Abstract
Analyses of human mortality data classified according to cause of death frequently are based on competing risk theory. In particular, the times to death for different causes often are assumed to be independent. In this paper, a competing risk model with a weaker assumption of conditional independence of the times to death, given an assumed stochastic covariate process, is developed and applied to cause specific mortality data from the Framingham Heart Study. The results generated under this conditional independence model are compared with analogous results under the standard marginal independence model. Under the assumption that this conditional independence model is valid, the comparison suggests that the standard model overestimates by 4% the effect on life expectancy at age 30 due to the hypothetical elimination of cancer and by 7% the effect for cardiovascular/cerebrovascular disease. By age 80 the overestimates were 11% for cancer and 16% for heart disease. These results suggest the importance of avoiding the marginal independence assumption when appropriate data are available — especially when focusing on mortality at advanced ages.
This is a preview of subscription content, log in via an institution to check access.
References
Abraham, S.: Total serum cholesterol levels of adults 18–74 years. Vital and Health Statistics Series 11. Data from the National Health Survey, No. 205, DHEW Pub. No. (PHS) 78-1652, NCHS, Hyattsville, MD, 1978
Arnold, B. C., Brockett, P. L.: Identifiability for dependent multiple decrement/competing risk models. Scand. Actuarial J. 117–127 (1983)
Baum, H. M., Manton, K. G.: Cerebrovascular disease mortality: the relationship of underlying and associated causes of death. In review at American J. Public Health (1985)
Chiang, C. L.: Introduction to stochastic processes in biostatistics. New York: Wiley 1968
Cohen, J., Liu, L.: Competing risks without independence. Manuscript (1984)
Dellacherie, C.: Capacities and stochastic processes. Berlin Heidelberg New York: Springer 1972
Economos, A. C.: Rate of aging, rate of dying and the mechanism of mortality. Arch. Gerontol. Geriatr. 1, 3–27 (1982)
Greville, T. N.: Mortality tables analyzed by cause of death. Record American Inst. Actuaries 37, 283–294 (1948)
Jacod, J.: Calcul stochastique et problèmes de martingales. In: Lect. Notes Math. 714. Berlin Heidelberg New York: Springer 1979
Keyfitz, N.: What difference would it make if cancer were eradicated? An examination of the Taeuber paradox. Demography 14, 411–418 (1977)
Klebba, A. J.: Mortality from diseases associated with smoking. Vital and Health Statistics Series 20. Data from the National Vital Statistics System, No. 17 DHHS Pub. No. (PHS) 82-1854, NCHS, Hyattsville, MD, 1982
Liptser, R. S., Shiryayev, A. N.: Statistics of random processes. Berlin Heidelberg New York: Springer 1977
Manton, K. G., Stallard, E., Woodbury, M. A.: Chronic disease evolution and human aging: a general model for assessing the impact of chronic disease in human populations. Internat. J. Math. Modeling (special issue on math, modeling of diseases, M. Wirten, ed.), forthcoming (1986)
Peterson, A. V.: Bounds for a joint distribution function with fixed sub-distribution functions: application to competing risks. Proc. Natl. Acad. Sci. USA 73, 11–13 (1976)
Pollard, J. H.: The expectation of life and its relationship to mortality. J. Inst. Actuaries 109, 225–240 (1982)
Roberts, J.: Blood pressure levels in persons 6–74 years, United States, 1971–1974. Vital and Health Statistics Series 11. Data from the National Health Survey, No. 203, DHEW Pub. No. (HRA) 78-1648, NCHS, Hyattsville, MD, 1977
Shock, N. W., Greulich, R. C., Andres, R., Arenberg, D., Costa, P. T., Lakatta, E. G., Tobin, J. D.: Normal human aging: the Baltimore longitudinal study of aging. DHHS, Pub. No. (NIH) 84-2450. Washington, D.C.; USGPO 1984
Tsiatis, A.: A nonidentifiability aspect of the problem of competing risks. Proc. Natl. Acad. Sci. USA 72, 20–22 (1975)
Tolley, H. D., Manton, K. G.: Comparing mortality risks with chronic conditions. Proc. American Stat. Assoc., Toronto Meeting (1984)
Tolley, H. D., Manton, K. G.: Multiple cause models of disease dependency. Scand. Actuarial J. 211–226 (1983)
Woodbury, M. A., Manton, K. G.: A random walk model of human mortality and aging. Theor. Popul. Biol. 11, 37–48 (1977)
Woodbury, M. A., Manton, K. G.: A theoretical model of the physiological dynamics of circulatory disease in human populations. Human Biol. 55, 417–441 (1983)
Yashin, A. I.: Hazard rates and probability distributions: Representation of random intensities. WP-84-21, International Institute for Applied Systems Analysis, Laxenburg, Austria (1984)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Yashin, A.I., Manton, K.G. & Stallard, E. Dependent competing risks: a stochastic process model. J. Math. Biology 24, 119–140 (1986). https://doi.org/10.1007/BF00275995
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00275995