261.
Ancel Keys, Henry L. Taylor, Henry Blackburn, Joseph Brozek, Joseph T. Anderson, and Ernst Simonson, “Mortality and Coronary Heart Disease Among Men Studied for 23 Years,” Archives of Internal Medicine 128 (1971): 211.
Jack L. VanDerhei, “Multivariate Analysis of Underwriting Risk Factors and Mortality” (Ph.D. dissertation, University of Pennsylvania, 1982 ), pp. 7–61.
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Cummins, J.D., Smith, B.D., Vance, R.N., VanDerhei, J.L. (1983). A Multiple logistic Methodology for the Estimation of Risk Classification Models. In: Risk Classification in Life Insurance. Huebner International Series on Risk, Insurance, and Economic Security, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2911-6_13
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DOI: https://doi.org/10.1007/978-94-017-2911-6_13
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